Package 'pcds'

Title:	Proximity Catch Digraphs and Their Applications
Description:	Contains the functions for construction and visualization of various families of the proximity catch digraphs (PCDs) (see (Ceyhan (2005) ISBN:978-3-639-19063-2), for computing the graph invariants for testing the patterns of segregation and association against complete spatial randomness (CSR) or uniformity in one, two and three dimensional cases. The package also has tools for generating points from these spatial patterns. The graph invariants used in testing spatial point data are the domination number (Ceyhan (2011) <doi:10.1080/03610921003597211>) and arc density (Ceyhan et al. (2006) <doi:10.1016/j.csda.2005.03.002>; Ceyhan et al. (2007) <doi:10.1002/cjs.5550350106>). The PCD families considered are Arc-Slice PCDs, Proportional-Edge PCDs, and Central Similarity PCDs.
Authors:	Elvan Ceyhan
Maintainer:	Elvan Ceyhan <[email protected]>
License:	GPL-2
Version:	0.1.7
Built:	2024-11-15 05:23:54 UTC
Source:	https://github.com/elvanceyhan/pcds

Help Index

.onAttach start message
.onLoad getOption package settings
The angles to draw arcs between two line segments
The angle between two line segments
The arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for a 2D data set - multiple triangle case
The arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for 2D data - one triangle case
The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D data - multiple triangle case
The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - multiple interval case
The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - end interval case
The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - one interval case
The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - middle intervals case
The arcs of Central Similarity Proximity Catch Digraphs (CS-PCD) for 2D data - one triangle case
The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - multiple triangle case
The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - multiple interval case
The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - end interval case
The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - one interval case
The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - middle intervals case
The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - one triangle case
The area of a polygon in R^2
The labels of the vertices of a triangle in the basic triangle form
Arc density of Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case
Centers for non-degenerate asymptotic distribution of domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs)
Parameterized center of an interval
Parameterized centers of intervals
Circumcenter of a standard basic triangle form
Circumcenter of a general tetrahedron
Circumcenter of a general triangle
The closest points to circumcenter in each CC-vertex region in a triangle
The closest points to circumcenter in each CC-vertex region in a standard basic triangle
The closest points in a data set to edges in the standard equilateral triangle
The closest points among a data set in the vertex regions to the corresponding edges in a standard basic triangle
The closest points in a data set to edges in each CC-vertex region in a triangle
The closest points in a data set to edges in each CM-vertex region in a triangle
The closest points among a data set in the vertex regions to the respective edges in a triangle
The closest points among a data set in the vertex regions to the respective faces in a tetrahedron
The closest points to center in each vertex region in an interval
A test of segregation/association based on arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D data
A test of uniformity of 1D data in a given interval based on Central Similarity Proximity Catch Digraph (CS-PCD)
A test of segregation/association based on arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data
Arc density of Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case
The dimension of a vector or matrix or a data frame
The distance between two vectors, matrices, or data frames
The distance from a point to a line defined by two points
The distance from a point to a plane spanned by three 3D points
Distance from a point to a set of finite cardinality
Exact domination number (i.e., domination number by the exact algorithm)
Approximate domination number and approximate dominating set by the greedy algorithm
The vertices of the CM-edge region in a triangle that contains the point
The furthest points in a data set from edges in each CM-edge region in the standard equilateral triangle
The furthest points in a data set from vertices in each CC-vertex region in a triangle
The furthest points from vertices in each CC-vertex region in a standard basic triangle
The lines joining two vertices to the center of mass in standard equilateral triangle
The lines joining the three vertices of the standard equilateral triangle to a center, M, of it
Converts of a point in Cartesian coordinates to Barycentric coordinates and vice versa
Each function is for the presence of an arc from a point in one of the edge regions to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case
The function gammakCSstd.tri is for k (k=2,3,4,5) points constituting a dominating set for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case
Each function is for the presence of an arc from a point in one of the edge regions to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case with t=1
Functions provide the indices of the Delaunay triangles where the points reside
Returning the mean and (asymptotic) variance of arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - middle interval case
Returns the mean and (asymptotic) variance of arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D uniform data in one triangle
Returns the mean and (asymptotic) variance of arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - end interval case
Returns the mean and (asymptotic) variance of arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - middle interval case
Returns the mean and (asymptotic) variance of arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D uniform data in one triangle
Returns the mean and (asymptotic) variance of arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - end interval case
The functions for probability of domination number = 2 for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case
Returns the ranks and orders of points in decreasing distance to the edges of the triangle
Two functions lineD1CCinTb and lineD2CCinTb which are of class "TriLines" — The lines joining the midpoints of edges to the circumcenter (CC) in the standard basic triangle.
The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case
The indicator for the presence of an arc from a point in set S to the point p for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case
The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case
The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - first one-sixth of the standard equilateral triangle case
The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard basic triangle case
The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case
The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - end interval case
The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one interval case
The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - middle interval case
The indicator for the presence of an arc from a point in set S to the point p for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case
The indicator for the presence of an arc from a point in set S to the point p for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case
The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case
The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case with t=1
The indicator for the presence of an arc from one point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs)
An alternative to the function IarcCStri which yields the indicator for the presence of an arc from one point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs)
The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic triangle case
The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - end interval case
The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one interval case
The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case
The indicator for the presence of an arc from a point in set S to the point p or Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case
The indicator for the presence of an arc from a point in set S to the point p for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case
The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case
The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case
The indicator for the presence of an arc from one 3D point to another 3D point for Proportional Edge Proximity Catch Digraphs (PE-PCDs)
The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case
Indicator for an upper bound for the domination number by the exact algorithm
The indicator for a point being a dominating point for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case
The indicator for a point being a dominating point for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case
The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - first one-sixth of the standard equilateral triangle case
The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) for an interval
The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case
The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case with t=1
The indicator for a point being a dominating point or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic triangle case
The indicator for a point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) for an interval
The indicator for a 3D point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case
The indicator for a 3D point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case
The indicator for a point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case
The indicator for two points being a dominating set for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case
The indicator for two points constituting a dominating set for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case
The indicator for two points constituting a dominating set for Central Similarity Proximity Catch Digraphs (CS-PCDs) - first one-sixth of the standard equilateral triangle case
The indicator for two points being a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic triangle case
The indicator for two 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case
The indicator for two 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case
The indicator for two points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case
The indicator for three 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case
The indicator for three 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case
Indicator for an upper bound for the domination number of Arc Slice Proximity Catch Digraph (AS-PCD) by the exact algorithm - one triangle case
The indicator for k being an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - standard equilateral triangle case
Indicator for an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - one triangle case
The indicator for the set of points S being a dominating set or not for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case
The indicator for the set of points S being a dominating set or not for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case
The indicator for the set of points S being a dominating set or not for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case
The indicator for the set of points S being a dominating set or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case
The indicator for the set of points S being a dominating set or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case
Check whether a point is inside a circle
Check whether a point is inside a tetrahedron
Check whether all points in a data set are inside the triangle
Check whether a point is inside a triangle
Incidence matrix for Arc Slice Proximity Catch Digraphs (AS-PCDs) - multiple triangle case
Incidence matrix for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case
Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) - multiple triangle case
Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) for 1D data - multiple interval case
Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) for 1D data - one interval case
Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case
Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case
Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - multiple triangle case
Incidence matrix for Proportional-Edge Proximity Catch Digraphs (PE-PCDs) for 1D data - multiple interval case
Incidence matrix for Proportional-Edge Proximity Catch Digraphs (PE-PCDs) for 1D data - one interval case
Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case
Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case
Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case
Region index inside the Gamma-1 region
The points of intersection of a line and a circle
The point of intersection of a line and a plane
The point of intersection of two lines defined by two pairs of points
Indices of the intervals where the 1D point(s) reside
Check a point belong to a given data set
Check the argument is a point of a given dimension
Check whether a triangle is a standard equilateral triangle
The k furthest points in a data set from vertices in each CC-vertex region in a triangle
The line joining two distinct 2D points a and b
The line crossing 3D point p in the direction of vector v (or if v is a point, in direction of v-r_0)
The vertices of the Arc Slice (AS) Proximity Region in the standard basic triangle
The vertices of the Arc Slice (AS) Proximity Region in a general triangle
The end points of the Central Similarity (CS) Proximity Region for a point - one interval case
The vertices of the Central Similarity (CS) Proximity Region in a general triangle
The vertices of the Proportional Edge (PE) Proximity Region in a standard basic triangle
The end points of the Proportional Edge (PE) Proximity Region for a point - one interval case
The vertices of the Proportional Edge (PE) Proximity Region in the standard regular tetrahedron
The vertices of the Proportional Edge (PE) Proximity Region in a tetrahedron
The vertices of the Proportional Edge (PE) Proximity Region in a general triangle
Number of arcs of Arc Slice Proximity Catch Digraphs (AS-PCDs) and related quantities of the induced subdigraphs for points in the Delaunay triangles - multiple triangle case
Number of arcs of Arc Slice Proximity Catch Digraphs (AS-PCDs) and quantities related to the triangle - one triangle case
Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and related quantities of the induced subdigraphs for points in the Delaunay triangles - multiple triangle case
Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and related quantities of the induced subdigraphs for points in the partition intervals - multiple interval case
Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) - end interval case
Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the interval - one interval case
Number of Arcs of of Central Similarity Proximity Catch Digraphs (CS-PCDs) - middle interval case
Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the triangle - standard equilateral triangle case
Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the triangle - one triangle case
Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and related quantities of the induced subdigraphs for points in the Delaunay triangles - multiple triangle case
Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and related quantities of the induced subdigraphs for points in the partition intervals - multiple interval case
Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - end interval case
Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the interval - one interval case
Number of Arcs for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case
Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the triangle - standard equilateral triangle case
Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the tetrahedron - one tetrahedron case
Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the triangle - one triangle case
Number of Delaunay triangles based on a 2D data set
The line at a point p parallel to the line segment joining two distinct 2D points a and b
The line crossing the 3D point p and parallel to line joining 3D points a and b
The plane at a point and parallel to the plane spanned by three distinct 3D points a, b, and c
pcds: A package for Proximity Catch Digraphs and Their Applications
The asymptotic probability of domination number = 2 for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case
Asymptotic probability that domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs) equals 2 where vertices of the digraph are uniform points in a triangle
A test of segregation/association based on arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data
A test of uniformity of 1D data in a given interval based on Proportional Edge Proximity Catch Digraph (PE-PCD)
A test of segregation/association based on arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data
Arc density of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case
Arc density of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case
The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - multiple triangle case
A test of segregation/association based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - Binomial Approximation
A test of segregation/association based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - Binomial Approximation
A test of uniformity for 1D data based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - Binomial Approximation
The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) with non-degeneracy centers - multiple triangle case
A test of segregation/association based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - Normal Approximation
The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - one tetrahedron case
The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - one triangle case
The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data
The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) with non-degeneracy centers - multiple interval case
The line passing through a point and perpendicular to the line segment joining two points
The line crossing the 3D point p and perpendicular to the plane spanned by 3D points a, b, and c
The plane passing through three distinct 3D points a, b, and c
Plot an Extrema object
Plot a Lines object
Plot a Lines3D object
Plot a NumArcs object
Plot a Patterns object
Plot a PCDs object
Plot a Planes object
Plot a TriLines object
Plot a Uniform object
The plot of the arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for a 2D data set - multiple triangle case
The plot of the arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for a 2D data set - one triangle case
The plot of the Arc Slice (AS) Proximity Regions for a 2D data set - multiple triangle case
The plot of the Arc Slice (AS) Proximity Regions for a 2D data set - one triangle case
The plot of the arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for a 2D data set - multiple triangle case
The plot of the arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) for 1D data (vertices jittered along y-coordinate) - one interval case
The plot of the arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for a 2D data set - one triangle case
The plot of the arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) for 1D data (vertices jittered along y-coordinate) - multiple interval case
The plot of the Central Similarity (CS) Proximity Regions for a 2D data set - multiple triangle case
The plot of the Central Similarity (CS) Proximity Regions for a general interval (vertices jittered along y-coordinate) - one interval case
The plot of the Central Similarity (CS) Proximity Regions for a 2D data set - one triangle case
The plot of the Central Similarity (CS) Proximity Regions (vertices jittered along y-coordinate) - multiple interval case
The scatterplot of points from one class and plot of the Delaunay triangulation of the other class
The plot of the subintervals based on Yp points together with Xp points
The plot of the arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for a 2D data set - multiple triangle case
The plot of the arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) for 1D data (vertices jittered along y-coordinate) - one interval case
The plot of the arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for a 2D data set - one triangle case
The plot of the arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) for 1D data (vertices jittered along y-coordinate) - multiple interval case
The plot of the Proportional Edge (PE) Proximity Regions for a 2D data set - multiple triangle case
The plot of the Proportional Edge (PE) Proximity Regions for a general interval (vertices jittered along y-coordinate) - one interval case
The plot of the Proportional Edge (PE) Proximity Regions for a 3D data set - standard regular tetrahedron case
The plot of the Proportional Edge (PE) Proximity Regions for a 3D data set - one tetrahedron case
The plot of the Proportional Edge (PE) Proximity Regions for a 2D data set - one triangle case
The plot of the Proportional Edge (PE) Proximity Regions (vertices jittered along y-coordinate) - multiple interval case
Print a Extrema object
Print a Lines object
Print a Lines3D object
Print a NumArcs object
Print a Patterns object
Print a PCDs object
Print a Planes object
Print a summary of a Extrema object
Print a summary of a Lines object
Print a summary of a Lines3D object
Print a summary of a NumArcs object
Print a summary of a Patterns object
Print a summary of a PCDs object
Print a summary of a Planes object
Print a summary of a TriLines object
Print a summary of a Uniform object
Print a TriLines object
Print a Uniform object
Projections of a point inside a triangle to its edges
Projections of a point inside the standard basic triangle form to its edges
Projections of Centers for non-degenerate asymptotic distribution of domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs) to its edges
The radii of points from one class with respect to points from the other class
The radius of a point from one class with respect to points from the other class
Generation of points associated (in a radial or circular fashion) with a given set of points
Generation of points associated (in a Matern-like fashion) to a given set of points
Generation of points associated (in a Type I fashion) with a given set of points
Generation of points associated (in a Type I fashion) with the vertices of T_e
Generation of points associated (in a Type I fashion) with the vertices of a triangle
Generation of points associated (in a Type II fashion) with the edges of T_e
The index of the edge region in a standard basic triangle form that contains a point
The index of the CM-edge region in a standard basic triangle form that contains a point
The index of the edge region in the standard equilateral triangle that contains a point
The index of the edge region in a triangle that contains the point
The index of the CM-edge region in a triangle that contains the point
The indices of the M-edge regions in a triangle that contains the points in a give data set
The indices of the CM-edge regions in a triangle that contains the points in a give data set
The index of the vertex region in a standard basic triangle form that contains a given point
The index of the CC-vertex region in a standard basic triangle form that contains a point
The index of the CM-vertex region in a standard basic triangle form that contains a point
The index of the vertex region in an end-interval that contains a given point
The index of the vertex region in a middle interval that contains a given point
The index of the vertex region in the standard equilateral triangle that contains a given point
The index of the CM-vertex region in the standard equilateral triangle that contains a given point
The index of the CC-vertex region in a tetrahedron that contains a point
The index of the CM-vertex region in a tetrahedron that contains a point
The index of the vertex region in a triangle that contains a given point
The index of the CC-vertex region in a triangle that contains a point
The index of the CM-vertex region in a triangle that contains a given point
The indices of the vertex regions in a triangle that contains the points in a give data set
The indices of the vertex regions in a triangle that contains the points in a give data set
The indices of the CC-vertex regions in a triangle that contains the points in a give data set.
The indices of the CM-vertex regions in a triangle that contains the points in a give data set
The alternative function for the indices of the M-vertex regions in a triangle that contains the points in a give data set
Generation of points segregated (in a radial or circular fashion) from a given set of points
Generation of points segregated (in a Type I fashion) from a given set of points
Generation of points segregated (in a Type I fashion) from the vertices of T_e
Generation of points segregated (in a Type I fashion) from the vertices of a triangle
Generation of points segregated (in a Type II fashion) from the vertices of T_e
Generation of Uniform Points in the standard basic triangle
Generation of Uniform Points in the Convex Hull of Points
Generation of Uniform Points in the Standard Regular Tetrahedron T_h
Generation of Uniform Points in the Standard Equilateral Triangle
Generation of Uniform Points in the first one-sixth of standard equilateral triangle
Generation of Uniform Points in a tetrahedron
Generation of Uniform Points in a Triangle
The auxiliary triangle to define the support of type I segregation
The closest points among a data set in the standard equilateral triangle to the median lines in the six half edge regions
The slope of a line
Return a summary of a Extrema object
Return a summary of a Lines object
Return a summary of a Lines3D object
Return a summary of a NumArcs object
Return a summary of a Patterns object
Return a summary of a PCDs object
Return a summary of a Planes object
Return a summary of a TriLines object
Return a summary of a Uniform object
Tree Species in a Swamp Forest
Converting a triangle to the standard basic triangle form form
Points from one class inside the convex hull of the points from the other class

.onLoad getOption package settings

Description

.onLoad getOption package settings

Usage

.onLoad(libname, pkgname)
.onLoad(libname, pkgname)

Arguments

`libname`	defunct
`pkgname`	defunct

Value

invisible()

Examples

getOption("pcds.name")
getOption("pcds.name")

The angles to draw arcs between two line segments

Description

Gives the two pairs of angles in radians or degrees to draw arcs between two vectors or line segments for the draw.arc function in the plotrix package. The angles are provided with respect to the $x$ -axis in the coordinate system. The line segments are $[ba]$ and $[bc]$ when the argument is given as a,b,c in the function.

radian is a logical argument (default=TRUE) which yields the angle in radians if TRUE, and in degrees if FALSE. The first pair of angles is for drawing arcs in the smaller angle between $[ba]$ and $[bc]$ and the second pair of angles is for drawing arcs in the counter-clockwise order from $[ba]$ to $[bc]$ .

Usage

angle.str2end(a, b, c, radian = TRUE)
angle.str2end(a, b, c, radian = TRUE)

Arguments

`a`, `b`, `c`	Three 2D points which represent the intersecting line segments $[ba]$ and $[bc]$ .
`radian`	A logical argument (default=`TRUE`). If `TRUE`, the smaller angle or counter-clockwise angle between the line segments $[ba]$ and $[bc]$ is provided in radians, else it is provided in degrees.

Value

A list with two elements

`small.arc.angles`	Angles of $[ba]$ and $[bc]$ with the $x$ -axis so that difference between them is the smaller angle between $[ba]$ and $[bc]$
`ccw.arc.angles`	Angles of $[ba]$ and $[bc]$ with the $x$ -axis so that difference between them is the counter-clockwise angle between $[ba]$ and $[bc]$

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(.3,.2); B<-c(.6,.3); C<-c(1,1)

pts<-rbind(A,B,C)

Xp<-c(B[1]+max(abs(C[1]-B[1]),abs(A[1]-B[1])),0)

angle.str2end(A,B,C)
angle.str2end(A,B,A)

angle.str2end(A,B,C,radian=FALSE)

#plot of the line segments
ang.rad<-angle.str2end(A,B,C,radian=TRUE); ang.rad
ang.deg<-angle.str2end(A,B,C,radian=FALSE); ang.deg
ang.deg1<-ang.deg$s; ang.deg1
ang.deg2<-ang.deg$c; ang.deg2

rad<-min(Dist(A,B),Dist(B,C))

Xlim<-range(pts[,1],Xp[1],B+Xp,B[1]+c(+rad,-rad))
Ylim<-range(pts[,2],B[2]+c(+rad,-rad))
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot for the smaller arc
plot(pts,pch=1,asp=1,xlab="x",ylab="y",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
L<-rbind(B,B,B); R<-rbind(A,C,B+Xp)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
plotrix::draw.arc(B[1],B[2],radius=.3*rad,angle1=ang.rad$s[1],angle2=ang.rad$s[2])
plotrix::draw.arc(B[1],B[2],radius=.6*rad,angle1=0, angle2=ang.rad$s[1],lty=2,col=2)
plotrix::draw.arc(B[1],B[2],radius=.9*rad,angle1=0,angle2=ang.rad$s[2],col=3)
txt<-rbind(A,B,C)
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.02,nrow(txt))),c("A","B","C"))

text(rbind(B)+.5*rad*c(cos(mean(ang.rad$s)),sin(mean(ang.rad$s))),
     paste(abs(round(ang.deg1[2]-ang.deg1[1],2))," degrees",sep=""))
text(rbind(B)+.6*rad*c(cos(ang.rad$s[1]/2),sin(ang.rad$s[1]/2)),paste(round(ang.deg1[1],2)),col=2)
text(rbind(B)+.9*rad*c(cos(ang.rad$s[2]/2),sin(ang.rad$s[2]/2)),paste(round(ang.deg1[2],2)),col=3)

#plot for the counter-clockwise arc
plot(pts,pch=1,asp=1,xlab="x",ylab="y",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
L<-rbind(B,B,B); R<-rbind(A,C,B+Xp)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
plotrix::draw.arc(B[1],B[2],radius=.3*rad,angle1=ang.rad$c[1],angle2=ang.rad$c[2])
plotrix::draw.arc(B[1],B[2],radius=.6*rad,angle1=0, angle2=ang.rad$s[1],lty=2,col=2)
plotrix::draw.arc(B[1],B[2],radius=.9*rad,angle1=0,angle2=ang.rad$s[2],col=3)
txt<-pts
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.02,nrow(txt))),c("A","B","C"))

text(rbind(B)+.5*rad*c(cos(mean(ang.rad$c)),sin(mean(ang.rad$c))),
     paste(abs(round(ang.deg2[2]-ang.deg2[1],2))," degrees",sep=""))
text(rbind(B)+.6*rad*c(cos(ang.rad$s[1]/2),sin(ang.rad$s[1]/2)),paste(round(ang.deg1[1],2)),col=2)
text(rbind(B)+.9*rad*c(cos(ang.rad$s[2]/2),sin(ang.rad$s[2]/2)),paste(round(ang.deg1[2],2)),col=3)

## End(Not run)

## Not run: 
A<-c(.3,.2); B<-c(.6,.3); C<-c(1,1)

pts<-rbind(A,B,C)

Xp<-c(B[1]+max(abs(C[1]-B[1]),abs(A[1]-B[1])),0)

angle.str2end(A,B,C)
angle.str2end(A,B,A)

angle.str2end(A,B,C,radian=FALSE)

#plot of the line segments
ang.rad<-angle.str2end(A,B,C,radian=TRUE); ang.rad
ang.deg<-angle.str2end(A,B,C,radian=FALSE); ang.deg
ang.deg1<-ang.deg$s; ang.deg1
ang.deg2<-ang.deg$c; ang.deg2

rad<-min(Dist(A,B),Dist(B,C))

Xlim<-range(pts[,1],Xp[1],B+Xp,B[1]+c(+rad,-rad))
Ylim<-range(pts[,2],B[2]+c(+rad,-rad))
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot for the smaller arc
plot(pts,pch=1,asp=1,xlab="x",ylab="y",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
L<-rbind(B,B,B); R<-rbind(A,C,B+Xp)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
plotrix::draw.arc(B[1],B[2],radius=.3*rad,angle1=ang.rad$s[1],angle2=ang.rad$s[2])
plotrix::draw.arc(B[1],B[2],radius=.6*rad,angle1=0, angle2=ang.rad$s[1],lty=2,col=2)
plotrix::draw.arc(B[1],B[2],radius=.9*rad,angle1=0,angle2=ang.rad$s[2],col=3)
txt<-rbind(A,B,C)
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.02,nrow(txt))),c("A","B","C"))

text(rbind(B)+.5*rad*c(cos(mean(ang.rad$s)),sin(mean(ang.rad$s))),
     paste(abs(round(ang.deg1[2]-ang.deg1[1],2))," degrees",sep=""))
text(rbind(B)+.6*rad*c(cos(ang.rad$s[1]/2),sin(ang.rad$s[1]/2)),paste(round(ang.deg1[1],2)),col=2)
text(rbind(B)+.9*rad*c(cos(ang.rad$s[2]/2),sin(ang.rad$s[2]/2)),paste(round(ang.deg1[2],2)),col=3)

#plot for the counter-clockwise arc
plot(pts,pch=1,asp=1,xlab="x",ylab="y",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
L<-rbind(B,B,B); R<-rbind(A,C,B+Xp)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
plotrix::draw.arc(B[1],B[2],radius=.3*rad,angle1=ang.rad$c[1],angle2=ang.rad$c[2])
plotrix::draw.arc(B[1],B[2],radius=.6*rad,angle1=0, angle2=ang.rad$s[1],lty=2,col=2)
plotrix::draw.arc(B[1],B[2],radius=.9*rad,angle1=0,angle2=ang.rad$s[2],col=3)
txt<-pts
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.02,nrow(txt))),c("A","B","C"))

text(rbind(B)+.5*rad*c(cos(mean(ang.rad$c)),sin(mean(ang.rad$c))),
     paste(abs(round(ang.deg2[2]-ang.deg2[1],2))," degrees",sep=""))
text(rbind(B)+.6*rad*c(cos(ang.rad$s[1]/2),sin(ang.rad$s[1]/2)),paste(round(ang.deg1[1],2)),col=2)
text(rbind(B)+.9*rad*c(cos(ang.rad$s[2]/2),sin(ang.rad$s[2]/2)),paste(round(ang.deg1[2],2)),col=3)

## End(Not run)

The angle between two line segments

Description

Returns the angle in radians or degrees between two vectors or line segments at the point of intersection. The line segments are $[ba]$ and $[bc]$ when the arguments of the function are given as a,b,c. radian is a logical argument (default=TRUE) which yields the angle in radians if TRUE, and in degrees if FALSE. The smaller of the angle between the line segments is provided by the function.

Usage

angle3pnts(a, b, c, radian = TRUE)
angle3pnts(a, b, c, radian = TRUE)

Arguments

`a`, `b`, `c`	Three 2D points which represent the intersecting line segments $[ba]$ and $[bc]$ . The smaller angle between these line segments is to be computed.
`radian`	A logical argument (default=`TRUE`). If `TRUE`, the (smaller) angle between the line segments $[ba]$ and $[bc]$ is provided in radians, else it is provided in degrees.

Value

angle in radians or degrees between the line segments $[ba]$ and $[bc]$

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(.3,.2); B<-c(.6,.3); C<-c(1,1)
pts<-rbind(A,B,C)

angle3pnts(A,B,C)

angle3pnts(A,B,A)
round(angle3pnts(A,B,A),7)

angle3pnts(A,B,C,radian=FALSE)

#plot of the line segments
Xlim<-range(pts[,1])
Ylim<-range(pts[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

ang1<-angle3pnts(A,B,C,radian=FALSE)
ang2<-angle3pnts(B+c(1,0),B,C,radian=FALSE)

sa<-angle.str2end(A,B,C,radian=FALSE)$s #small arc angles
ang1<-sa[1]
ang2<-sa[2]

plot(pts,asp=1,pch=1,xlab="x",ylab="y",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
L<-rbind(B,B); R<-rbind(A,C)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
plotrix::draw.arc(B[1],B[2],radius=xd*.1,deg1=ang1,deg2=ang2)
txt<-rbind(A,B,C)
text(txt+cbind(rep(xd*.05,nrow(txt)),rep(-xd*.02,nrow(txt))),c("A","B","C"))

text(rbind(B)+.15*xd*c(cos(pi*(ang2+ang1)/360),sin(pi*(ang2+ang1)/360)),
paste(round(abs(ang1-ang2),2)," degrees"))

## End(Not run)

## Not run: 
A<-c(.3,.2); B<-c(.6,.3); C<-c(1,1)
pts<-rbind(A,B,C)

angle3pnts(A,B,C)

angle3pnts(A,B,A)
round(angle3pnts(A,B,A),7)

angle3pnts(A,B,C,radian=FALSE)

#plot of the line segments
Xlim<-range(pts[,1])
Ylim<-range(pts[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

ang1<-angle3pnts(A,B,C,radian=FALSE)
ang2<-angle3pnts(B+c(1,0),B,C,radian=FALSE)

sa<-angle.str2end(A,B,C,radian=FALSE)$s #small arc angles
ang1<-sa[1]
ang2<-sa[2]

plot(pts,asp=1,pch=1,xlab="x",ylab="y",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
L<-rbind(B,B); R<-rbind(A,C)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
plotrix::draw.arc(B[1],B[2],radius=xd*.1,deg1=ang1,deg2=ang2)
txt<-rbind(A,B,C)
text(txt+cbind(rep(xd*.05,nrow(txt)),rep(-xd*.02,nrow(txt))),c("A","B","C"))

text(rbind(B)+.15*xd*c(cos(pi*(ang2+ang1)/360),sin(pi*(ang2+ang1)/360)),
paste(round(abs(ang1-ang2),2)," degrees"))

## End(Not run)

The arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for a 2D data set - multiple triangle case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) of AS-PCD whose vertices are the data set Xp and related parameters and the quantities of the digraph.

AS proximity regions are defined with respect to the Delaunay triangles based on Yp points, i.e., AS proximity regions are defined only for Xp points inside the convex hull of Yp points. That is, arcs may exist for points only inside the convex hull of Yp points. It also provides various descriptions and quantities about the arcs of the AS-PCD such as number of arcs, arc density, etc.

Vertex regions are based on the center M="CC" for circumcenter of each Delaunay triangle or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle; default is M="CC" i.e., circumcenter of each triangle. M must be entered in barycentric coordinates unless it is the circumcenter.

See (Ceyhan (2005, 2010)) for more on AS PCDs. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

arcsAS(Xp, Yp, M = "CC")
arcsAS(Xp, Yp, M = "CC")

Arguments

`Xp`	A set of 2D points which constitute the vertices of the AS-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangulation. The Delaunay triangles partition the convex hull of `Yp` points.
`M`	The center of the triangle. `"CC"` represents the circumcenter of each Delaunay triangle or 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle; default is `M="CC"` i.e., the circumcenter of each triangle. `M` must be entered in barycentric coordinates unless it is the circumcenter.

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, it is the center used to construct the vertex regions, default is circumcenter, denoted as `"CC"`, otherwise given in barycentric coordinates.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the Delaunay triangulation based on `Yp` points.
`tess.name`	Name of data set used in tessellation, i.e., `Yp`
`vertices`	Vertices of the digraph, `Xp`.
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of AS-PCD for 2D data set `Xp` in the multiple triangle case as the vertices of the digraph
`E`	Heads (or arrow ends) of the arcs of AS-PCD for 2D data set `Xp` in the multiple triangle case as the vertices of the digraph
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx=20; nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

Arcs<-arcsAS(Xp,Yp,M) #try also the default M with Arcs<-arcsAS(Xp,Yp)
Arcs
summary(Arcs)
plot(Arcs)

arcsAS(Xp,Yp[1:3,],M)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx=20; nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

Arcs<-arcsAS(Xp,Yp,M) #try also the default M with Arcs<-arcsAS(Xp,Yp)
Arcs
summary(Arcs)
plot(Arcs)

arcsAS(Xp,Yp[1:3,],M)

## End(Not run)

The arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for 2D data - one triangle case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) for data set Xp as the vertices of AS-PCD and related parameters and the quantities of the digraph.

AS proximity regions are constructed with respect to the triangle tri, i.e., arcs may exist for points only inside tri. It also provides various descriptions and quantities about the arcs of the AS-PCD such as number of arcs, arc density, etc.

Vertex regions are based on the center M="CC" for circumcenter of tri; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is M="CC" the circumcenter of tri. The different consideration of circumcenter vs any other interior center of the triangle is because the projections from circumcenter are orthogonal to the edges, while projections of M on the edges are on the extensions of the lines connecting M and the vertices.

See also (Ceyhan (2005, 2010)).

Usage

arcsAStri(Xp, tri, M = "CC")
arcsAStri(Xp, tri, M = "CC")

Arguments

`Xp`	A set of 2D points which constitute the vertices of the AS-PCD.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$ ; default is `M="CC"` i.e., the circumcenter of `tri`.

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, it is the center used to construct the vertex regions.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle.
`tess.name`	Name of data set used in tessellation (i.e., vertices of the triangle).
`vertices`	Vertices of the digraph, `Xp`.
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of AS-PCD for 2D data set `Xp` as vertices of the digraph
`E`	Heads (or arrow ends) of the arcs of AS-PCD for 2D data set `Xp` as vertices of the digraph
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also  M<-c(1.6,1.2) or M<-circumcenter.tri(Tr)

Arcs<-arcsAStri(Xp,Tr,M) #try also Arcs<-arcsAStri(Xp,Tr)
#uses the default center, namely circumcenter for M
Arcs
summary(Arcs)
plot(Arcs) #use plot(Arcs,asp=1) if M=CC

#can add vertex regions
#but we first need to determine center is the circumcenter or not,
#see the description for more detail.
CC<-circumcenter.tri(Tr)
M = as.numeric(Arcs$parameters[[1]])
if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

#now we add the vertex names and annotation
txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]+c(-.02,.03,.02,.03,.04,-.03,-.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.04,.05,-.07)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also  M<-c(1.6,1.2) or M<-circumcenter.tri(Tr)

Arcs<-arcsAStri(Xp,Tr,M) #try also Arcs<-arcsAStri(Xp,Tr)
#uses the default center, namely circumcenter for M
Arcs
summary(Arcs)
plot(Arcs) #use plot(Arcs,asp=1) if M=CC

#can add vertex regions
#but we first need to determine center is the circumcenter or not,
#see the description for more detail.
CC<-circumcenter.tri(Tr)
M = as.numeric(Arcs$parameters[[1]])
if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

#now we add the vertex names and annotation
txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]+c(-.02,.03,.02,.03,.04,-.03,-.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.04,.05,-.07)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D data - multiple triangle case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) of Central Similarity Proximity Catch Digraph (CS-PCD) whose vertices are the data points in Xp in the multiple triangle case and related parameters and the quantities of the digraph.

CS proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter $t>0$ and edge regions in each triangle are based on the center $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle (default for $M=(1,1,1)$ which is the center of mass of the triangle). Each Delaunay triangle is first converted to an (nonscaled) basic triangle so that M will be the same type of center for each Delaunay triangle (this conversion is not necessary when M is $CM$ ).

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points). For the number of arcs, loops are not allowed so arcs are only possible for points inside the convex hull of Yp points.

See (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)) for more on CS-PCDs. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

arcsCS(Xp, Yp, t, M = c(1, 1, 1))
arcsCS(Xp, Yp, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle, default for $M=(1,1,1)$ which is the center of mass of each triangle.

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, it is the center used to construct the edge regions.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is Delaunay triangulation based on `Yp` points.
`tess.name`	Name of data set used in tessellation, it is `Yp` for this function
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of CS-PCD for 2D data set `Xp` as vertices of the digraph
`E`	Heads (or arrow ends) of the arcs of CS-PCD for 2D data set `Xp` as vertices of the digraph
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of triangles, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2014). “Comparison of Relative Density of Two Random Geometric Digraph Families in Testing Spatial Clustering.” TEST, 23(1), 100-134.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

tau<-1.5  #try also tau<-2

Arcs<-arcsCS(Xp,Yp,tau,M)
#or use the default center Arcs<-arcsCS(Xp,Yp,tau)
Arcs
summary(Arcs)
plot(Arcs)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

tau<-1.5  #try also tau<-2

Arcs<-arcsCS(Xp,Yp,tau,M)
#or use the default center Arcs<-arcsCS(Xp,Yp,tau)
Arcs
summary(Arcs)
plot(Arcs)

## End(Not run)

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - multiple interval case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) for 1D data set Xp as the vertices of CS-PCD and related parameters and the quantities of the digraph. Yp determines the end points of the intervals.

For this function, CS proximity regions are constructed data points inside or outside the intervals based on Yp points with expansion parameter $t>0$ and centrality parameter $c \in (0,1)$ . That is, for this function, arcs may exist for points in the middle or end intervals. It also provides various descriptions and quantities about the arcs of the CS-PCD such as number of arcs, arc density, etc.

Equivalent to function arcsCS1D.

See also (Ceyhan (2016)).

Usage

arcsCS1D(Xp, Yp, t, c = 0.5)
arcsCS1D(Xp, Yp, t, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the CS-PCD.
`Yp`	A set or `vector` of 1D points which constitute the end points of the intervals.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, they are expansion and centrality parameters.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the intervalization of the real line based on `Yp` points.
`tess.name`	Name of data set used in tessellation, it is `Yp` for this function
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of CS-PCD for 1D data
`E`	Heads (or arrow ends) of the arcs of CS-PCD for 1D data
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

t<-2
c<-.4
a<-0; b<-10;

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xr<-range(a,b)
xf<-(xr[2]-xr[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Arcs<-arcsCS1D(Xp,Yp,t,c)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

arcsCS1D(Xp,Yp,t,c)

arcsCS1D(Xp,Yp+10,t,c)

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(a,b,Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),
main="arcs of CS-PCD for points (jittered along y-axis)\n in middle intervals ",
xlab=" ", ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit),pch=".")
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

t<-2
c<-.4
a<-0; b<-10;
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

arcsCS1D(Xp,Yp,t,c)

t<-2
c<-.4
a<-0; b<-10;

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xr<-range(a,b)
xf<-(xr[2]-xr[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Arcs<-arcsCS1D(Xp,Yp,t,c)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

arcsCS1D(Xp,Yp,t,c)

arcsCS1D(Xp,Yp+10,t,c)

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(a,b,Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),
main="arcs of CS-PCD for points (jittered along y-axis)\n in middle intervals ",
xlab=" ", ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit),pch=".")
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

t<-2
c<-.4
a<-0; b<-10;
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

arcsCS1D(Xp,Yp,t,c)

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - end interval case

Description

For this function, CS proximity regions are constructed data points outside the intervals based on Yp points with expansion parameter $t>0$ . That is, for this function, arcs may exist for points only inside end intervals. It also provides various descriptions and quantities about the arcs of the CS-PCD such as number of arcs, arc density, etc.

See also (Ceyhan (2016)).

Usage

arcsCSend.int(Xp, Yp, t)
arcsCSend.int(Xp, Yp, t)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the CS-PCD.
`Yp`	A set or `vector` of 1D points which constitute the end points of the intervals.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, it is the expansion parameter.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the intervalization based on `Yp`.
`tess.name`	Name of data set used in tessellation, it is `Yp` for this function
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitutes the vertices of the digraph
`S`	Tails (or sources) of the arcs of CS-PCD for 1D data in the end intervals
`E`	Heads (or arrow ends) of the arcs of CS-PCD for 1D data in the end intervals
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals (which is 2 for end intervals), number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

t<-1.5
a<-0; b<-10; int<-c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xr<-range(a,b)
xf<-(xr[2]-xr[1])*.5

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

arcsCSend.int(Xp,Yp,t)

Arcs<-arcsCSend.int(Xp,Yp,t)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(a,b,Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),pch=".",
main="arcs of CS-PCD with vertices (jittered along y-axis)\n in end intervals ",
     xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit))
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

arcsCSend.int(Xp,Yp,t)

t<-1.5
a<-0; b<-10; int<-c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xr<-range(a,b)
xf<-(xr[2]-xr[1])*.5

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

arcsCSend.int(Xp,Yp,t)

Arcs<-arcsCSend.int(Xp,Yp,t)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(a,b,Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),pch=".",
main="arcs of CS-PCD with vertices (jittered along y-axis)\n in end intervals ",
     xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit))
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

arcsCSend.int(Xp,Yp,t)

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - one interval case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) for 1D data set Xp as the vertices of CS-PCD. int determines the end points of the interval.

For this function, CS proximity regions are constructed data points inside or outside the interval based on int points with expansion parameter $t > 0$ and centrality parameter $c \in (0,1)$ . That is, for this function, arcs may exist for points in the middle or end intervals. It also provides various descriptions and quantities about the arcs of the CS-PCD such as number of arcs, arc density, etc.

Usage

arcsCSint(Xp, int, t, c = 0.5)
arcsCSint(Xp, int, t, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the CS-PCD.
`int`	A `vector` of two 1D points which constitutes the end points of the interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, they are expansion and centrality parameters.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the intervalization of the real line based on `int` points.
`tess.name`	Name of data set used in tessellation, it is `int` for this function
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of CS-PCD for 1D data
`E`	Heads (or arrow ends) of the arcs of CS-PCD for 1D data
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

There are no references for Rd macro ⁠\insertAllCites⁠ on this help page.

Examples

tau<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#n is number of X points
n<-10; #try also n<-20

xf<-(int[2]-int[1])*.1

set.seed(1)
Xp<-runif(n,a-xf,b+xf)

Arcs<-arcsCSint(Xp,int,tau,c)
Arcs
summary(Arcs)
plot(Arcs)

Xp<-runif(n,a+10,b+10)
Arcs=arcsCSint(Xp,int,tau,c)
Arcs
summary(Arcs)
plot(Arcs)

tau<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#n is number of X points
n<-10; #try also n<-20

xf<-(int[2]-int[1])*.1

set.seed(1)
Xp<-runif(n,a-xf,b+xf)

Arcs<-arcsCSint(Xp,int,tau,c)
Arcs
summary(Arcs)
plot(Arcs)

Xp<-runif(n,a+10,b+10)
Arcs=arcsCSint(Xp,int,tau,c)
Arcs
summary(Arcs)
plot(Arcs)

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - middle intervals case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) for 1D data set Xp as the vertices of CS-PCD and related parameters and the quantities of the digraph.

For this function, CS proximity regions are constructed with respect to the intervals based on Yp points with expansion parameter $t>0$ and centrality parameter $c \in (0,1)$ . That is, for this function, arcs may exist for points only inside the intervals. It also provides various descriptions and quantities about the arcs of the CS-PCD such as number of arcs, arc density, etc.

Vertex regions are based on center $M_c$ of each middle interval.

See also (Ceyhan (2016)).

Usage

arcsCSmid.int(Xp, Yp, t, c = 0.5)
arcsCSmid.int(Xp, Yp, t, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the CS-PCD.
`Yp`	A set or `vector` of 1D points which constitute the end points of the intervals.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, they are expansion and centrality parameters.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the intervalization based on `Yp` points.
`tess.name`	Name of data set used in tessellation, it is `Yp` for this function
`vertices`	Vertices of the digraph, i.e., `Xp` points
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of CS-PCD for 1D data in the middle intervals
`E`	Heads (or arrow ends) of the arcs of CS-PCD for 1D data in the middle intervals
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

t<-1.5
c<-.4
a<-0; b<-10

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

arcsCSmid.int(Xp,Yp,t,c)
arcsCSmid.int(Xp,Yp+10,t,c)

Arcs<-arcsCSmid.int(Xp,Yp,t,c)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),
main="arcs of CS-PCD whose vertices (jittered along y-axis)\n in middle intervals ",
xlab=" ", ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit),pch=".")
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

t<-.5
c<-.4
a<-0; b<-10;
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

arcsCSmid.int(Xp,Yp,t,c)

t<-1.5
c<-.4
a<-0; b<-10

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

arcsCSmid.int(Xp,Yp,t,c)
arcsCSmid.int(Xp,Yp+10,t,c)

Arcs<-arcsCSmid.int(Xp,Yp,t,c)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),
main="arcs of CS-PCD whose vertices (jittered along y-axis)\n in middle intervals ",
xlab=" ", ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit),pch=".")
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

t<-.5
c<-.4
a<-0; b<-10;
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

arcsCSmid.int(Xp,Yp,t,c)

The arcs of Central Similarity Proximity Catch Digraphs (CS-PCD) for 2D data - one triangle case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) for data set Xp as the vertices of CS-PCD and related parameters and the quantities of the digraph.

CS proximity regions are constructed with respect to the triangle tri with expansion parameter $t>0$ , i.e., arcs may exist for points only inside tri. It also provides various descriptions and quantities about the arcs of the CS-PCD such as number of arcs, arc density, etc.

Edge regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is $M=(1,1,1)$ i.e., the center of mass of tri.

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

arcsCStri(Xp, tri, t, M = c(1, 1, 1))
arcsCStri(Xp, tri, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`.

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, the center `M` used to construct the edge regions and the expansion parameter `t`.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle.
`tess.name`	Name of data set used in tessellation (i.e., vertices of the triangle)
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of CS-PCD for 2D data set `Xp` as vertices of the digraph
`E`	Heads (or arrow ends) of the arcs of CS-PCD for 2D data set `Xp` as vertices of the digraph
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of triangles, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-1.5  #try also t<-2

Arcs<-arcsCStri(Xp,Tr,t,M)
#or try with the default center Arcs<-arcsCStri(Xp,Tr,t); M= (Arcs$param)$c
Arcs
summary(Arcs)
plot(Arcs)

#can add edge regions
L<-rbind(M,M,M); R<-Tr
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

#now we can add the vertex names and annotation
txt<-rbind(Tr,M)
xc<-txt[,1]+c(-.02,.03,.02,.03)
yc<-txt[,2]+c(.02,.02,.03,.06)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-1.5  #try also t<-2

Arcs<-arcsCStri(Xp,Tr,t,M)
#or try with the default center Arcs<-arcsCStri(Xp,Tr,t); M= (Arcs$param)$c
Arcs
summary(Arcs)
plot(Arcs)

#can add edge regions
L<-rbind(M,M,M); R<-Tr
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

#now we can add the vertex names and annotation
txt<-rbind(Tr,M)
xc<-txt[,1]+c(-.02,.03,.02,.03)
yc<-txt[,2]+c(.02,.02,.03,.06)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

## End(Not run)

The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - multiple triangle case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) of Proportional Edge Proximity Catch Digraph (PE-PCD) whose vertices are the data points in Xp in the multiple triangle case and related parameters and the quantities of the digraph.

PE proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter $r \ge 1$ and vertex regions in each triangle are based on the center $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle or based on circumcenter of each Delaunay triangle (default for $M=(1,1,1)$ which is the center of mass of the triangle). Each Delaunay triangle is first converted to an (nonscaled) basic triangle so that M will be the same type of center for each Delaunay triangle (this conversion is not necessary when M is $CM$ ).

See (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)) for more on the PE-PCDs. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

arcsPE(Xp, Yp, r, M = c(1, 1, 1))
arcsPE(Xp, Yp, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as `M="CC"`), default for $M=(1,1,1)$ which is the center of mass of each triangle.

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, the center used to construct the vertex regions and the expansion parameter.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is Delaunay triangulation based on `Yp` points.
`tess.name`	Name of data set used in tessellation, it is `Yp` for this function
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of PE-PCD for 2D data set `Xp` as vertices of the digraph
`E`	Heads (or arrow ends) of the arcs of PE-PCD for 2D data set `Xp` as vertices of the digraph
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of triangles, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

r<-1.5  #try also r<-2

Arcs<-arcsPE(Xp,Yp,r,M)
#or try with the default center Arcs<-arcsPE(Xp,Yp,r)
Arcs
summary(Arcs)
plot(Arcs)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

r<-1.5  #try also r<-2

Arcs<-arcsPE(Xp,Yp,r,M)
#or try with the default center Arcs<-arcsPE(Xp,Yp,r)
Arcs
summary(Arcs)
plot(Arcs)

## End(Not run)

The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - multiple interval case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) for 1D data set Xp as the vertices of PE-PCD and related parameters and the quantities of the digraph. Yp determines the end points of the intervals.

For this function, PE proximity regions are constructed data points inside or outside the intervals based on Yp points with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ . That is, for this function, arcs may exist for points in the middle or end intervals. It also provides various descriptions and quantities about the arcs of the PE-PCD such as number of arcs, arc density, etc.

See also (Ceyhan (2012)).

Usage

arcsPE1D(Xp, Yp, r, c = 0.5)
arcsPE1D(Xp, Yp, r, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the PE-PCD.
`Yp`	A set or `vector` of 1D points which constitute the end points of the intervals.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, they are expansion and centrality parameters.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the intervalization of the real line based on `Yp` points.
`tess.name`	Name of data set used in tessellation, it is `Yp` for this function
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of PE-PCD for 1D data
`E`	Heads (or arrow ends) of the arcs of PE-PCD for 1D data
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Arcs<-arcsPE1D(Xp,Yp,r,c)
Arcs
summary(Arcs)
plot(Arcs)

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Arcs<-arcsPE1D(Xp,Yp,r,c)
Arcs
summary(Arcs)
plot(Arcs)

## End(Not run)

The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - end interval case

Description

For this function, PE proximity regions are constructed data points outside the intervals based on Yp points with expansion parameter $r \ge 1$ . That is, for this function, arcs may exist for points only inside end intervals. It also provides various descriptions and quantities about the arcs of the PE-PCD such as number of arcs, arc density, etc.

See also (Ceyhan (2012)).

Usage

arcsPEend.int(Xp, Yp, r)
arcsPEend.int(Xp, Yp, r)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the PE-PCD.
`Yp`	A set or `vector` of 1D points which constitute the end points of the intervals.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, it is the expansion parameter.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the intervalization based on `Yp`.
`tess.name`	Name of data set used in tessellation, it is `Yp` for this function
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitutes the vertices of the digraph
`S`	Tails (or sources) of the arcs of PE-PCD for 1D data in the end intervals
`E`	Heads (or arrow ends) of the arcs of PE-PCD for 1D data in the end intervals
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals (which is 2 for end intervals), number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.5

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)  #try also Yp<-runif(ny,a,b)+c(-10,10)

Arcs<-arcsPEend.int(Xp,Yp,r)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(a,b,Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),pch=".",
main="arcs of PE-PCDs for points (jittered along y-axis)\n in end intervals ",
xlab=" ", ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit))
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

## End(Not run)

## Not run: 
r<-2
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.5

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)  #try also Yp<-runif(ny,a,b)+c(-10,10)

Arcs<-arcsPEend.int(Xp,Yp,r)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(a,b,Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),pch=".",
main="arcs of PE-PCDs for points (jittered along y-axis)\n in end intervals ",
xlab=" ", ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit))
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

## End(Not run)

The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - one interval case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) for 1D data set Xp as the vertices of PE-PCD. int determines the end points of the interval.

For this function, PE proximity regions are constructed data points inside or outside the interval based on int points with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ . That is, for this function, arcs may exist for points in the middle or end intervals. It also provides various descriptions and quantities about the arcs of the PE-PCD such as number of arcs, arc density, etc.

See also (Ceyhan (2012)).

Usage

arcsPEint(Xp, int, r, c = 0.5)
arcsPEint(Xp, int, r, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the PE-PCD.
`int`	A `vector` of two 1D points which constitutes the end points of the interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, they are expansion and centrality parameters.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the intervalization of the real line based on `int` points.
`tess.name`	Name of data set used in tessellation, it is `int` for this function
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of PE-PCD for 1D data
`E`	Heads (or arrow ends) of the arcs of PE-PCD for 1D data
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#n is number of X points
n<-10; #try also n<-20

xf<-(int[2]-int[1])*.1

set.seed(1)
Xp<-runif(n,a-xf,b+xf)

Arcs<-arcsPEint(Xp,int,r,c)
Arcs
summary(Arcs)
plot(Arcs)

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#n is number of X points
n<-10; #try also n<-20

xf<-(int[2]-int[1])*.1

set.seed(1)
Xp<-runif(n,a-xf,b+xf)

Arcs<-arcsPEint(Xp,int,r,c)
Arcs
summary(Arcs)
plot(Arcs)

## End(Not run)

The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - middle intervals case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) for 1D data set Xp as the vertices of PE-PCD.

For this function, PE proximity regions are constructed with respect to the intervals based on Yp points with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ . That is, for this function, arcs may exist for points only inside the intervals. It also provides various descriptions and quantities about the arcs of the PE-PCD such as number of arcs, arc density, etc.

Vertex regions are based on center $M_c$ of each middle interval.

See also (Ceyhan (2012)).

Usage

arcsPEmid.int(Xp, Yp, r, c = 0.5)
arcsPEmid.int(Xp, Yp, r, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the PE-PCD.
`Yp`	A set or `vector` of 1D points which constitute the end points of the intervals.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, here, they are expansion and centrality parameters.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the intervalization based on `Yp` points.
`tess.name`	Name of data set used in tessellation, it is `Yp` for this function
`vertices`	Vertices of the digraph, i.e., `Xp` points
`vert.name`	Name of the data set which constitute the vertices of the digraph
`S`	Tails (or sources) of the arcs of PE-PCD for 1D data in the middle intervals
`E`	Heads (or arrow ends) of the arcs of PE-PCD for 1D data in the middle intervals
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of intervals, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10;

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

Arcs<-arcsPEmid.int(Xp,Yp,r,c)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

arcsPEmid.int(Xp,Yp,r,c)
arcsPEmid.int(Xp,Yp+10,r,c)

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),
main="arcs of PE-PCD for points (jittered along y-axis)\n in middle intervals ",
xlab=" ", ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit),pch=".")
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10;

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

Arcs<-arcsPEmid.int(Xp,Yp,r,c)
Arcs
summary(Arcs)
plot(Arcs)

S<-Arcs$S
E<-Arcs$E

arcsPEmid.int(Xp,Yp,r,c)
arcsPEmid.int(Xp,Yp+10,r,c)

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),
main="arcs of PE-PCD for points (jittered along y-axis)\n in middle intervals ",
xlab=" ", ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit),pch=".")
abline(h=0,lty=1)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
arrows(S, yjit, E, yjit, length = .05, col= 4)

## End(Not run)

The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - one triangle case

Description

An object of class "PCDs". Returns arcs as tails (or sources) and heads (or arrow ends) for data set Xp as the vertices of PE-PCD and related parameters and the quantities of the digraph.

PE proximity regions are constructed with respect to the triangle tri with expansion parameter $r \ge 1$ , i.e., arcs may exist only for points inside tri. It also provides various descriptions and quantities about the arcs of the PE-PCD such as number of arcs, arc density, etc.

Vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri or based on the circumcenter of tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri. When the center is the circumcenter, CC, the vertex regions are constructed based on the orthogonal projections to the edges, while with any interior center M, the vertex regions are constructed using the extensions of the lines combining vertices with M. M-vertex regions are recommended spatial inference, due to geometry invariance property of the arc density and domination number the PE-PCDs based on uniform data.

See also (Ceyhan (2005); Ceyhan et al. (2006)).

Usage

arcsPEtri(Xp, tri, r, M = c(1, 1, 1))
arcsPEtri(Xp, tri, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.

Value

A list with the elements

`type`	A description of the type of the digraph
`parameters`	Parameters of the digraph, the center `M` used to construct the vertex regions and the expansion parameter `r`.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle.
`tess.name`	Name of data set (i.e. points from the non-target class) used in the tessellation of the space (here, vertices of the triangle)
`vertices`	Vertices of the digraph, `Xp` points
`vert.name`	Name of the data set which constitutes the vertices of the digraph
`S`	Tails (or sources) of the arcs of PE-PCD for 2D data set `Xp` as vertices of the digraph
`E`	Heads (or arrow ends) of the arcs of PE-PCD for 2D data set `Xp` as vertices of the digraph
`mtitle`	Text for `"main"` title in the plot of the digraph
`quant`	Various quantities for the digraph: number of vertices, number of partition points, number of triangles, number of arcs, and arc density.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5  #try also r<-2

Arcs<-arcsPEtri(Xp,Tr,r,M)
#or try with the default center Arcs<-arcsPEtri(Xp,Tr,r); M= (Arcs$param)$cent
Arcs
summary(Arcs)
plot(Arcs)

#can add vertex regions
#but we first need to determine center is the circumcenter or not,
#see the description for more detail.
CC<-circumcenter.tri(Tr)
if (isTRUE(all.equal(M,CC)))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

#now we can add the vertex names and annotation
txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]+c(-.02,.02,.02,.02,.03,-.03,-.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.04,.05,-.07)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5  #try also r<-2

Arcs<-arcsPEtri(Xp,Tr,r,M)
#or try with the default center Arcs<-arcsPEtri(Xp,Tr,r); M= (Arcs$param)$cent
Arcs
summary(Arcs)
plot(Arcs)

#can add vertex regions
#but we first need to determine center is the circumcenter or not,
#see the description for more detail.
CC<-circumcenter.tri(Tr)
if (isTRUE(all.equal(M,CC)))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

#now we can add the vertex names and annotation
txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]+c(-.02,.02,.02,.02,.03,-.03,-.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.04,.05,-.07)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The area of a polygon in $R^2$

Description

Returns the area of the polygon, h, in the real plane $R^{2}$ ; the vertices of the polygon h must be provided in clockwise or counter-clockwise order, otherwise the function does not yield the area of the polygon. Also, the polygon could be convex or non-convex. See (Weisstein (2019)).

Usage

area.polygon(h)
area.polygon(h)

Arguments

`h`	A `vector` of $n$ 2D points, stacked row-wise, each row representing a vertex of the polygon, where $n$ is the number of vertices of the polygon.

Value

area of the polygon h

Author(s)

Elvan Ceyhan

References

Weisstein EW (2019). “Polygon Area.” From MathWorld — A Wolfram Web Resource, http://mathworld.wolfram.com/PolygonArea.html.

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(0.5,.8);
Tr<-rbind(A,B,C);
area.polygon(Tr)

A<-c(0,0); B<-c(1,0); C<-c(.7,.6); D<-c(0.3,.8);
h1<-rbind(A,B,C,D);
#try also h1<-rbind(A,B,D,C) or h1<-rbind(A,C,B,D) or h1<-rbind(A,D,C,B);
area.polygon(h1)

Xlim<-range(h1[,1])
Ylim<-range(h1[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(h1,xlab="",ylab="",main="A Convex Polygon with Four Vertices",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(h1)
xc<-rbind(A,B,C,D)[,1]+c(-.03,.03,.02,-.01)
yc<-rbind(A,B,C,D)[,2]+c(.02,.02,.02,.03)
txt.str<-c("A","B","C","D")
text(xc,yc,txt.str)

#when the triangle is degenerate, it gives zero area
B<-A+2*(C-A);
T2<-rbind(A,B,C)
area.polygon(T2)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(0.5,.8);
Tr<-rbind(A,B,C);
area.polygon(Tr)

A<-c(0,0); B<-c(1,0); C<-c(.7,.6); D<-c(0.3,.8);
h1<-rbind(A,B,C,D);
#try also h1<-rbind(A,B,D,C) or h1<-rbind(A,C,B,D) or h1<-rbind(A,D,C,B);
area.polygon(h1)

Xlim<-range(h1[,1])
Ylim<-range(h1[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(h1,xlab="",ylab="",main="A Convex Polygon with Four Vertices",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(h1)
xc<-rbind(A,B,C,D)[,1]+c(-.03,.03,.02,-.01)
yc<-rbind(A,B,C,D)[,2]+c(.02,.02,.02,.03)
txt.str<-c("A","B","C","D")
text(xc,yc,txt.str)

#when the triangle is degenerate, it gives zero area
B<-A+2*(C-A);
T2<-rbind(A,B,C)
area.polygon(T2)

## End(Not run)

The labels of the vertices of a triangle in the basic triangle form

Description

Labels the vertices of triangle, tri, as $ABC$ so that $AB$ is the longest edge, $BC$ is the second longest and $AC$ is the shortest edge (the order is as in the basic triangle).

The standard basic triangle form is $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ . Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle is useful for simulation studies under the uniformity hypothesis.

The option scaled a logical argument for scaling the resulting triangle or not. If scaled=TRUE, then the resulting triangle is scaled to be a regular basic triangle, i.e., longest edge having unit length, else (i.e., if scaled=FALSE which is the default), the new triangle $T(A,B,C)$ is nonscaled, i.e., the longest edge $AB$ may not be of unit length. The vertices of the resulting triangle (whether scaled or not) is presented in the order of vertices of the corresponding basic triangle, however when scaled the triangle is equivalent to the basic triangle $T_b$ up to translation and rotation. That is, this function converts any triangle to a basic triangle (up to translation and rotation), so that the output triangle is $T(A',B',C')$ so that edges in decreasing length are $A'B'$, $B'C'$, and $A'C'$. Most of the times, the resulting triangle will still need to be translated and/or rotated to be in the standard basic triangle form.

Usage

as.basic.tri(tri, scaled = FALSE)
as.basic.tri(tri, scaled = FALSE)

Arguments

`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`scaled`	A logical argument for scaling the resulting basic triangle. If `scaled=TRUE`, then the resulting triangle is scaled to be a regular basic triangle, i.e., longest edge having unit length, else the new triangle $T(A,B,C)$ is nonscaled. The default is `scaled=FALSE`.

Value

A list with three elements

`tri`	The vertices of the basic triangle stacked row-wise and labeled row-wise as $A$ , $B$ , $C$ .
`desc`	Description of the edges based on the vertices, i.e., `"Edges (in decreasing length are) AB, BC, and AC"`.
`orig.order`	Row order of the input triangle, `tri`, when converted to the scaled version of the basic triangle

Author(s)

Elvan Ceyhan

Examples

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);

as.basic.tri(rbind(A,B,C))
as.basic.tri(rbind(B,C,A))

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
as.basic.tri(rbind(A,B,C))
as.basic.tri(rbind(A,C,B))
as.basic.tri(rbind(B,A,C))

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);

as.basic.tri(rbind(A,B,C))
as.basic.tri(rbind(B,C,A))

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
as.basic.tri(rbind(A,B,C))
as.basic.tri(rbind(A,C,B))
as.basic.tri(rbind(B,A,C))

## End(Not run)

Arc density of Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns the arc density of AS-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

AS proximity regions is defined with respect to tri and vertex regions are defined with the center M="CC" for circumcenter of tri; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is M="CC" i.e., circumcenter of tri. For the number of arcs, loops are not allowed so arcs are only possible for points inside tri for this function.

tri.cor is a logical argument for triangle correction (default is TRUE), if TRUE, only the points inside the triangle are considered (i.e., digraph induced by these vertices are considered) in computing the arc density, otherwise all points are considered (for the number of vertices in the denominator of arc density).

See also (Ceyhan (2005, 2010)).

Usage

ASarc.dens.tri(Xp, tri, M = "CC", tri.cor = FALSE)
ASarc.dens.tri(Xp, tri, M = "CC", tri.cor = FALSE)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the AS-PCD.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of `tri`; default is `M="CC"` i.e., the circumcenter of `tri`.
`tri.cor`	A logical argument for computing the arc density for only the points inside the triangle, `tri` (default is `tri.cor=FALSE`), i.e., if `tri.cor=TRUE` only the induced digraph with the vertices inside `tri` are considered in the computation of arc density.

Value

Arc density of AS-PCD whose vertices are the 2D numerical data set, Xp; AS proximity regions are defined with respect to the triangle tri and $CC$ -vertex regions.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

set.seed(1)
n<-10  #try also n<-20

Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

num.arcsAStri(Xp,Tr,M)
ASarc.dens.tri(Xp,Tr,M)
ASarc.dens.tri(Xp,Tr,M,tri.cor = FALSE)

ASarc.dens.tri(Xp,Tr,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

set.seed(1)
n<-10  #try also n<-20

Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

num.arcsAStri(Xp,Tr,M)
ASarc.dens.tri(Xp,Tr,M)
ASarc.dens.tri(Xp,Tr,M,tri.cor = FALSE)

ASarc.dens.tri(Xp,Tr,M)

## End(Not run)

Centers for non-degenerate asymptotic distribution of domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs)

Description

Returns the centers which yield nondegenerate asymptotic distribution for the domination number of PE-PCD for uniform data in a triangle, tri $=T(v_1,v_2,v_3)$ .

PE proximity region is defined with respect to the triangle tri with expansion parameter r in $(1,1.5]$ .

Vertex regions are defined with the centers that are output of this function. Centers are stacked row-wise with row number is corresponding to the vertex row number in tri (see the examples for an illustration). The center labels 1,2,3 correspond to the vertices $M_1$ , $M_2$ , and $M_3$ (which are the three centers for r in $(1,1.5)$ which becomes center of mass for $r=1.5$ .).

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

center.nondegPE(tri, r)
center.nondegPE(tri, r)

Arguments

`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,1.5]$ for this function.

Value

The centers (stacked row-wise) which give nondegenerate asymptotic distribution for the domination number of PE-PCD for uniform data in a triangle, tri.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
r<-1.35

Ms<-center.nondegPE(Tr,r)
Ms

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Centers of nondegeneracy\n for the PE-PCD in a triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Ms,pch=".",col=1)
polygon(Ms,lty = 2)

xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.03)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

xc<-Ms[,1]+c(-.04,.04,.03)
yc<-Ms[,2]+c(.02,.02,.05)
txt.str<-c("M1","M2","M3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
r<-1.35

Ms<-center.nondegPE(Tr,r)
Ms

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Centers of nondegeneracy\n for the PE-PCD in a triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Ms,pch=".",col=1)
polygon(Ms,lty = 2)

xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.03)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

xc<-Ms[,1]+c(-.04,.04,.03)
yc<-Ms[,2]+c(.02,.02,.05)
txt.str<-c("M1","M2","M3")
text(xc,yc,txt.str)

## End(Not run)

Parameterized center of an interval

Description

Returns the (parameterized) center, $M_c$ , of the interval, int $=(a,b)$ , parameterized by $c \in (0,1)$ so that $100c$ % of the length of interval is to the left of $M_c$ and $100(1-c)$ % of the length of the interval is to the right of $M_c$ . That is, for the interval, int $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

See also (Ceyhan (2012, 2016)).

Usage

centerMc(int, c = 0.5)
centerMc(int, c = 0.5)

Arguments

`int`	A `vector` with two entries representing an interval.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

(parameterized) center inside int

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

c<-.4
a<-0; b<-10
int = c(a,b)
centerMc(int,c)

c<-.3
a<-2; b<-4; int<-c(a,b)
centerMc(int,c)

c<-.4
a<-0; b<-10
int = c(a,b)
centerMc(int,c)

c<-.3
a<-2; b<-4; int<-c(a,b)
centerMc(int,c)

Parameterized centers of intervals

Description

Returns the centers of the intervals based on 1D points in Yp parameterized by $c \in (0,1)$ so that $100c$ % of the length of interval is to the left of $M_c$ and $100(1-c)$ % of the length of the interval is to the right of $M_c$ . That is, for an interval $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ Yp is a vector of 1D points, not necessarily sorted.

See also (Ceyhan (2012, 2016)).

Usage

centersMc(Yp, c = 0.5)
centersMc(Yp, c = 0.5)

Arguments

`Yp`	A `vector` real numbers that constitute the end points of intervals.
`c`	A positive real number in $(0,1)$ parameterizing the centers inside the intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

(parameterized) centers of the intervals based on Yp points as a vector

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
n<-10
c<-.4  #try also c<-runif(1)
Yp<-runif(n)
centersMc(Yp,c)

c<-.3  #try also c<-runif(1)
Yp<-runif(n,0,10)
centersMc(Yp,c)

## End(Not run)

## Not run: 
n<-10
c<-.4  #try also c<-runif(1)
Yp<-runif(n)
centersMc(Yp,c)

c<-.3  #try also c<-runif(1)
Yp<-runif(n,0,10)
centersMc(Yp,c)

## End(Not run)

Circumcenter of a standard basic triangle form

Description

Returns the circumcenter of a standard basic triangle form $T_b=T((0,0),(1,0),(c_1,c_2))$ given $c_1$ , $c_2$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Any given triangle can be mapped to the standard basic triangle form by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle form is useful for simulation studies under the uniformity hypothesis.

See (Weisstein (2019); Ceyhan (2010)) for triangle centers and (Ceyhan et al. (2006); Ceyhan et al. (2007); Ceyhan (2011)) for the standard basic triangle form.

Usage

circumcenter.basic.tri(c1, c2)
circumcenter.basic.tri(c1, c2)

Arguments

c1, c2

Positive real numbers representing the top vertex in standard basic triangle form $T_b=T((0,0),(1,0),(c_1,c_2))$ , $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Value

circumcenter of the standard basic triangle form $T_b=T((0,0),(1,0),(c_1,c_2))$ given $c_1$ , $c_2$ as the arguments of the function.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Weisstein EW (2019). “Triangle Centers.” From MathWorld — A Wolfram Web Resource, http://mathworld.wolfram.com/TriangleCenter.html.

Examples

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
#the vertices of the standard basic triangle form Tb
Tb<-rbind(A,B,C)
CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
CC

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1])
Ylim<-range(Tb[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

par(pty = "s")
plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(rbind(CC))
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.04,.03,.06,.06,-.03,0)
yc<-txt[,2]+c(.02,.02,.03,-.03,.02,.04,-.03)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

#for an obtuse triangle
c1<-.4; c2<-.3;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
#the vertices of the standard basic triangle form Tb
Tb<-rbind(A,B,C)
CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
CC

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1],CC[1])
Ylim<-range(Tb[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

par(pty = "s")
plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(rbind(CC))
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.03,.03,.07,.07,-.05,0)
yc<-txt[,2]+c(.02,.02,.04,-.03,.03,.04,.06)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
#the vertices of the standard basic triangle form Tb
Tb<-rbind(A,B,C)
CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
CC

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1])
Ylim<-range(Tb[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

par(pty = "s")
plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(rbind(CC))
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.04,.03,.06,.06,-.03,0)
yc<-txt[,2]+c(.02,.02,.03,-.03,.02,.04,-.03)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

#for an obtuse triangle
c1<-.4; c2<-.3;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
#the vertices of the standard basic triangle form Tb
Tb<-rbind(A,B,C)
CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
CC

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1],CC[1])
Ylim<-range(Tb[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

par(pty = "s")
plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(rbind(CC))
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.03,.03,.07,.07,-.05,0)
yc<-txt[,2]+c(.02,.02,.04,-.03,.03,.04,.06)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

Circumcenter of a general tetrahedron

Description

Returns the circumcenter a given tetrahedron th with vertices stacked row-wise.

Usage

circumcenter.tetra(th)
circumcenter.tetra(th)

Arguments

`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.

Value

circumcenter of the tetrahedron th

Author(s)

Elvan Ceyhan

Examples

## Not run: 
set.seed(123)
A<-c(0,0,0)+runif(3,-.2,.2);
B<-c(1,0,0)+runif(3,-.2,.2);
C<-c(1/2,sqrt(3)/2,0)+runif(3,-.2,.2);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3)+runif(3,-.2,.2);
tetra<-rbind(A,B,C,D)

CC<-circumcenter.tetra(tetra)
CC

Xlim<-range(tetra[,1],CC[1])
Ylim<-range(tetra[,2],CC[2])
Zlim<-range(tetra[,3],CC[3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(tetra[,1],tetra[,2],tetra[,3], phi =0,theta=40, bty = "g",
main="Illustration of the Circumcenter\n in a Tetrahedron",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(CC[1],CC[2],CC[3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CC,6),byrow = TRUE,ncol=3)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty = 2)

plot3D::text3D(CC[1],CC[2],CC[3], labels="CC", add=TRUE)

## End(Not run)

## Not run: 
set.seed(123)
A<-c(0,0,0)+runif(3,-.2,.2);
B<-c(1,0,0)+runif(3,-.2,.2);
C<-c(1/2,sqrt(3)/2,0)+runif(3,-.2,.2);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3)+runif(3,-.2,.2);
tetra<-rbind(A,B,C,D)

CC<-circumcenter.tetra(tetra)
CC

Xlim<-range(tetra[,1],CC[1])
Ylim<-range(tetra[,2],CC[2])
Zlim<-range(tetra[,3],CC[3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(tetra[,1],tetra[,2],tetra[,3], phi =0,theta=40, bty = "g",
main="Illustration of the Circumcenter\n in a Tetrahedron",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(CC[1],CC[2],CC[3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CC,6),byrow = TRUE,ncol=3)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty = 2)

plot3D::text3D(CC[1],CC[2],CC[3], labels="CC", add=TRUE)

## End(Not run)

Circumcenter of a general triangle

Description

Returns the circumcenter a given triangle, tri, with vertices stacked row-wise. See (Weisstein (2019); Ceyhan (2010)) for triangle centers.

Usage

circumcenter.tri(tri)
circumcenter.tri(tri)

Arguments

tri

A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

circumcenter of the triangle tri

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Weisstein EW (2019). “Triangle Centers.” From MathWorld — A Wolfram Web Resource, http://mathworld.wolfram.com/TriangleCenter.html.

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);  #the vertices of the triangle Tr

CC<-circumcenter.tri(Tr)  #the circumcenter
CC

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],CC[1])
Ylim<-range(Tr[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,asp=1,pch=".",xlab="",ylab="",main="Circumcenter of a triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(rbind(CC))
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.08,.08,.08,.12,-.09,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.03,-.06,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);  #the vertices of the equilateral triangle Te
circumcenter.tri(Te)  #the circumcenter

A<-c(0,0); B<-c(0,1); C<-c(2,0);
Tr<-rbind(A,B,C);  #the vertices of the triangle T
circumcenter.tri(Tr)  #the circumcenter

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);  #the vertices of the triangle Tr

CC<-circumcenter.tri(Tr)  #the circumcenter
CC

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],CC[1])
Ylim<-range(Tr[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,asp=1,pch=".",xlab="",ylab="",main="Circumcenter of a triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(rbind(CC))
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.08,.08,.08,.12,-.09,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.03,-.06,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);  #the vertices of the equilateral triangle Te
circumcenter.tri(Te)  #the circumcenter

A<-c(0,0); B<-c(0,1); C<-c(2,0);
Tr<-rbind(A,B,C);  #the vertices of the triangle T
circumcenter.tri(Tr)  #the circumcenter

## End(Not run)

The closest points to circumcenter in each $CC$ -vertex region in a triangle

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, to circumcenter, $CC$ , in each $CC$ -vertex region in the triangle tri $=T(A,B,C)=$ (vertex 1,vertex 2,vertex 3).

ch.all.intri is for checking whether all data points are inside tri (default is FALSE). If some of the data points are not inside tri and ch.all.intri=TRUE, then the function yields an error message. If some of the data points are not inside tri and ch.all.intri=FALSE, then the function yields the closest points to $CC$ among the data points in each $CC$ -vertex region of tri (yields NA if there are no data points inside tri).

See also (Ceyhan (2005, 2012)).

Usage

cl2CCvert.reg(Xp, tri, ch.all.intri = FALSE)
cl2CCvert.reg(Xp, tri, ch.all.intri = FALSE)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`ch.all.intri`	A logical argument (default=`FALSE`) to check whether all data points are inside the triangle `tri`. So, if it is `TRUE`, the function checks if all data points are inside the closure of the triangle (i.e., interior and boundary combined) else it does not.

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances from closest points to CC ..."`
`type`	Type of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, closest points to $CC$ in each $CC$ -vertex region
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is `tri`
`cent`	The center point used for construction of vertex regions
`ncent`	Name of the center, `cent`, it is `"CC"` for this function
`regions`	Vertex regions inside the triangle, `tri`, provided as a `list`
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside `tri`
`dist2ref`	Distances from closest points in each $CC$ -vertex region to CC.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-cl2CCvert.reg(Xp,Tr)
Ext
summary(Ext)
plot(Ext)

c2CC<-Ext

CC<-circumcenter.tri(Tr)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main="Closest Points in CC-Vertex Regions \n to the Circumcenter",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(c2CC$ext,pch=4,col=2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.07,.08,.06,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.03,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

Xp2<-rbind(Xp,c(.2,.4))
cl2CCvert.reg(Xp2,Tr,ch.all.intri = FALSE)
#gives an error message if ch.all.intri = TRUE since not all points are in the triangle

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-cl2CCvert.reg(Xp,Tr)
Ext
summary(Ext)
plot(Ext)

c2CC<-Ext

CC<-circumcenter.tri(Tr)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main="Closest Points in CC-Vertex Regions \n to the Circumcenter",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(c2CC$ext,pch=4,col=2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.07,.08,.06,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.03,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

Xp2<-rbind(Xp,c(.2,.4))
cl2CCvert.reg(Xp2,Tr,ch.all.intri = FALSE)
#gives an error message if ch.all.intri = TRUE since not all points are in the triangle

## End(Not run)

The closest points to circumcenter in each $CC$ -vertex region in a standard basic triangle

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, to circumcenter, $CC$ , in each $CC$ -vertex region in the standard basic triangle $T_b = T(A=(0,0),B=(1,0),C=(c_1,c_2))=$ (vertex 1,vertex 2,vertex 3). ch.all.intri is for checking whether all data points are inside $T_b$ (default is FALSE).

See also (Ceyhan (2005, 2012)).

Usage

cl2CCvert.reg.basic.tri(Xp, c1, c2, ch.all.intri = FALSE)
cl2CCvert.reg.basic.tri(Xp, c1, c2, ch.all.intri = FALSE)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle. adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$
`ch.all.intri`	A logical argument for checking whether all data points are inside $T_b$ (default is `FALSE`).

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances from closest points to ..."`.
`type`	Type of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, closest points to $CC$ in each vertex region.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is $T_b$ .
`cent`	The center point used for construction of vertex regions
`ncent`	Name of the center, `cent`, it is `"CC"` for this function.
`regions`	Vertex regions inside the triangle, $T_b$ , provided as a `list`.
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside $T_b$ .
`dist2ref`	Distances from closest points in each vertex region to CC.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-15

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

Ext<-cl2CCvert.reg.basic.tri(Xp,c1,c2)
Ext
summary(Ext)
plot(Ext)

c2CC<-Ext

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main="Closest Points in CC-Vertex Regions \n to the Circumcenter",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(c2CC$ext,pch=4,col=2)

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,.02,.07,.06,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,-.01,.03,.03,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

Xp2<-rbind(Xp,c(.2,.4))
cl2CCvert.reg.basic.tri(Xp2,c1,c2,ch.all.intri = FALSE)
#gives an error message if ch.all.intri = TRUE
#since not all points are in the standard basic triangle

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-15

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

Ext<-cl2CCvert.reg.basic.tri(Xp,c1,c2)
Ext
summary(Ext)
plot(Ext)

c2CC<-Ext

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main="Closest Points in CC-Vertex Regions \n to the Circumcenter",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(c2CC$ext,pch=4,col=2)

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,.02,.07,.06,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,-.01,.03,.03,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

Xp2<-rbind(Xp,c(.2,.4))
cl2CCvert.reg.basic.tri(Xp2,c1,c2,ch.all.intri = FALSE)
#gives an error message if ch.all.intri = TRUE
#since not all points are in the standard basic triangle

## End(Not run)

The closest points in a data set to edges in the standard equilateral triangle

Description

An object of class "Extrema". Returns the closest points from the 2D data set, Xp, to the edges in the standard equilateral triangle $T_e=T(A=(0,0),B=(1,0),C=(1/2,\sqrt{3}/2))$ .

ch.all.intri is for checking whether all data points are inside $T_e$ (default is FALSE).

If some of the data points are not inside $T_e$ and ch.all.intri=TRUE, then the function yields an error message. If some of the data points are not inside $T_e$ and ch.all.intri=FALSE, then the function yields the closest points to edges among the data points inside $T_e$ (yields NA if there are no data points inside $T_e$ ).

See also (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan and Priebe (2007)).

Usage

cl2edges.std.tri(Xp, ch.all.intri = FALSE)
cl2edges.std.tri(Xp, ch.all.intri = FALSE)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`ch.all.intri`	A logical argument (default=`FALSE`) to check whether all data points are inside the standard equilateral triangle $T_e$ . So, if it is `TRUE`, the function checks if all data points are inside the closure of the triangle (i.e., interior and boundary combined) else it does not.

Value

A list with the elements

`txt1`	Edge labels as $AB=3$ , $BC=1$ , and $AC=2$ for $T_e$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances to Edges ..."`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, i.e., closest points to edges
`X`	The input data, `Xp`, which can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, i.e., the standard equilateral triangle $T_e$
`cent`	The center point used for construction of edge regions, not required for this extrema, hence it is `NULL` for this function
`ncent`	Name of the center, `cent`, not required for this extrema, hence it is `NULL` for this function
`regions`	Edge regions inside the triangle, $T_e$ , not required for this extrema, hence it is `NULL` for this function
`region.names`	Names of the edge regions, not required for this extrema, hence it is `NULL` for this function
`region.centers`	Centers of mass of the edge regions inside $T_e$ , not required for this extrema, hence it is `NULL` for this function
`dist2ref`	Distances from closest points in each edge region to the corresponding edge

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
n<-20  #try also n<-100
Xp<-runif.std.tri(n)$gen.points

Ext<-cl2edges.std.tri(Xp)
Ext
summary(Ext)
plot(Ext,asp=1)

ed.clo<-Ext

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
CM<-(A+B+C)/3
p1<-(A+B)/2
p2<-(B+C)/2
p3<-(A+C)/2

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,xlab="",ylab="")
points(ed.clo$ext,pty=2,pch=4,col="red")

txt<-rbind(Te,p1,p2,p3)
xc<-txt[,1]+c(-.03,.03,.03,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,0,0,0)
txt.str<-c("A","B","C","re=1","re=2","re=3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
n<-20  #try also n<-100
Xp<-runif.std.tri(n)$gen.points

Ext<-cl2edges.std.tri(Xp)
Ext
summary(Ext)
plot(Ext,asp=1)

ed.clo<-Ext

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
CM<-(A+B+C)/3
p1<-(A+B)/2
p2<-(B+C)/2
p3<-(A+C)/2

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,xlab="",ylab="")
points(ed.clo$ext,pty=2,pch=4,col="red")

txt<-rbind(Te,p1,p2,p3)
xc<-txt[,1]+c(-.03,.03,.03,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,0,0,0)
txt.str<-c("A","B","C","re=1","re=2","re=3")
text(xc,yc,txt.str)

## End(Not run)

The closest points among a data set in the vertex regions to the corresponding edges in a standard basic triangle

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, to edge $i$ in M-vertex region $i$ for $i=1,2,3$ in the standard basic triangle $T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ . Vertex labels are $A=1$ , $B=2$ , and $C=3$ , and corresponding edge labels are $BC=1$ , $AC=2$ , and $AB=3$ .

Vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the standard basic triangle $T_b$ or based on the circumcenter of $T_b$ .

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010)).

Usage

cl2edges.vert.reg.basic.tri(Xp, c1, c2, M)
cl2edges.vert.reg.basic.tri(Xp, c1, c2, M)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle $T_b$ or the circumcenter of $T_b$ .

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances to Edges in the Respective \eqn{M}-Vertex Regions"`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, closest points to edges in the corresponding vertex region.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is $T_b$ .
`cent`	The center point used for construction of vertex regions
`ncent`	Name of the center, `cent`, it is `"M"` or `"CC"` for this function
`regions`	Vertex regions inside the triangle, $T_b$ .
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside $T_b$ .
`dist2ref`	Distances of closest points in the vertex regions to corresponding edges.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4;  c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

set.seed(1)
n<-20
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)

Ext<-cl2edges.vert.reg.basic.tri(Xp,c1,c2,M)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

Ds<-prj.cent2edges.basic.tri(c1,c2,M)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,pch=".",xlab="",ylab="",
main="Closest Points in M-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(Xp,pch=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

xc<-Tb[,1]+c(-.02,.02,0.02)
yc<-Tb[,2]+c(.02,.02,.02)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.03,0)
yc<-txt[,2]+c(-.02,.02,.02,-.03)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4;  c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

set.seed(1)
n<-20
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)

Ext<-cl2edges.vert.reg.basic.tri(Xp,c1,c2,M)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

Ds<-prj.cent2edges.basic.tri(c1,c2,M)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,pch=".",xlab="",ylab="",
main="Closest Points in M-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(Xp,pch=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

xc<-Tb[,1]+c(-.02,.02,0.02)
yc<-Tb[,2]+c(.02,.02,.02)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.03,0)
yc<-txt[,2]+c(-.02,.02,.02,-.03)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The closest points in a data set to edges in each $CC$ -vertex region in a triangle

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, to edge $j$ in $CC$ -vertex region $j$ for $j=1,2,3$ in the triangle, tri $=T(A,B,C)$ , where $CC$ stands for circumcenter. Vertex labels are $A=1$ , $B=2$ , and $C=3$ , and corresponding edge labels are $BC=1$ , $AC=2$ , and $AB=3$ . Function yields NA if there are no data points in a $CC$ -vertex region.

See also (Ceyhan (2005, 2010)).

Usage

cl2edgesCCvert.reg(Xp, tri)
cl2edgesCCvert.reg(Xp, tri)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances to Edges in the Respective CC-Vertex Regions"`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, closest points to edges in the respective vertex region.
`ind.ext`	Indices of the extrema points,`ext`.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is `tri`
`cent`	The center point used for construction of vertex regions
`ncent`	Name of the center, `cent`, it is `"CC"` for this function
`regions`	Vertex regions inside the triangle, `tri`, provided as a `list`
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside `tri`
`dist2ref`	Distances of closest points in the vertex regions to corresponding edges

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-20  #try also n<-100
set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-cl2edgesCCvert.reg(Xp,Tr)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

CC<-circumcenter.tri(Tr);
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1],CC[1])
Ylim<-range(Tr[,2],Xp[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,asp=1,pch=".",xlab="",ylab="",
main="Closest Points in CC-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)

xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

points(Xp,pch=1,col=1)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

txt<-rbind(CC,Ds)
xc<-txt[,1]+c(-.04,.04,-.03,0)
yc<-txt[,2]+c(-.05,.04,.06,-.08)
txt.str<-c("CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-20  #try also n<-100
set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-cl2edgesCCvert.reg(Xp,Tr)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

CC<-circumcenter.tri(Tr);
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1],CC[1])
Ylim<-range(Tr[,2],Xp[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,asp=1,pch=".",xlab="",ylab="",
main="Closest Points in CC-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)

xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

points(Xp,pch=1,col=1)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

txt<-rbind(CC,Ds)
xc<-txt[,1]+c(-.04,.04,-.03,0)
yc<-txt[,2]+c(-.05,.04,.06,-.08)
txt.str<-c("CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The closest points in a data set to edges in each $CM$ -vertex region in a triangle

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, to edge $j$ in $CM$ -vertex region $j$ for $j=1,2,3$ in the triangle, tri $=T(A,B,C)$ , where $CM$ stands for center of mass. Vertex labels are $A=1$ , $B=2$ , and $C=3$ , and corresponding edge labels are $BC=1$ , $AC=2$ , and $AB=3$ . Function yields NA if there are no data points in a $CM$ -vertex region.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2010, 2011)).

Usage

cl2edgesCMvert.reg(Xp, tri)
cl2edgesCMvert.reg(Xp, tri)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances to Edges in the Respective CM-Vertex Regions"`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, closest points to edges in the respective vertex region.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is `tri`
`cent`	The center point used for construction of vertex regions
`ncent`	Name of the center, `cent`, it is `"CM"` for this function
`regions`	Vertex regions inside the triangle, `tri`, provided as a `list`
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside `tri`
`dist2ref`	Distances of closest points in the vertex regions to corresponding edges

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-20  #try also n<-100
set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-cl2edgesCMvert.reg(Xp,Tr)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

CM<-(A+B+C)/3;
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Closest Points in CM-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)

xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

points(Xp,pch=1,col=1)
L<-matrix(rep(CM,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

txt<-rbind(CM,Ds)
xc<-txt[,1]+c(-.04,.04,-.03,0)
yc<-txt[,2]+c(-.05,.04,.06,-.08)
txt.str<-c("CM","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-20  #try also n<-100
set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-cl2edgesCMvert.reg(Xp,Tr)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

CM<-(A+B+C)/3;
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Closest Points in CM-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)

xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

points(Xp,pch=1,col=1)
L<-matrix(rep(CM,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

txt<-rbind(CM,Ds)
xc<-txt[,1]+c(-.04,.04,-.03,0)
yc<-txt[,2]+c(-.05,.04,.06,-.08)
txt.str<-c("CM","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The closest points among a data set in the vertex regions to the respective edges in a triangle

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, to edge $i$ in M-vertex region $i$ for $i=1,2,3$ in the triangle tri $=T(A,B,C)$ . Vertex labels are $A=1$ , $B=2$ , and $C=3$ , and corresponding edge labels are $BC=1$ , $AC=2$ , and $AB=3$ .

Two methods of finding these extrema are provided in the function, which can be chosen in the logical argument alt, whose default is alt=FALSE. When alt=FALSE, the function sequentially finds the vertex region of the data point and then updates the minimum distance to the opposite edge and the relevant extrema objects, and when alt=TRUE, it first partitions the data set according which vertex regions they reside, and then finds the minimum distance to the opposite edge and the relevant extrema on each partition. Both options yield equivalent results for the extrema points and indices, with the default being slightly ~ 20

See also (Ceyhan (2005, 2010)).

Usage

cl2edgesMvert.reg(Xp, tri, M, alt = FALSE)
cl2edgesMvert.reg(Xp, tri, M, alt = FALSE)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri`; which may be entered as "CC" as well;
`alt`	A logical argument for alternative method of finding the closest points to the edges, default `alt=FALSE`. When `alt=FALSE`, the function sequentially finds the vertex region of the data point and then the minimum distance to the opposite edge and the relevant extrema objects, and when `alt=TRUE`, it first partitions the data set according which vertex regions they reside, and then finds the minimum distance to the opposite edge and the relevant extrema on each partition.

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances to Edges in the Respective \eqn{M}-Vertex Regions"`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, closest points to edges in the respective vertex region.
`ind.ext`	The data indices of extrema points, `ext`.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is `tri`
`cent`	The center point used for construction of vertex regions
`ncent`	Name of the center, `cent`, it is `"M"` or `"CC"` for this function
`regions`	Vertex regions inside the triangle, `tri`, provided as a `list`
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside `tri`
`dist2ref`	Distances of closest points in the `M`-vertex regions to corresponding edges.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-20  #try also n<-100

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Ext<-cl2edgesMvert.reg(Xp,Tr,M)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

Ds<-prj.cent2edges(Tr,M)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Closest Points in M-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

xc<-Tr[,1]+c(-.02,.03,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.05,-.02,-.01)
yc<-txt[,2]+c(-.03,.02,.08,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-20  #try also n<-100

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Ext<-cl2edgesMvert.reg(Xp,Tr,M)
Ext
summary(Ext)
plot(Ext)

cl2e<-Ext

Ds<-prj.cent2edges(Tr,M)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Closest Points in M-Vertex Regions \n to the Opposite Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(cl2e$ext,pch=3,col=2)

xc<-Tr[,1]+c(-.02,.03,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.05,-.02,-.01)
yc<-txt[,2]+c(-.03,.02,.08,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The closest points among a data set in the vertex regions to the respective faces in a tetrahedron

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, to face $i$ in M-vertex region $i$ for $i=1,2,3,4$ in the tetrahedron $th=T(A,B,C,D)$ . Vertex labels are $A=1$ , $B=2$ , $C=3$ , and $D=4$ and corresponding face labels are $BCD=1$ , $ACD=2$ , $ABD=3$ , and $ABC=4$ .

Vertex regions are based on center M which can be the center of mass ("CM") or circumcenter ("CC") of th.

Usage

cl2faces.vert.reg.tetra(Xp, th, M = "CM")
cl2faces.vert.reg.tetra(Xp, th, M = "CM")

Arguments

`Xp`	A set of 3D points representing the set of data points.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , $C=3$ , and $D=4$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances from Closest Points to Faces ..."`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, closest points to faces in the respective vertex region.
`ind.ext`	The data indices of extrema points, `ext`.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is `th`
`cent`	The center point used for construction of vertex regions, it is circumcenter of center of mass for this function
`ncent`	Name of the center, it is circumcenter `"CC"` or center of mass `"CM"` for this function.
`regions`	Vertex regions inside the tetrahedron `th` provided as a list.
`region.names`	Names of the vertex regions as `"vr=1","vr=2","vr=3","vr=4"`
`region.centers`	Centers of mass of the vertex regions inside `th`.
`dist2ref`	Distances from closest points in each vertex region to the corresponding face.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
set.seed(1)
tetra<-rbind(A,B,C,D)+matrix(runif(12,-.25,.25),ncol=3)
n<-10  #try also n<-20
Cent<-"CC"  #try also "CM"

n<-20  #try also n<-100
Xp<-runif.tetra(n,tetra)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))

Ext<-cl2faces.vert.reg.tetra(Xp,tetra,Cent)
Ext
summary(Ext)
plot(Ext)

clf<-Ext$ext

if (Cent=="CC") {M<-circumcenter.tetra(tetra)}
if (Cent=="CM") {M<-apply(tetra,2,mean)}

Xlim<-range(tetra[,1],Xp[,1],M[1])
Ylim<-range(tetra[,2],Xp[,2],M[2])
Zlim<-range(tetra[,3],Xp[,3],M[3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi =0,theta=40, bty = "g",
main="Closest Pointsin CC-Vertex Regions \n to the Opposite Faces",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
plot3D::points3D(clf[,1],clf[,2],clf[,3], pch=4,col="red", add=TRUE)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

#for center of mass use #Cent<-apply(tetra,2,mean)
D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2;
D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-rbind(M,M,M,M,M,M)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
set.seed(1)
tetra<-rbind(A,B,C,D)+matrix(runif(12,-.25,.25),ncol=3)
n<-10  #try also n<-20
Cent<-"CC"  #try also "CM"

n<-20  #try also n<-100
Xp<-runif.tetra(n,tetra)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))

Ext<-cl2faces.vert.reg.tetra(Xp,tetra,Cent)
Ext
summary(Ext)
plot(Ext)

clf<-Ext$ext

if (Cent=="CC") {M<-circumcenter.tetra(tetra)}
if (Cent=="CM") {M<-apply(tetra,2,mean)}

Xlim<-range(tetra[,1],Xp[,1],M[1])
Ylim<-range(tetra[,2],Xp[,2],M[2])
Zlim<-range(tetra[,3],Xp[,3],M[3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi =0,theta=40, bty = "g",
main="Closest Pointsin CC-Vertex Regions \n to the Opposite Faces",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
plot3D::points3D(clf[,1],clf[,2],clf[,3], pch=4,col="red", add=TRUE)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

#for center of mass use #Cent<-apply(tetra,2,mean)
D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2;
D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-rbind(M,M,M,M,M,M)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2)

## End(Not run)

The closest points to center in each vertex region in an interval

Description

An object of class "Extrema". Returns the closest data points among the data set, Xp, in each $M_c$ -vertex region i.e., finds the closest points from right and left to $M_c$ among points of the 1D data set Xp which reside in in the interval int $=(a,b)$ .

$M_c$ is based on the centrality parameter $c \in (0,1)$ , so that $100c$ % of the length of interval is to the left of $M_c$ and $100(1-c)$ % of the length of the interval is to the right of $M_c$ . That is, for the interval $(a,b)$ , $M_c=a+c(b-a)$ . If there are no points from Xp to the left of $M_c$ in the interval, then it yields NA, and likewise for the right of $M_c$ in the interval.

See also (Ceyhan (2012)).

Usage

cl2Mc.int(Xp, int, c)
cl2Mc.int(Xp, int, c)

Arguments

`Xp`	A set or `vector` of 1D points from which closest points to $M_c$ are found in the interval `int`.
`int`	A `vector` of two real numbers representing an interval.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ . For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`txt1`	Vertex Labels are $a=1$ and $b=2$ for the interval $(a,b)$ .
`txt2`	A short description of the distances as `"Distances from ..."`
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, closest points to $M_c$ in each vertex region
`ind.ext`	The data indices of extrema points, `ext`.
`X`	The input data vector, `Xp`.
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is `int`.
`cent`	The (parameterized) center point used for construction of vertex regions.
`ncent`	Name of the (parameterized) center, `cent`, it is `"Mc"` for this function.
`regions`	Vertex regions inside the interval, `int`, provided as a list.
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`
`region.centers`	Centers of mass of the vertex regions inside `int`.
`dist2ref`	Distances from closest points in each vertex region to $M_c$ .

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
c<-.4
a<-0; b<-10; int<-c(a,b)

Mc<-centerMc(int,c)

nx<-10
xr<-range(a,b,Mc)
xf<-(xr[2]-xr[1])*.5

Xp<-runif(nx,a,b)

Ext<-cl2Mc.int(Xp,int,c)
Ext
summary(Ext)
plot(Ext)

cMc<-Ext

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",
main=paste("Closest Points in Mc-Vertex Regions \n to the Center Mc = ",Mc,sep=""),
  xlim=Xlim+xd*c(-.05,.05))
  abline(h=0)
abline(v=c(a,b,Mc),col=c(1,1,2),lty=2)
points(cbind(Xp,0))
points(cbind(c(cMc$ext),0),pch=4,col=2)
text(cbind(c(a,b,Mc)-.02*xd,-0.05),c("a","b",expression(M[c])))

## End(Not run)

## Not run: 
c<-.4
a<-0; b<-10; int<-c(a,b)

Mc<-centerMc(int,c)

nx<-10
xr<-range(a,b,Mc)
xf<-(xr[2]-xr[1])*.5

Xp<-runif(nx,a,b)

Ext<-cl2Mc.int(Xp,int,c)
Ext
summary(Ext)
plot(Ext)

cMc<-Ext

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",
main=paste("Closest Points in Mc-Vertex Regions \n to the Center Mc = ",Mc,sep=""),
  xlim=Xlim+xd*c(-.05,.05))
  abline(h=0)
abline(v=c(a,b,Mc),col=c(1,1,2),lty=2)
points(cbind(Xp,0))
points(cbind(c(cMc$ext),0),pch=4,col=2)
text(cbind(c(a,b,Mc)-.02*xd,-0.05),c("a","b",expression(M[c])))

## End(Not run)

A test of segregation/association based on arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D data

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of Xp points in the convex hull of Yp points against the alternatives of segregation (where Xp points cluster away from Yp points) and association (where Xp points cluster around Yp points) based on the normal approximation of the arc density of the CS-PCD for uniform 2D data in the convex hull of Yp points.

The function yields the test statistic, $p$ -value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is the arc density), and method and name of the data set used.

Under the null hypothesis of uniformity of Xp points in the convex hull of Yp points, arc density of CS-PCD whose vertices are Xp points equals to its expected value under the uniform distribution and alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association) or right-sided (i.e., data is accumulated around the centers of the triangles, or segregation).

CS proximity region is constructed with the expansion parameter $t>0$ and $CM$ -edge regions (i.e., the test is not available for a general center $M$ at this version of the function).

**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are assumed to be fixed (i.e., the test is conditional on Yp points). Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points, say at least 5 times more. This test is more appropriate when supports of Xp and Yp has a substantial overlap. Currently, the Xp points outside the convex hull of Yp points are handled with a convex hull correction factor (see the description below and the function code.) However, in the special case of no Xp points in the convex hull of Yp points, arc density is taken to be 1, as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is an ongoing line of research of the author of the package.

ch.cor is for convex hull correction (default is "no convex hull correction", i.e., ch.cor=FALSE) which is recommended when both Xp and Yp have the same rectangular support.

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

CSarc.dens.test(
  Xp,
  Yp,
  t,
  ch.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
CSarc.dens.test(
  Xp,
  Yp,
  t,
  ch.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`ch.cor`	A logical argument for convex hull correction, default `ch.cor=FALSE`, recommended when both `Xp` and `Yp` have the same rectangular support.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the arc density of CS-PCD based on the 2D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`
`conf.int`	Confidence interval for the arc density at the given confidence level `conf.level` and depends on the type of `alternative`.
`estimate`	Estimate of the parameter, i.e., arc density
`null.value`	Hypothesized value for the parameter, i.e., the null arc density, which is usually the mean arc density under uniform distribution.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab = "")

CSarc.dens.test(Xp,Yp,t=.5)
CSarc.dens.test(Xp,Yp,t=.5,ch=TRUE)
#try also t=1.0 and 1.5 above

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab = "")

CSarc.dens.test(Xp,Yp,t=.5)
CSarc.dens.test(Xp,Yp,t=.5,ch=TRUE)
#try also t=1.0 and 1.5 above

## End(Not run)

A test of uniformity of 1D data in a given interval based on Central Similarity Proximity Catch Digraph (CS-PCD)

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of uniformity of 1D data in one interval based on the normal approximation of the arc density of the CS-PCD with expansion parameter $t>0$ and centrality parameter $c \in (0,1)$ .

The null hypothesis is that data is uniform in a finite interval (i.e., arc density of CS-PCD equals to its expected value under uniform distribution) and alternative could be two-sided, or left-sided (i.e., data is accumulated around the end points) or right-sided (i.e., data is accumulated around the mid point or center $M_c$ ).

See also (Ceyhan (2016)).

Usage

CSarc.dens.test.int(
  Xp,
  int,
  t,
  c = 0.5,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
CSarc.dens.test.int(
  Xp,
  int,
  t,
  c = 0.5,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of CS-PCD.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the arc density of CS-PCD based on the 1D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`
`conf.int`	Confidence interval for the arc density at the given level `conf.level` and depends on the type of `alternative`.
`estimate`	Estimate of the parameter, i.e., arc density
`null.value`	Hypothesized value for the parameter, i.e., the null arc density, which is usually the mean arc density under uniform distribution.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples


c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

n<-10
Xp<-runif(n,a,b)

num.arcsCSmid.int(Xp,int,t,c)
CSarc.dens.test.int(Xp,int,t,c)

num.arcsCSmid.int(Xp,int,t,c=.3)
CSarc.dens.test.int(Xp,int,t,c=.3)

num.arcsCSmid.int(Xp,int,t=1.5,c)
CSarc.dens.test.int(Xp,int,t=1.5,c)

Xp<-runif(n,a-1,b+1)
num.arcsCSmid.int(Xp,int,t,c)
CSarc.dens.test.int(Xp,int,t,c)

c<-.4
t<-.5
a<-0; b<-10; int<-c(a,b)
n<-10  #try also n<-20
Xp<-runif(n,a,b)

CSarc.dens.test.int(Xp,int,t,c)


c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

n<-10
Xp<-runif(n,a,b)

num.arcsCSmid.int(Xp,int,t,c)
CSarc.dens.test.int(Xp,int,t,c)

num.arcsCSmid.int(Xp,int,t,c=.3)
CSarc.dens.test.int(Xp,int,t,c=.3)

num.arcsCSmid.int(Xp,int,t=1.5,c)
CSarc.dens.test.int(Xp,int,t=1.5,c)

Xp<-runif(n,a-1,b+1)
num.arcsCSmid.int(Xp,int,t,c)
CSarc.dens.test.int(Xp,int,t,c)

c<-.4
t<-.5
a<-0; b<-10; int<-c(a,b)
n<-10  #try also n<-20
Xp<-runif(n,a,b)

CSarc.dens.test.int(Xp,int,t,c)

A test of segregation/association based on arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of Xp points in the range (i.e., range) of Yp points against the alternatives of segregation (where Xp points cluster away from Yp points) and association (where Xp points cluster around Yp points) based on the normal approximation of the arc density of the CS-PCD for uniform 1D data.

Under the null hypothesis of uniformity of Xp points in the range of Yp points, arc density of CS-PCD whose vertices are Xp points equals to its expected value under the uniform distribution and alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association) or right-sided (i.e., data is accumulated around the centers of the intervals, or segregation).

CS proximity region is constructed with the expansion parameter $t > 0$ and centrality parameter c which yields $M$ -vertex regions. More precisely, for a middle interval $(y_{(i)},y_{(i+1)})$ , the center is $M=y_{(i)}+c(y_{(i+1)}-y_{(i)})$ for the centrality parameter $c \in (0,1)$ . This test is more appropriate when supports of Xp and Yp has a substantial overlap.

end.int.cor is for end interval correction, (default is "no end interval correction", i.e., end.int.cor=FALSE), recommended when both Xp and Yp have the same interval support.

Usage

CSarc.dens.test1D(
  Xp,
  Yp,
  t,
  c = 0.5,
  support.int = NULL,
  end.int.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
CSarc.dens.test1D(
  Xp,
  Yp,
  t,
  c = 0.5,
  support.int = NULL,
  end.int.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set of 1D points which constitute the vertices of the CS-PCD.
`Yp`	A set of 1D points which constitute the end points of the partition intervals.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number which serves as the centrality parameter in CS proximity region; must be in $(0,1)$ (default `c=.5`).
`support.int`	Support interval $(a,b)$ with $a<b$ . Uniformity of `Xp` points in this interval is tested. Default is `NULL`.
`end.int.cor`	A logical argument for end interval correction, default is `FALSE`, recommended when both `Xp` and `Yp` have the same interval support.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the arc density CS-PCD whose vertices are the 1D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`.
`conf.int`	Confidence interval for the arc density at the given confidence level `conf.level` and depends on the type of `alternative`.
`estimate`	Estimate of the parameter, i.e., arc density
`null.value`	Hypothesized value for the parameter, i.e., the null arc density, which is usually the mean arc density under uniform distribution.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

There are no references for Rd macro ⁠\insertAllCites⁠ on this help page.

Examples

tau<-2
c<-.4
a<-0; b<-10; int=c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

CSarc.dens.test1D(Xp,Yp,tau,c,int)
CSarc.dens.test1D(Xp,Yp,tau,c,int,alt="l")
CSarc.dens.test1D(Xp,Yp,tau,c,int,alt="g")

CSarc.dens.test1D(Xp,Yp,tau,c,int,end.int.cor = TRUE)

Yp2<-runif(ny,a,b)+11
CSarc.dens.test1D(Xp,Yp2,tau,c,int)

n<-10  #try also n<-20
Xp<-runif(n,a,b)
CSarc.dens.test1D(Xp,Yp,tau,c,int)

tau<-2
c<-.4
a<-0; b<-10; int=c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

CSarc.dens.test1D(Xp,Yp,tau,c,int)
CSarc.dens.test1D(Xp,Yp,tau,c,int,alt="l")
CSarc.dens.test1D(Xp,Yp,tau,c,int,alt="g")

CSarc.dens.test1D(Xp,Yp,tau,c,int,end.int.cor = TRUE)

Yp2<-runif(ny,a,b)+11
CSarc.dens.test1D(Xp,Yp2,tau,c,int)

n<-10  #try also n<-20
Xp<-runif(n,a,b)
CSarc.dens.test1D(Xp,Yp,tau,c,int)

Arc density of Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case

Description

Returns the arc density of CS-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

CS proximity regions is defined with respect to tri with expansion parameter $t>0$ and edge regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is $M=(1,1,1)$ i.e., the center of mass of tri. The function also provides arc density standardized by the mean and asymptotic variance of the arc density of CS-PCD for uniform data in the triangle tri only when M is the center of mass. For the number of arcs, loops are not allowed.

See (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)) for more on CS-PCDs.

Usage

CSarc.dens.tri(Xp, tri, t, M = c(1, 1, 1), tri.cor = FALSE)
CSarc.dens.tri(Xp, tri, t, M = c(1, 1, 1), tri.cor = FALSE)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`.
`tri.cor`	A logical argument for computing the arc density for only the points inside the triangle, `tri` (default is `tri.cor=FALSE`), i.e., if `tri.cor=TRUE` only the induced digraph with the vertices inside `tri` are considered in the computation of arc density.

Value

A list with the elements

`arc.dens`	Arc density of CS-PCD whose vertices are the 2D numerical data set, `Xp`; CS proximity regions are defined with respect to the triangle `tri` and `M`-edge regions
`std.arc.dens`	Arc density standardized by the mean and asymptotic variance of the arc density of CS-PCD for uniform data in the triangle `tri`.This will only be returned if `M` is the center of mass.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

CSarc.dens.tri(Xp,Tr,t=.5,M)
CSarc.dens.tri(Xp,Tr,t=.5,M,tri.cor = FALSE)
#try also t=1 and t=1.5 above

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

CSarc.dens.tri(Xp,Tr,t=.5,M)
CSarc.dens.tri(Xp,Tr,t=.5,M,tri.cor = FALSE)
#try also t=1 and t=1.5 above

## End(Not run)

The dimension of a `vector` or matrix or a data frame

Description

Returns the dimension (i.e., number of columns) of x, which is a matrix or a vector or a data frame. This is different than the dim function in base R, in the sense that, dimension gives only the number of columns of the argument x, while dim gives the number of rows and columns of x. dimension also works for a scalar or a vector, while dim yields NULL for such arguments.

Usage

dimension(x)
dimension(x)

Arguments

`x`	A `vector` or a matrix or a data frame whose dimension is to be determined.

Value

Dimension (i.e., number of columns) of x

Author(s)

Elvan Ceyhan

Examples

## Not run: 
dimension(3)
dim(3)

A<-c(1,2)
dimension(A)
dim(A)

B<-c(2,3)
dimension(rbind(A,B,A))
dimension(cbind(A,B,A))

M<-matrix(runif(20),ncol=5)
dimension(M)
dim(M)

dimension(c("a","b"))

## End(Not run)

## Not run: 
dimension(3)
dim(3)

A<-c(1,2)
dimension(A)
dim(A)

B<-c(2,3)
dimension(rbind(A,B,A))
dimension(cbind(A,B,A))

M<-matrix(runif(20),ncol=5)
dimension(M)
dim(M)

dimension(c("a","b"))

## End(Not run)

The distance between two vectors, matrices, or data frames

Description

Returns the Euclidean distance between x and y which can be vectors or matrices or data frames of any dimension (x and y should be of same dimension).

This function is different from the dist function in the stats package of the standard R distribution. dist requires its argument to be a data matrix and dist computes and returns the distance matrix computed by using the specified distance measure to compute the distances between the rows of a data matrix (Becker et al. (1988)), while Dist needs two arguments to find the distances between. For two data matrices $A$ and $B$ , dist(rbind(as.vector(A), as.vector(B))) and Dist(A,B) yield the same result.

Usage

Dist(x, y)
Dist(x, y)

Arguments

x, y

Vectors, matrices or data frames (both should be of the same type).

Value

Euclidean distance between x and y

Author(s)

Elvan Ceyhan

References

Becker RA, Chambers JM, Wilks AR (1988). The New S Language. Wadsworth & Brooks/Cole.

Examples

## Not run: 
B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Dist(B,C);
dist(rbind(B,C))

x<-runif(10)
y<-runif(10)
Dist(x,y)

xm<-matrix(x,ncol=2)
ym<-matrix(y,ncol=2)
Dist(xm,ym)
dist(rbind(as.vector(xm),as.vector(ym)))

Dist(xm,xm)

## End(Not run)

## Not run: 
B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Dist(B,C);
dist(rbind(B,C))

x<-runif(10)
y<-runif(10)
Dist(x,y)

xm<-matrix(x,ncol=2)
ym<-matrix(y,ncol=2)
Dist(xm,ym)
dist(rbind(as.vector(xm),as.vector(ym)))

Dist(xm,xm)

## End(Not run)

The distance from a point to a line defined by two points

Description

Returns the distance from a point p to the line joining points a and b in 2D space.

Usage

dist.point2line(p, a, b)
dist.point2line(p, a, b)

Arguments

`p`	A 2D point, distance from `p` to the line passing through points `a` and `b` are to be computed.
`a`, `b`	2D points that determine the straight line (i.e., through which the straight line passes).

Value

A list with two elements

`dis`	Distance from point `p` to the line passing through `a` and `b`
`cl2p`	The closest point on the line passing through `a` and `b` to the point `p`

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1,2); B<-c(2,3); P<-c(3,1.5)

dpl<-dist.point2line(P,A,B);
dpl
C<-dpl$cl2p
pts<-rbind(A,B,C,P)

xr<-range(pts[,1])
xf<-(xr[2]-xr[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
lnAB<-Line(A,B,x)
y<-lnAB$y
int<-lnAB$intercept #intercept
sl<-lnAB$slope #slope

xsq<-seq(min(A[1],B[1],P[1])-xf,max(A[1],B[1],P[1])+xf,l=5)
#try also l=10, 20, or 100
pline<-(-1/sl)*(xsq-P[1])+P[2]
#line passing thru P and perpendicular to AB

Xlim<-range(pts[,1],x)
Ylim<-range(pts[,2],y)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(rbind(P),asp=1,pch=1,xlab="x",ylab="y",
main="Illustration of the distance from P \n to the Line Crossing Points A and B",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(rbind(A,B),pch=1)
lines(x,y,lty=1,xlim=Xlim,ylim=Ylim)
int<-round(int,2); sl<-round(sl,2)
text(rbind((A+B)/2+xd*c(-.01,-.01)),ifelse(sl==0,paste("y=",int),
ifelse(sl==1,paste("y=x+",int),
ifelse(int==0,paste("y=",sl,"x"),paste("y=",sl,"x+",int)))))
text(rbind(A+xd*c(0,-.01),B+xd*c(.0,-.01),P+xd*c(.01,-.01)),c("A","B","P"))
lines(xsq,pline,lty=2)
segments(P[1],P[2], C[1], C[2], lty=1,col=2,lwd=2)
text(rbind(C+xd*c(-.01,-.01)),"C")
text(rbind((P+C)/2),col=2,paste("d=",round(dpl$dis,2)))

## End(Not run)

## Not run: 
A<-c(1,2); B<-c(2,3); P<-c(3,1.5)

dpl<-dist.point2line(P,A,B);
dpl
C<-dpl$cl2p
pts<-rbind(A,B,C,P)

xr<-range(pts[,1])
xf<-(xr[2]-xr[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
lnAB<-Line(A,B,x)
y<-lnAB$y
int<-lnAB$intercept #intercept
sl<-lnAB$slope #slope

xsq<-seq(min(A[1],B[1],P[1])-xf,max(A[1],B[1],P[1])+xf,l=5)
#try also l=10, 20, or 100
pline<-(-1/sl)*(xsq-P[1])+P[2]
#line passing thru P and perpendicular to AB

Xlim<-range(pts[,1],x)
Ylim<-range(pts[,2],y)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(rbind(P),asp=1,pch=1,xlab="x",ylab="y",
main="Illustration of the distance from P \n to the Line Crossing Points A and B",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(rbind(A,B),pch=1)
lines(x,y,lty=1,xlim=Xlim,ylim=Ylim)
int<-round(int,2); sl<-round(sl,2)
text(rbind((A+B)/2+xd*c(-.01,-.01)),ifelse(sl==0,paste("y=",int),
ifelse(sl==1,paste("y=x+",int),
ifelse(int==0,paste("y=",sl,"x"),paste("y=",sl,"x+",int)))))
text(rbind(A+xd*c(0,-.01),B+xd*c(.0,-.01),P+xd*c(.01,-.01)),c("A","B","P"))
lines(xsq,pline,lty=2)
segments(P[1],P[2], C[1], C[2], lty=1,col=2,lwd=2)
text(rbind(C+xd*c(-.01,-.01)),"C")
text(rbind((P+C)/2),col=2,paste("d=",round(dpl$dis,2)))

## End(Not run)

The distance from a point to a plane spanned by three 3D points

Description

Returns the distance from a point p to the plane passing through points a, b, and c in 3D space.

Usage

dist.point2plane(p, a, b, c)
dist.point2plane(p, a, b, c)

Arguments

`p`	A 3D point, distance from `p` to the plane passing through points `a`, `b`, and `c` are to be computed.
`a`, `b`, `c`	3D points that determine the plane (i.e., through which the plane is passing).

Value

A list with two elements

`dis`	Distance from point `p` to the plane spanned by 3D points `a`, `b`, and `c`
`cl2pl`	The closest point on the plane spanned by 3D points `a`, `b`, and `c` to the point `p`

Author(s)

Elvan Ceyhan

Examples

## Not run: 
P<-c(5,2,40)
P1<-c(1,2,3); P2<-c(3,9,12); P3<-c(1,1,3);

dis<-dist.point2plane(P,P1,P2,P3);
dis
Pr<-dis$proj #projection on the plane

xseq<-seq(0,10,l=5)  #try also l=10, 20, or 100
yseq<-seq(0,10,l=5)  #try also l=10, 20, or 100

pl.grid<-Plane(P1,P2,P3,xseq,yseq)$z

plot3D::persp3D(z = pl.grid, x = xseq, y = yseq, theta =225, phi = 30,
ticktype = "detailed",
expand = 0.7, facets = FALSE, scale = TRUE,
main="Point P and its Orthogonal Projection \n on the Plane Defined by P1, P2, P3")
#plane spanned by points P1, P2, P3
#add the vertices of the tetrahedron
plot3D::points3D(P[1],P[2],P[3], add=TRUE)
plot3D::points3D(Pr[1],Pr[2],Pr[3], add=TRUE)
plot3D::segments3D(P[1], P[2], P[3], Pr[1], Pr[2],Pr[3], add=TRUE,lwd=2)

plot3D::text3D(P[1]-.5,P[2],P[3]+1, c("P"),add=TRUE)
plot3D::text3D(Pr[1]-.5,Pr[2],Pr[3]+2, c("Pr"),add=TRUE)

persp(xseq,yseq,pl.grid, xlab="x",ylab="y",zlab="z",theta = -30,
phi = 30, expand = 0.5, col = "lightblue",
      ltheta = 120, shade = 0.05, ticktype = "detailed")


## End(Not run)

## Not run: 
P<-c(5,2,40)
P1<-c(1,2,3); P2<-c(3,9,12); P3<-c(1,1,3);

dis<-dist.point2plane(P,P1,P2,P3);
dis
Pr<-dis$proj #projection on the plane

xseq<-seq(0,10,l=5)  #try also l=10, 20, or 100
yseq<-seq(0,10,l=5)  #try also l=10, 20, or 100

pl.grid<-Plane(P1,P2,P3,xseq,yseq)$z

plot3D::persp3D(z = pl.grid, x = xseq, y = yseq, theta =225, phi = 30,
ticktype = "detailed",
expand = 0.7, facets = FALSE, scale = TRUE,
main="Point P and its Orthogonal Projection \n on the Plane Defined by P1, P2, P3")
#plane spanned by points P1, P2, P3
#add the vertices of the tetrahedron
plot3D::points3D(P[1],P[2],P[3], add=TRUE)
plot3D::points3D(Pr[1],Pr[2],Pr[3], add=TRUE)
plot3D::segments3D(P[1], P[2], P[3], Pr[1], Pr[2],Pr[3], add=TRUE,lwd=2)

plot3D::text3D(P[1]-.5,P[2],P[3]+1, c("P"),add=TRUE)
plot3D::text3D(Pr[1]-.5,Pr[2],Pr[3]+2, c("Pr"),add=TRUE)

persp(xseq,yseq,pl.grid, xlab="x",ylab="y",zlab="z",theta = -30,
phi = 30, expand = 0.5, col = "lightblue",
      ltheta = 120, shade = 0.05, ticktype = "detailed")


## End(Not run)

Distance from a point to a set of finite cardinality

Description

Returns the Euclidean distance between a point p and set of points Yp and the closest point in set Yp to p. Distance between a point and a set is by definition the distance from the point to the closest point in the set. p should be of finite dimension and Yp should be of finite cardinality and p and elements of Yp must have the same dimension.

Usage

dist.point2set(p, Yp)
dist.point2set(p, Yp)

Arguments

`p`	A `vector` (i.e., a point in $R^d$ ).
`Yp`	A set of $d$ -dimensional points.

Value

A list with the elements

`distance`	Distance from point `p` to set `Yp`
`ind.cl.point`	Index of the closest point in set `Yp` to the point `p`
`closest.point`	The closest point in set `Yp` to the point `p`

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
dist.point2set(c(1,2),Te)

X2<-cbind(runif(10),runif(10))
dist.point2set(c(1,2),X2)

x<-runif(1)
y<-as.matrix(runif(10))
dist.point2set(x,y)
#this works, because x is a 1D point, and y is treated as a set of 10 1D points
#but will give an error message if y<-runif(10) is used above

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
dist.point2set(c(1,2),Te)

X2<-cbind(runif(10),runif(10))
dist.point2set(c(1,2),X2)

x<-runif(1)
y<-as.matrix(runif(10))
dist.point2set(x,y)
#this works, because x is a 1D point, and y is treated as a set of 10 1D points
#but will give an error message if y<-runif(10) is used above

## End(Not run)

Exact domination number (i.e., domination number by the exact algorithm)

Description

Returns the (exact) domination number based on the incidence matrix Inc.Mat of a graph or a digraph and the indices (i.e., row numbers of Inc.Mat) for the corresponding (exact) minimum dominating set. Here the row number in the incidence matrix corresponds to the index of the vertex (i.e., index of the data point). The function works whether loops are allowed or not (i.e., whether the first diagonal is all 1 or all 0). It takes a rather long time for large number of vertices (i.e., large number of row numbers).

Usage

dom.num.exact(Inc.Mat)
dom.num.exact(Inc.Mat)

Arguments

Inc.Mat

A square matrix consisting of 0's and 1's which represents the incidence matrix of a graph or digraph.

Value

A list with two elements

`dom.num`	The cardinality of the (exact) minimum dominating set, i.e., (exact) domination number of the graph or digraph whose incidence matrix `Inc.Mat` is given as input.
`ind.mds`	The vector of indices of the rows in the incidence matrix `Inc.Mat` for the (exact) minimum dominating set. The row numbers in the incidence matrix correspond to the indices of the vertices (i.e., indices of the data points).

Author(s)

Elvan Ceyhan

Examples

## Not run: 
n<-10
M<-matrix(sample(c(0,1),n^2,replace=TRUE),nrow=n)
diag(M)<-1

dom.num.greedy(M)
Idom.num.up.bnd(M,2)
dom.num.exact(M)

## End(Not run)

## Not run: 
n<-10
M<-matrix(sample(c(0,1),n^2,replace=TRUE),nrow=n)
diag(M)<-1

dom.num.greedy(M)
Idom.num.up.bnd(M,2)
dom.num.exact(M)

## End(Not run)

Approximate domination number and approximate dominating set by the greedy algorithm

Description

Returns the (approximate) domination number and the indices (i.e., row numbers) for the corresponding (approximate) minimum dominating set based on the incidence matrix Inc.Mat of a graph or a digraph by using the greedy algorithm (Chvatal (1979)). Here the row number in the incidence matrix corresponds to the index of the vertex (i.e., index of the data point). The function works whether loops are allowed or not (i.e., whether the first diagonal is all 1 or all 0). This function may yield the actual domination number or overestimates it.

Usage

dom.num.greedy(Inc.Mat)
dom.num.greedy(Inc.Mat)

Arguments

Inc.Mat

A square matrix consisting of 0's and 1's which represents the incidence matrix of a graph or digraph.

Value

A list with two elements

`dom.num`	The cardinality of the (approximate) minimum dominating set found by the greedy algorithm. i.e., (approximate) domination number of the graph or digraph whose incidence matrix `Inc.Mat` is given as input.
`ind.dom.set`	Indices of the rows in the incidence matrix `Inc.Mat` for the ((approximate) minimum dominating set). The row numbers in the incidence matrix correspond to the indices of the vertices (i.e., indices of the data points).

Author(s)

Elvan Ceyhan

References

Chvatal V (1979). “A greedy heuristic for the set-covering problem.” Mathematics of Operations Research, 4(3), 233 — 235.

Examples

n<-5
M<-matrix(sample(c(0,1),n^2,replace=TRUE),nrow=n)
diag(M)<-1

dom.num.greedy(M)

n<-5
M<-matrix(sample(c(0,1),n^2,replace=TRUE),nrow=n)
diag(M)<-1

dom.num.greedy(M)

The vertices of the $CM$ -edge region in a triangle that contains the point

Description

Returns the edge whose region contains point, p, in the triangle tri $=T(A,B,C)$ with edge regions based on center of mass $CM=(A+B+C)/3$ .

This function is related to rel.edge.triCM, but unlike rel.edge.triCM the related edges are given as vertices ABC for $re=3$ , as BCA for $re=1$ and as CAB for $re=2$ where edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ . The vertices are given one vertex in each row in the output, e.g., $ABC$ is printed as rbind(A,B,C), where row 1 has the entries of vertex A, row 2 has the entries of vertex B, and row 3 has the entries of vertex C.

If the point, p, is not inside tri, then the function yields NA as output.

Edge region for BCA is the triangle $T(B,C,CM)$ , edge region CAB is $T(A,C,CM)$ , and edge region ABC is $T(A,B,CM)$ .

See also (Ceyhan (2005, 2010)).

Usage

edge.reg.triCM(p, tri)
edge.reg.triCM(p, tri)

Arguments

`p`	A 2D point for which $CM$ -edge region it resides in is to be determined in the triangle `tri`.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

The $CM$ -edge region that contains point, p in the triangle tri. The related edges are given as vertices ABC for $re=3$ , as BCA for $re=1$ and as CAB for $re=2$ where edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ .

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(.4,.2)  #try also P<-as.numeric(runif.tri(1,Tr)$g)
edge.reg.triCM(P,Tr)

P<-c(1.8,.5)
edge.reg.triCM(P,Tr)

CM<-(A+B+C)/3
p1<-(A+B+CM)/3
p2<-(B+C+CM)/3
p3<-(A+C+CM)/3

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-Tr; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,CM,p1,p2,p3)
xc<-txt[,1]+c(-.02,.02,.02,-.05,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.02,0,0,0)
txt.str<-c("A","B","C","CM","re=T(A,B,CM)","re=T(B,C,CM)","re=T(A,C,CM)")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(.4,.2)  #try also P<-as.numeric(runif.tri(1,Tr)$g)
edge.reg.triCM(P,Tr)

P<-c(1.8,.5)
edge.reg.triCM(P,Tr)

CM<-(A+B+C)/3
p1<-(A+B+CM)/3
p2<-(B+C+CM)/3
p3<-(A+C+CM)/3

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-Tr; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,CM,p1,p2,p3)
xc<-txt[,1]+c(-.02,.02,.02,-.05,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.02,0,0,0)
txt.str<-c("A","B","C","CM","re=T(A,B,CM)","re=T(B,C,CM)","re=T(A,C,CM)")
text(xc,yc,txt.str)

## End(Not run)

The furthest points in a data set from edges in each $CM$ -edge region in the standard equilateral triangle

Description

An object of class "Extrema". Returns the furthest data points among the data set, Xp, in each $CM$ -edge region from the edge in the standard equilateral triangle $T_e=T(A=(0,0),B=(1,0),C=(1/2,\sqrt{3}/2))$ .

ch.all.intri is for checking whether all data points are inside $T_e$ (default is FALSE).

Usage

fr2edgesCMedge.reg.std.tri(Xp, ch.all.intri = FALSE)
fr2edgesCMedge.reg.std.tri(Xp, ch.all.intri = FALSE)

Arguments

`Xp`	A set of 2D points, some could be inside and some could be outside standard equilateral triangle $T_e$ .
`ch.all.intri`	A logical argument used for checking whether all data points are inside $T_e$ (default is `FALSE`).

Value

A list with the elements

`txt1`	Edge labels as $AB=3$ , $BC=1$ , and $AC=2$ for $T_e$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances to Edges"`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, furthest points from edges in each edge region.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is $T_e$ .
`cent`	The center point used for construction of edge regions.
`ncent`	Name of the center, `cent`, it is center of mass `"CM"` for this function.
`regions`	Edge regions inside the triangle, $T_e$ , provided as a list.
`region.names`	Names of the edge regions as `"er=1"`, `"er=2"`, and `"er=3"`.
`region.centers`	Centers of mass of the edge regions inside $T_e$ .
`dist2ref`	Distances from furthest points in each edge region to the corresponding edge.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
n<-20
Xp<-runif.std.tri(n)$gen.points

Ext<-fr2edgesCMedge.reg.std.tri(Xp)
Ext
summary(Ext)
plot(Ext,asp=1)

ed.far<-Ext

Xp2<-rbind(Xp,c(.8,.8))
fr2edgesCMedge.reg.std.tri(Xp2)
fr2edgesCMedge.reg.std.tri(Xp2,ch.all.intri = FALSE)
#gives error if ch.all.intri = TRUE

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
CM<-(A+B+C)/3
p1<-(A+B)/2
p2<-(B+C)/2
p3<-(A+C)/2

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",
main="Furthest Points in CM-Edge Regions \n of Std Equilateral Triangle from its Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp,xlab="",ylab="")
points(ed.far$ext,pty=2,pch=4,col="red")

txt<-rbind(Te,CM,p1,p2,p3)
xc<-txt[,1]+c(-.03,.03,.03,-.06,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.02,0,0,0)
txt.str<-c("A","B","C","CM","re=2","re=3","re=1")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
n<-20
Xp<-runif.std.tri(n)$gen.points

Ext<-fr2edgesCMedge.reg.std.tri(Xp)
Ext
summary(Ext)
plot(Ext,asp=1)

ed.far<-Ext

Xp2<-rbind(Xp,c(.8,.8))
fr2edgesCMedge.reg.std.tri(Xp2)
fr2edgesCMedge.reg.std.tri(Xp2,ch.all.intri = FALSE)
#gives error if ch.all.intri = TRUE

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
CM<-(A+B+C)/3
p1<-(A+B)/2
p2<-(B+C)/2
p3<-(A+C)/2

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",
main="Furthest Points in CM-Edge Regions \n of Std Equilateral Triangle from its Edges",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp,xlab="",ylab="")
points(ed.far$ext,pty=2,pch=4,col="red")

txt<-rbind(Te,CM,p1,p2,p3)
xc<-txt[,1]+c(-.03,.03,.03,-.06,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.02,0,0,0)
txt.str<-c("A","B","C","CM","re=2","re=3","re=1")
text(xc,yc,txt.str)

## End(Not run)

The furthest points in a data set from vertices in each $CC$ -vertex region in a triangle

Description

An object of class "Extrema". Returns the furthest data points among the data set, Xp, in each $CC$ -vertex region from the vertex in the triangle, tri $=T(A,B,C)$ . Vertex region labels/numbers correspond to the row number of the vertex in tri. ch.all.intri is for checking whether all data points are inside tri (default is FALSE).

If some of the data points are not inside tri and ch.all.intri=TRUE, then the function yields an error message. If some of the data points are not inside tri and ch.all.intri=FALSE, then the function yields the closest points to edges among the data points inside tri (yields NA if there are no data points inside tri).

See also (Ceyhan (2005, 2012)).

Usage

fr2vertsCCvert.reg(Xp, tri, ch.all.intri = FALSE)
fr2vertsCCvert.reg(Xp, tri, ch.all.intri = FALSE)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`ch.all.intri`	A logical argument (default=`FALSE`) to check whether all data points are inside the triangle `tri`. So, if it is `TRUE`, the function checks if all data points are inside the closure of the triangle (i.e., interior and boundary combined) else it does not.

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances from furthest points to ..."`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, furthest points from vertices in each $CC$ -vertex region in the triangle `tri`.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is the triangle `tri` for this function.
`cent`	The center point used for construction of edge regions.
`ncent`	Name of the center, `cent`, it is circumcenter `"CC"` for this function
`regions`	CC-Vertex regions inside the triangle, `tri`, provided as a `list`
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside `tri`
`dist2ref`	Distances from furthest points in each vertex region to the corresponding vertex

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-fr2vertsCCvert.reg(Xp,Tr)
Ext
summary(Ext)
plot(Ext)

f2v<-Ext

CC<-circumcenter.tri(Tr)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,xlab="",asp=1,ylab="",pch=".",
main="Furthest Points in CC-Vertex Regions \n from the Vertices",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(rbind(f2v$ext),pch=4,col=2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.06,.08,.05,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.05,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

Xp2<-rbind(Xp,c(.2,.4))
fr2vertsCCvert.reg(Xp2,Tr,ch.all.intri = FALSE)
#gives an error message if ch.all.intri = TRUE
#since not all points in the data set are in the triangle

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-fr2vertsCCvert.reg(Xp,Tr)
Ext
summary(Ext)
plot(Ext)

f2v<-Ext

CC<-circumcenter.tri(Tr)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,xlab="",asp=1,ylab="",pch=".",
main="Furthest Points in CC-Vertex Regions \n from the Vertices",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(rbind(f2v$ext),pch=4,col=2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.06,.08,.05,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.05,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

Xp2<-rbind(Xp,c(.2,.4))
fr2vertsCCvert.reg(Xp2,Tr,ch.all.intri = FALSE)
#gives an error message if ch.all.intri = TRUE
#since not all points in the data set are in the triangle

## End(Not run)

The furthest points from vertices in each $CC$ -vertex region in a standard basic triangle

Description

An object of class "Extrema". Returns the furthest data points among the data set, Xp, in each $CC$ -vertex region from the corresponding vertex in the standard basic triangle $T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2))$ .

ch.all.intri is for checking whether all data points are inside $T_b$ (default is FALSE).

See also (Ceyhan (2005, 2012)).

An object of class "Extrema". Returns the k furthest data points among the data set, Xp, in each $CC$ -vertex region from the vertex in the standard basic triangle $T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2))$ .

ch.all.intri is for checking whether all data points are inside $T_b$ (default is FALSE). In the extrema, $ext$ , in the output, the first k entries are the k furthest points from vertex 1, second k entries are k furthest points are from vertex 2, and last k entries are the k furthest points from vertex 3 If data size does not allow, NA's are inserted for some or all of the k furthest points for each vertex.

Usage

fr2vertsCCvert.reg.basic.tri(Xp, c1, c2, k, ch.all.intri = FALSE)

fr2vertsCCvert.reg.basic.tri(Xp, c1, c2, k, ch.all.intri = FALSE)
fr2vertsCCvert.reg.basic.tri(Xp, c1, c2, k, ch.all.intri = FALSE)

fr2vertsCCvert.reg.basic.tri(Xp, c1, c2, k, ch.all.intri = FALSE)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle. adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$
`k`	A positive integer. `k` furthest data points in each $CC$ -vertex region are to be found if exists, else `NA` are provided for (some of) the `k` furthest points.
`ch.all.intri`	A logical argument for checking whether all data points are inside $T_b$ (default is `FALSE`).

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances from furthest points to ..."`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, furthest points from vertices in each vertex region.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is $T_b$ .
`cent`	The center point used for construction of edge regions.
`ncent`	Name of the center, `cent`, it is circumcenter `"CC"` for this function.
`regions`	Vertex regions inside the triangle, $T_b$ , provided as a `list`.
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside $T_b$ .
`dist2ref`	Distances from furthest points in each vertex region to the corresponding vertex.

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A shorter description of the distances as `"Distances of k furthest points in the vertex regions to Vertices"`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, `k` furthest points from vertices in each vertex region.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is $T_b$ .
`cent`	The center point used for construction of edge regions.
`ncent`	Name of the center, `cent`, it is circumcenter `"CC"` for this function.
`regions`	Vertex regions inside the triangle, $T_b$ , provided as a `list`.
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside $T_b$ .
`dist2ref`	Distances from `k` furthest points in each vertex region to the corresponding vertex (each row representing a vertex).

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-20

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

Ext<-fr2vertsCCvert.reg.basic.tri(Xp,c1,c2)
Ext
summary(Ext)
plot(Ext)

f2v<-Ext

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main="Furthest Points in CC-Vertex Regions \n from the Vertices",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(rbind(f2v$ext),pch=4,col=2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.03,0.02,.07,.06,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.01,.02,.02,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-20
k<-3

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

Ext<-fr2vertsCCvert.reg.basic.tri(Xp,c1,c2,k)
Ext
summary(Ext)
plot(Ext)

kf2v<-Ext

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main=paste(k," Furthest Points in CC-Vertex Regions \n from the Vertices",sep=""),
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(kf2v$ext,pch=4,col=2)

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,.02,.07,.06,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,-.02,.02,.03,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-20

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

Ext<-fr2vertsCCvert.reg.basic.tri(Xp,c1,c2)
Ext
summary(Ext)
plot(Ext)

f2v<-Ext

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main="Furthest Points in CC-Vertex Regions \n from the Vertices",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(rbind(f2v$ext),pch=4,col=2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.03,0.02,.07,.06,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.01,.02,.02,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-20
k<-3

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

Ext<-fr2vertsCCvert.reg.basic.tri(Xp,c1,c2,k)
Ext
summary(Ext)
plot(Ext)

kf2v<-Ext

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main=paste(k," Furthest Points in CC-Vertex Regions \n from the Vertices",sep=""),
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(kf2v$ext,pch=4,col=2)

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,.02,.07,.06,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,-.02,.02,.03,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The lines joining two vertices to the center of mass in standard equilateral triangle

Description

Two functions, lineA2CMinTe and lineB2CMinTe of class "TriLines". Returns the equation, slope, intercept, and $y$ -coordinates of the lines joining $A$ and $CM$ and also $B$ and $CM$ .

lineA2CMinTe is the line joining $A$ to the center of mass, $CM$ , and lineB2CMinTe is the line joining $B$ to the center of mass, $CM$ , in the standard equilateral triangle $T_e=(A,B,C)$ with $A=(0,0)$ , $B=(1,0)$ , $C=(1/2,\sqrt{3}/2)$ ; $x$ -coordinates are provided in vector x.

Usage

lineA2CMinTe(x)

lineB2CMinTe(x)
lineA2CMinTe(x)

lineB2CMinTe(x)

Arguments

`x`	A single scalar or a `vector` of scalars which is the argument of the functions `lineA2CMinTe` and `lineB2CMinTe`.

Value

A list with the elements

`txt1`	Longer description of the line.
`txt2`	Shorter description of the line (to be inserted over the line in the plot).
`mtitle`	The `"main"` title for the plot of the line.
`cent`	The center chosen inside the standard equilateral triangle.
`cent.name`	The name of the center inside the standard equilateral triangle. It is `"CM"` for these two functions.
`tri`	The triangle (it is the standard equilateral triangle for this function).
`x`	The input vector, can be a scalar or a `vector` of scalars, which constitute the $x$ -coordinates of the point(s) of interest on the line.
`y`	The output vector, will be a scalar if `x` is a scalar or a `vector` of scalars if `x` is a `vector` of scalar, constitutes the $y$ -coordinates of the point(s) of interest on the line.
`slope`	Slope of the line.
`intercept`	Intercept of the line.
`equation`	Equation of the line.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
#Examples for lineA2CMinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .1)  #try also by = .01

lnACM<-lineA2CMinTe(x)
lnACM
summary(lnACM)
plot(lnACM)

CM<-(A+B+C)/3;
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Te,CM,D1,D2,D3,c(.25,lineA2CMinTe(.25)$y),c(.75,lineB2CMinTe(.75)$y))
xc<-txt[,1]+c(-.02,.02,.02,.05,.05,-.03,.0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.02,0,.02,-.04,0,0)
txt.str<-c("A","B","C","CM","D1","D2","D3","lineA2CMinTe(x)","lineB2CMinTe(x)")
text(xc,yc,txt.str)

lineA2CMinTe(.25)$y

## End(Not run)

## Not run: 
#Examples for lineB2CMinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .1)  #try also by = .01

lnBCM<-lineB2CMinTe(x)
lnBCM
summary(lnBCM)
plot(lnBCM,xlab=" x",ylab="y")

lineB2CMinTe(.25)$y

## End(Not run)

## Not run: 
#Examples for lineA2CMinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .1)  #try also by = .01

lnACM<-lineA2CMinTe(x)
lnACM
summary(lnACM)
plot(lnACM)

CM<-(A+B+C)/3;
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Te,CM,D1,D2,D3,c(.25,lineA2CMinTe(.25)$y),c(.75,lineB2CMinTe(.75)$y))
xc<-txt[,1]+c(-.02,.02,.02,.05,.05,-.03,.0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.02,0,.02,-.04,0,0)
txt.str<-c("A","B","C","CM","D1","D2","D3","lineA2CMinTe(x)","lineB2CMinTe(x)")
text(xc,yc,txt.str)

lineA2CMinTe(.25)$y

## End(Not run)

## Not run: 
#Examples for lineB2CMinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .1)  #try also by = .01

lnBCM<-lineB2CMinTe(x)
lnBCM
summary(lnBCM)
plot(lnBCM,xlab=" x",ylab="y")

lineB2CMinTe(.25)$y

## End(Not run)

The lines joining the three vertices of the standard equilateral triangle to a center, `M`, of it

Description

Three functions, lineA2MinTe, lineB2MinTe and lineC2MinTe of class "TriLines". Returns the equation, slope, intercept, and $y$ -coordinates of the lines joining $A$ and M, $B$ and M, and also $C$ and M.

lineA2MinTe is the line joining $A$ to the center, M, lineB2MinTe is the line joining $B$ to M, and lineC2MinTe is the line joining C to M, in the standard equilateral triangle $T_e=(A,B,C)$ with $A=(0,0)$ , $B=(1,0)$ , $C=(1/2,\sqrt{3}/2)$ ; $x$ -coordinates are provided in vector x

Usage

lineA2MinTe(x, M)

lineB2MinTe(x, M)

lineC2MinTe(x, M)
lineA2MinTe(x, M)

lineB2MinTe(x, M)

lineC2MinTe(x, M)

Arguments

`x`	A single scalar or a `vector` of scalars.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle.

Value

A list with the elements

`txt1`	Longer description of the line.
`txt2`	Shorter description of the line (to be inserted over the line in the plot).
`mtitle`	The `"main"` title for the plot of the line.
`cent`	The center chosen inside the standard equilateral triangle.
`cent.name`	The name of the center inside the standard equilateral triangle.
`tri`	The triangle (it is the standard equilateral triangle for this function).
`x`	The input vector, can be a scalar or a `vector` of scalars, which constitute the $x$ -coordinates of the point(s) of interest on the line.
`y`	The output vector, will be a scalar if `x` is a scalar or a `vector` of scalars if `x` is a `vector` of scalar, constitutes the $y$ -coordinates of the point(s) of interest on the line.
`slope`	Slope of the line.
`intercept`	Intercept of the line.
`equation`	Equation of the line.

Examples

## Not run: 
#Examples for lineA2MinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)

M<-c(.65,.2)  #try also M<-c(1,1,1)

xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .1)  #try also by = .01

lnAM<-lineA2MinTe(x,M)
lnAM
summary(lnAM)
plot(lnAM)

Ds<-prj.cent2edges(Te,M)
#finds the projections from a point M=(m1,m2) to the edges on the
#extension of the lines joining M to the vertices in the triangle Te

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
L<-Ds; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 3,col=2)

txt<-rbind(Te,M,Ds,c(.25,lineA2MinTe(.25,M)$y),c(.4,lineB2MinTe(.4,M)$y),
c(.60,lineC2MinTe(.60,M)$y))
xc<-txt[,1]+c(-.02,.02,.02,.02,.04,-.03,.0,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.05,.02,.03,-.03,0,0,0)
txt.str<-c("A","B","C","M","D1","D2","D3","lineA2MinTe(x)","lineB2MinTe(x)","lineC2MinTe(x)")
text(xc,yc,txt.str)

lineA2MinTe(.25,M)

## End(Not run)

## Not run: 
#Examples for lineB2MinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)

M<-c(.65,.2)  #try also M<-c(1,1,1)

xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .5)  #try also by = .1

lnBM<-lineB2MinTe(x,M)
lnBM
summary(lnBM)
plot(lnBM)

## End(Not run)

## Not run: 
#Examples for lineC2MinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)

M<-c(.65,.2)  #try also M<-c(1,1,1)

xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .5)
#try also by = .1

lnCM<-lineC2MinTe(x,M)
lnCM
summary(lnCM)
plot(lnCM)

## End(Not run)

## Not run: 
#Examples for lineA2MinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)

M<-c(.65,.2)  #try also M<-c(1,1,1)

xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .1)  #try also by = .01

lnAM<-lineA2MinTe(x,M)
lnAM
summary(lnAM)
plot(lnAM)

Ds<-prj.cent2edges(Te,M)
#finds the projections from a point M=(m1,m2) to the edges on the
#extension of the lines joining M to the vertices in the triangle Te

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
L<-Ds; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 3,col=2)

txt<-rbind(Te,M,Ds,c(.25,lineA2MinTe(.25,M)$y),c(.4,lineB2MinTe(.4,M)$y),
c(.60,lineC2MinTe(.60,M)$y))
xc<-txt[,1]+c(-.02,.02,.02,.02,.04,-.03,.0,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.05,.02,.03,-.03,0,0,0)
txt.str<-c("A","B","C","M","D1","D2","D3","lineA2MinTe(x)","lineB2MinTe(x)","lineC2MinTe(x)")
text(xc,yc,txt.str)

lineA2MinTe(.25,M)

## End(Not run)

## Not run: 
#Examples for lineB2MinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)

M<-c(.65,.2)  #try also M<-c(1,1,1)

xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .5)  #try also by = .1

lnBM<-lineB2MinTe(x,M)
lnBM
summary(lnBM)
plot(lnBM)

## End(Not run)

## Not run: 
#Examples for lineC2MinTe
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)

M<-c(.65,.2)  #try also M<-c(1,1,1)

xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by = .5)
#try also by = .1

lnCM<-lineC2MinTe(x,M)
lnCM
summary(lnCM)
plot(lnCM)

## End(Not run)

Converts of a point in Cartesian coordinates to Barycentric coordinates and vice versa

Description

Two functions: cart2bary and bary2cart.

cart2bary converts Cartesian coordinates of a given point P $=(x,y)$ to barycentric coordinates (in the normalized form) with respect to the triangle tri $=(v_1,v_2,v_3)$ with vertex labeling done row-wise in tri (i.e., row $i$ corresponds to vertex $v_i$ for $i=1,2,3$ ).

bary2cart converts barycentric coordinates of the point P $=(t_1,t_2,t_3)$ (not necessarily normalized) to Cartesian coordinates according to the coordinates of the triangle, tri. For information on barycentric coordinates, see (Weisstein (2019)).

Usage

cart2bary(P, tri)

bary2cart(P, tri)
cart2bary(P, tri)

bary2cart(P, tri)

Arguments

`P`	A 2D point for `cart2bary`, and a `vector` of three `numeric` entries for `bary2cart`.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

cart2bary returns the barycentric coordinates of a given point P $=(x,y)$ and bary2cart returns the Cartesian coordinates of the point P $=(t_1,t_2,t_3)$ (not necessarily normalized).

Author(s)

Elvan Ceyhan

References

Weisstein EW (2019). “Barycentric Coordinates.” From MathWorld — A Wolfram Web Resource, http://mathworld.wolfram.com/BarycentricCoordinates.html.

Examples

## Not run: 
#Examples for cart2bary
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tr<-rbind(A,B,C)

cart2bary(A,Tr)
cart2bary(c(.3,.2),Tr)

## End(Not run)

## Not run: 
#Examples for bary2cart
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tr<-rbind(A,B,C)

bary2cart(c(.3,.2,.5),Tr)
bary2cart(c(6,2,4),Tr)

## End(Not run)

## Not run: 
#Examples for cart2bary
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tr<-rbind(A,B,C)

cart2bary(A,Tr)
cart2bary(c(.3,.2),Tr)

## End(Not run)

## Not run: 
#Examples for bary2cart
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tr<-rbind(A,B,C)

bary2cart(c(.3,.2,.5),Tr)
bary2cart(c(6,2,4),Tr)

## End(Not run)

Each function is for the presence of an arc from a point in one of the edge regions to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Three indicator functions: IarcCSstd.triRAB, IarcCSstd.triRBC and IarcCSstd.triRAC.

The function IarcCSstd.triRAB returns I(p2 is in $N_{CS}(p1,t)$ for p1 in $RAB$ (edge region for edge $AB$ , i.e., edge 3) in the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ ;

IarcCSstd.triRBC returns I(p2 is in $N_{CS}(p1,t)$ for p1 in $RBC$ (edge region for edge $BC$ , i.e., edge 1) in $T_e$ ; and

IarcCSstd.triRAC returns I(p2 is in $N_{CS}(p1,t)$ for p1 in $RAC$ (edge region for edge $AC$ , i.e., edge 2) in $T_e$ . That is, each function returns 1 if p2 is in $N_{CS}(p1,t)$ , returns 0 otherwise.

CS proximity region is defined with respect to $T_e$ whose vertices are also labeled as $T_e=T(v=1,v=2,v=3)$ with expansion parameter $t>0$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$

If p1 and p2 are distinct and p1 is outside the corresponding edge region and p2 is outside $T_e$ , it returns 0, but if they are identical, then it returns 1 regardless of their location (i.e., it allows loops).

See also (Ceyhan (2005, 2010)).

Usage

IarcCSstd.triRAB(p1, p2, t, M)

IarcCSstd.triRBC(p1, p2, t, M)

IarcCSstd.triRAC(p1, p2, t, M)
IarcCSstd.triRAB(p1, p2, t, M)

IarcCSstd.triRBC(p1, p2, t, M)

IarcCSstd.triRAC(p1, p2, t, M)

Arguments

`p1`	A 2D point whose CS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ .

Value

Each function returns $I($ p2 is in $N_{CS}(p1,t))$ for p1, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

Examples

## Not run: 
#Examples for IarcCSstd.triRAB
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T3<-rbind(A,B,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1

IarcCSstd.triRAB(Xp[1,],Xp[2,],t,M)
IarcCSstd.triRAB(c(.2,.5),Xp[2,],t,M)

## End(Not run)

## Not run: 
#Examples for IarcCSstd.triRBC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T1<-rbind(B,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1

IarcCSstd.triRBC(Xp[1,],Xp[2,],t,M)
IarcCSstd.triRBC(c(.2,.5),Xp[2,],t,M)

## End(Not run)

## Not run: 
#Examples for IarcCSstd.triRAC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T2<-rbind(A,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1

IarcCSstd.triRAC(Xp[1,],Xp[2,],t,M)
IarcCSstd.triRAC(c(.2,.5),Xp[2,],t,M)

## End(Not run)

## Not run: 
#Examples for IarcCSstd.triRAB
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T3<-rbind(A,B,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1

IarcCSstd.triRAB(Xp[1,],Xp[2,],t,M)
IarcCSstd.triRAB(c(.2,.5),Xp[2,],t,M)

## End(Not run)

## Not run: 
#Examples for IarcCSstd.triRBC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T1<-rbind(B,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1

IarcCSstd.triRBC(Xp[1,],Xp[2,],t,M)
IarcCSstd.triRBC(c(.2,.5),Xp[2,],t,M)

## End(Not run)

## Not run: 
#Examples for IarcCSstd.triRAC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T2<-rbind(A,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1

IarcCSstd.triRAC(Xp[1,],Xp[2,],t,M)
IarcCSstd.triRAC(c(.2,.5),Xp[2,],t,M)

## End(Not run)

The function `gammakCSstd.tri` is for $k$ ( $k=2,3,4,5$ ) points constituting a dominating set for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Four indicator functions: Idom.num2CSstd.tri, Idom.num3CSstd.tri, Idom.num4CSstd.tri, Idom.num5CSstd.tri and Idom.num6CSstd.tri.

The function gammakCSstd.tri returns I({p1,...,pk} is a dominating set of the CS-PCD) where vertices of CS-PCD are the 2D data set Xp, that is, returns 1 if {p1,...,pk} is a dominating set of CS-PCD, returns 0 otherwise for $k=2,3,4,5,6$ .

CS proximity region is constructed with respect to $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with expansion parameter $t>0$ and edge regions are based on center of mass $CM=(1/2,\sqrt{3}/6)$ .

ch.data.pnts is for checking whether points p1,...,pk are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1,...,pk would be a dominating set if they actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2CSstd.tri(p1, p2, Xp, t, ch.data.pnts = FALSE)

Idom.num3CSstd.tri(p1, p2, p3, Xp, t, ch.data.pnts = FALSE)

Idom.num4CSstd.tri(p1, p2, p3, p4, Xp, t, ch.data.pnts = FALSE)

Idom.num5CSstd.tri(p1, p2, p3, p4, p5, Xp, t, ch.data.pnts = FALSE)

Idom.num6CSstd.tri(p1, p2, p3, p4, p5, p6, Xp, t, ch.data.pnts = FALSE)
Idom.num2CSstd.tri(p1, p2, Xp, t, ch.data.pnts = FALSE)

Idom.num3CSstd.tri(p1, p2, p3, Xp, t, ch.data.pnts = FALSE)

Idom.num4CSstd.tri(p1, p2, p3, p4, Xp, t, ch.data.pnts = FALSE)

Idom.num5CSstd.tri(p1, p2, p3, p4, p5, Xp, t, ch.data.pnts = FALSE)

Idom.num6CSstd.tri(p1, p2, p3, p4, p5, p6, Xp, t, ch.data.pnts = FALSE)

Arguments

`p1`, `p2`, `p3`, `p4`, `p5`, `p6`	The points { $p1,\ldots,pk$ } are $k$ 2D points (for $k=2,3,4,5,6$ ) to be tested for constituting a dominating set of the CS-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the CS-PCD.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`ch.data.pnts`	A logical argument for checking whether points { $p1,\ldots,pk$ } are data points in `Xp` or not (default is `FALSE`).

Value

The function gammakCSstd.tri returns {p1,...,pk} is a dominating set of the CS-PCD) where vertices of the CS-PCD are the 2D data set Xp), that is, returns 1 if {p1,...,pk} is a dominating set of CS-PCD, returns 0 otherwise.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
set.seed(123)
#Examples for Idom.num2CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num2CSstd.tri(Xp[1,],Xp[2,],Xp,t)
Idom.num2CSstd.tri(c(.2,.2),Xp[2,],Xp,t)

ind.gam2<-vector()
for (i in 1:(n-1))
 for (j in (i+1):n)
 {if (Idom.num2CSstd.tri(Xp[i,],Xp[j,],Xp,t)==1)
  ind.gam2<-rbind(ind.gam2,c(i,j))}

ind.gam2

## End(Not run)

## Not run: 
#Examples for Idom.num3CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num3CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp,t)

ind.gam3<-vector()
for (i in 1:(n-2))
 for (j in (i+1):(n-1))
   for (k in (j+1):n)
   {if (Idom.num3CSstd.tri(Xp[i,],Xp[j,],Xp[k,],Xp,t)==1)
    ind.gam3<-rbind(ind.gam3,c(i,j,k))}

ind.gam3

## End(Not run)

## Not run: 
#Examples for Idom.num4CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num4CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp[4,],Xp,t)

ind.gam4<-vector()
for (i in 1:(n-3))
 for (j in (i+1):(n-2))
   for (k in (j+1):(n-1))
     for (l in (k+1):n)
     {if (Idom.num4CSstd.tri(Xp[i,],Xp[j,],Xp[k,],Xp[l,],Xp,t)==1)
      ind.gam4<-rbind(ind.gam4,c(i,j,k,l))}

ind.gam4

Idom.num4CSstd.tri(c(.2,.2),Xp[2,],Xp[3,],Xp[4,],Xp,t,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

## Not run: 
#Examples for Idom.num5CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num5CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp,t)

ind.gam5<-vector()
for (i1 in 1:(n-4))
 for (i2 in (i1+1):(n-3))
   for (i3 in (i2+1):(n-2))
     for (i4 in (i3+1):(n-1))
       for (i5 in (i4+1):n)
       {if (Idom.num5CSstd.tri(Xp[i1,],Xp[i2,],Xp[i3,],Xp[i4,],Xp[i5,],Xp,t)==1)
        ind.gam5<-rbind(ind.gam5,c(i1,i2,i3,i4,i5))}

ind.gam5

Idom.num5CSstd.tri(c(.2,.2),Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp,t,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

## Not run: 
#Examples for Idom.num6CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num6CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp[6,],Xp,t)

ind.gam6<-vector()
for (i1 in 1:(n-5))
 for (i2 in (i1+1):(n-4))
   for (i3 in (i2+1):(n-3))
     for (i4 in (i3+1):(n-2))
       for (i5 in (i4+1):(n-1))
         for (i6 in (i5+1):n)
         {if (Idom.num6CSstd.tri(Xp[i1,],Xp[i2,],Xp[i3,],Xp[i4,],Xp[i5,],Xp[i6,],Xp,t)==1)
          ind.gam6<-rbind(ind.gam6,c(i1,i2,i3,i4,i5,i6))}

ind.gam6

Idom.num6CSstd.tri(c(.2,.2),Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp[6,],Xp,t,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

## Not run: 
set.seed(123)
#Examples for Idom.num2CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num2CSstd.tri(Xp[1,],Xp[2,],Xp,t)
Idom.num2CSstd.tri(c(.2,.2),Xp[2,],Xp,t)

ind.gam2<-vector()
for (i in 1:(n-1))
 for (j in (i+1):n)
 {if (Idom.num2CSstd.tri(Xp[i,],Xp[j,],Xp,t)==1)
  ind.gam2<-rbind(ind.gam2,c(i,j))}

ind.gam2

## End(Not run)

## Not run: 
#Examples for Idom.num3CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num3CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp,t)

ind.gam3<-vector()
for (i in 1:(n-2))
 for (j in (i+1):(n-1))
   for (k in (j+1):n)
   {if (Idom.num3CSstd.tri(Xp[i,],Xp[j,],Xp[k,],Xp,t)==1)
    ind.gam3<-rbind(ind.gam3,c(i,j,k))}

ind.gam3

## End(Not run)

## Not run: 
#Examples for Idom.num4CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num4CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp[4,],Xp,t)

ind.gam4<-vector()
for (i in 1:(n-3))
 for (j in (i+1):(n-2))
   for (k in (j+1):(n-1))
     for (l in (k+1):n)
     {if (Idom.num4CSstd.tri(Xp[i,],Xp[j,],Xp[k,],Xp[l,],Xp,t)==1)
      ind.gam4<-rbind(ind.gam4,c(i,j,k,l))}

ind.gam4

Idom.num4CSstd.tri(c(.2,.2),Xp[2,],Xp[3,],Xp[4,],Xp,t,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

## Not run: 
#Examples for Idom.num5CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num5CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp,t)

ind.gam5<-vector()
for (i1 in 1:(n-4))
 for (i2 in (i1+1):(n-3))
   for (i3 in (i2+1):(n-2))
     for (i4 in (i3+1):(n-1))
       for (i5 in (i4+1):n)
       {if (Idom.num5CSstd.tri(Xp[i1,],Xp[i2,],Xp[i3,],Xp[i4,],Xp[i5,],Xp,t)==1)
        ind.gam5<-rbind(ind.gam5,c(i1,i2,i3,i4,i5))}

ind.gam5

Idom.num5CSstd.tri(c(.2,.2),Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp,t,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

## Not run: 
#Examples for Idom.num6CSstd.tri
t<-1.5
n<-10 #try also 10, 20 (it may take longer for larger n)

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num6CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp[6,],Xp,t)

ind.gam6<-vector()
for (i1 in 1:(n-5))
 for (i2 in (i1+1):(n-4))
   for (i3 in (i2+1):(n-3))
     for (i4 in (i3+1):(n-2))
       for (i5 in (i4+1):(n-1))
         for (i6 in (i5+1):n)
         {if (Idom.num6CSstd.tri(Xp[i1,],Xp[i2,],Xp[i3,],Xp[i4,],Xp[i5,],Xp[i6,],Xp,t)==1)
          ind.gam6<-rbind(ind.gam6,c(i1,i2,i3,i4,i5,i6))}

ind.gam6

Idom.num6CSstd.tri(c(.2,.2),Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp[6,],Xp,t,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

Each function is for the presence of an arc from a point in one of the edge regions to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case with $t=1$

Description

Three indicator functions: IarcCSt1.std.triRAB, IarcCSt1.std.triRBC and IarcCSt1.std.triRAC.

The function IarcCSt1.std.triRAB returns $I($ p2 is in $N_{CS}(p1,t=1)$ for p1 in $RAB$ (edge region for edge $AB$ , i.e., edge 3) in the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ ;

IarcCSt1.std.triRBC returns $I($ p2 is in $N_{CS}(p1,t=1)$ for p1 in $RBC$ (edge region for edge $BC$ , i.e., edge 1) in $T_e$ ; and

IarcCSt1.std.triRAC returns $I($ p2 is in $N_{CS}(p1,t=1)$ for p1 in $RAC$ (edge region for edge $AC$ , i.e., edge 2) in $T_e$ .

That is, each function returns 1 if p2 is in $N_{CS}(p1,t=1)$ , returns 0 otherwise, where $N_{CS}(x,t)$ is the CS proximity region for point $x$ with expansion parameter $t=1$ .

Usage

IarcCSt1.std.triRAB(p1, p2)

IarcCSt1.std.triRBC(p1, p2)

IarcCSt1.std.triRAC(p1, p2)
IarcCSt1.std.triRAB(p1, p2)

IarcCSt1.std.triRBC(p1, p2)

IarcCSt1.std.triRAC(p1, p2)

Arguments

`p1`	A 2D point whose CS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not.

Value

Each function returns $I($ p2 is in $N_{CS}(p1,t=1))$ for p1, that is, returns 1 if p2 is in $N_{CS}(p1,t=1)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

Examples

## Not run: 
#Examples for IarcCSt1.std.triRAB
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T3<-rbind(A,B,CM);

set.seed(1)
Xp<-runif.std.tri(10)$gen.points

IarcCSt1.std.triRAB(Xp[1,],Xp[2,])

IarcCSt1.std.triRAB(c(.2,.5),Xp[2,])

## End(Not run)

## Not run: 
#Examples for IarcCSt1.std.triRBC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T1<-rbind(B,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

IarcCSt1.std.triRBC(Xp[1,],Xp[2,])

IarcCSt1.std.triRBC(c(.2,.5),Xp[2,])

## End(Not run)

## Not run: 
#Examples for IarcCSt1.std.triRAC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T2<-rbind(A,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

IarcCSt1.std.triRAC(Xp[1,],Xp[2,])
IarcCSt1.std.triRAC(c(1,2),Xp[2,])

## End(Not run)

## Not run: 
#Examples for IarcCSt1.std.triRAB
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T3<-rbind(A,B,CM);

set.seed(1)
Xp<-runif.std.tri(10)$gen.points

IarcCSt1.std.triRAB(Xp[1,],Xp[2,])

IarcCSt1.std.triRAB(c(.2,.5),Xp[2,])

## End(Not run)

## Not run: 
#Examples for IarcCSt1.std.triRBC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T1<-rbind(B,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

IarcCSt1.std.triRBC(Xp[1,],Xp[2,])

IarcCSt1.std.triRBC(c(.2,.5),Xp[2,])

## End(Not run)

## Not run: 
#Examples for IarcCSt1.std.triRAC
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
T2<-rbind(A,C,CM);

set.seed(1)
Xp<-runif.std.tri(3)$gen.points

IarcCSt1.std.triRAC(Xp[1,],Xp[2,])
IarcCSt1.std.triRAC(c(1,2),Xp[2,])

## End(Not run)

Functions provide the indices of the Delaunay triangles where the points reside

Description

Two functions: index.delaunay.tri and indices.delaunay.tri.

index.delaunay.tri finds the index of the Delaunay triangle in which the given point, p, resides.

indices.delaunay.tri finds the indices of triangles for all the points in data set, Xp, as a vector.

Delaunay triangulation is based on Yp and DTmesh are the Delaunay triangles with default NULL. The function returns NA for a point not inside the convex hull of Yp. Number of Yp points (i.e., size of Yp) should be at least three and the points should be in general position so that Delaunay triangulation is (uniquely) defined.

If the number of Yp points is 3, then there is only one Delaunay triangle and the indices of all the points inside this triangle are all 1.

See (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

index.delaunay.tri(p, Yp, DTmesh = NULL)

indices.delaunay.tri(Xp, Yp, DTmesh = NULL)
index.delaunay.tri(p, Yp, DTmesh = NULL)

indices.delaunay.tri(Xp, Yp, DTmesh = NULL)

Arguments

`p`	A 2D point; the index of the Delaunay triangle in which `p` resides is to be determined. It is an argument for `index.delaunay.tri`.
`Yp`	A set of 2D points from which Delaunay triangulation is constructed.
`DTmesh`	Delaunay triangles based on `Yp`, default is `NULL`, which is computed via `tri.mesh` function in `interp` package. `triangles` function yields a triangulation data structure from the triangulation object created by `tri.mesh`.
`Xp`	A set of 2D points representing the set of data points for which the indices of the Delaunay triangles they reside is to be determined. It is an argument for `indices.delaunay.tri`.

Value

index.delaunay.tri returns the index of the Delaunay triangle in which the given point, p, resides and indices.delaunay.tri returns the vector of indices of the Delaunay triangles in which points in the data set, Xp, reside.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#Examples for index.delaunay.tri
nx<-20 #number of X points (target)
ny<-5 #number of Y points (nontarget)
set.seed(1)
Yp<-cbind(runif(ny),runif(ny))

Xp<-runif.multi.tri(nx,Yp)$g #data under CSR in the convex hull of Ypoints
#try also Xp<-cbind(runif(nx),runif(nx))

index.delaunay.tri(Xp[10,],Yp)

#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation
TRY<-interp::triangles(DTY)[,1:3];
index.delaunay.tri(Xp[10,],Yp,DTY)

ind.DT<-vector()
for (i in 1:nx)
 ind.DT<-c(ind.DT,index.delaunay.tri(Xp[i,],Yp))
ind.DT

Xlim<-range(Yp[,1],Xp[,1])
Ylim<-range(Yp[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points

#plot of the data in the convex hull of Y points together with the Delaunay triangulation
plot(Xp,main=" ", xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),type="n")
interp::plot.triSht(DTY, add=TRUE, do.points = TRUE,pch=16,col="blue")
points(Xp,pch=".",cex=3)
text(Xp,labels = factor(ind.DT))

## End(Not run)

## Not run: 
#Examples for indices.delaunay.tri
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
Xp<-runif.multi.tri(nx,Yp)$g #data under CSR in the convex hull of Ypoints
#try also Xp<-cbind(runif(nx),runif(nx))

tr.ind<-indices.delaunay.tri(Xp,Yp)  #indices of the Delaunay triangles
tr.ind

#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points
tr.ind<-indices.delaunay.tri(Xp,Yp,DTY)  #indices of the Delaunay triangles
tr.ind

Xlim<-range(Yp[,1],Xp[,1])
Ylim<-range(Yp[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot of the data in the convex hull of Y points together with the Delaunay triangulation

par(pty = "s")
plot(Xp,main=" ", xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),pch=".")
interp::plot.triSht(DTY, add=TRUE, do.points = TRUE,pch=16,col="blue")
text(Xp,labels = factor(tr.ind))

## End(Not run)

## Not run: 
#Examples for index.delaunay.tri
nx<-20 #number of X points (target)
ny<-5 #number of Y points (nontarget)
set.seed(1)
Yp<-cbind(runif(ny),runif(ny))

Xp<-runif.multi.tri(nx,Yp)$g #data under CSR in the convex hull of Ypoints
#try also Xp<-cbind(runif(nx),runif(nx))

index.delaunay.tri(Xp[10,],Yp)

#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation
TRY<-interp::triangles(DTY)[,1:3];
index.delaunay.tri(Xp[10,],Yp,DTY)

ind.DT<-vector()
for (i in 1:nx)
 ind.DT<-c(ind.DT,index.delaunay.tri(Xp[i,],Yp))
ind.DT

Xlim<-range(Yp[,1],Xp[,1])
Ylim<-range(Yp[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points

#plot of the data in the convex hull of Y points together with the Delaunay triangulation
plot(Xp,main=" ", xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),type="n")
interp::plot.triSht(DTY, add=TRUE, do.points = TRUE,pch=16,col="blue")
points(Xp,pch=".",cex=3)
text(Xp,labels = factor(ind.DT))

## End(Not run)

## Not run: 
#Examples for indices.delaunay.tri
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
Xp<-runif.multi.tri(nx,Yp)$g #data under CSR in the convex hull of Ypoints
#try also Xp<-cbind(runif(nx),runif(nx))

tr.ind<-indices.delaunay.tri(Xp,Yp)  #indices of the Delaunay triangles
tr.ind

#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points
tr.ind<-indices.delaunay.tri(Xp,Yp,DTY)  #indices of the Delaunay triangles
tr.ind

Xlim<-range(Yp[,1],Xp[,1])
Ylim<-range(Yp[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot of the data in the convex hull of Y points together with the Delaunay triangulation

par(pty = "s")
plot(Xp,main=" ", xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),pch=".")
interp::plot.triSht(DTY, add=TRUE, do.points = TRUE,pch=16,col="blue")
text(Xp,labels = factor(tr.ind))

## End(Not run)

Returning the mean and (asymptotic) variance of arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - middle interval case

Description

Two functions: muCS1D and asyvarCS1D.

muCS1D returns the mean of the (arc) density of CS-PCD and asyvarCS1D returns the (asymptotic) variance of the arc density of CS-PCD for a given centrality parameter $c \in (0,1)$ and an expansion parameter $t>0$ and 1D uniform data in a finite interval $(a,b)$ , i.e., data from $U(a,b)$ distribution.

See also (Ceyhan (2016)).

Usage

muCS1D(t, c)

asyvarCS1D(t, c)
muCS1D(t, c)

asyvarCS1D(t, c)

Arguments

`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ . For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

muCS1D returns the mean and asyvarCS1D returns the asymptotic variance of the arc density of CS-PCD for uniform data in an interval

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples


#Examples for muCS1D
muCS1D(1.2,.4)
muCS1D(1.2,.6)

tseq<-seq(0.01,5,by=.05)
cseq<-seq(0.01,.99,by=.05)

ltseq<-length(tseq)
lcseq<-length(cseq)

mu.grid<-matrix(0,nrow=ltseq,ncol=lcseq)
for (i in 1:ltseq)
  for (j in 1:lcseq)
  {
    mu.grid[i,j]<-muCS1D(tseq[i],cseq[j])
  }

persp(tseq,cseq,mu.grid, xlab="t", ylab="c", zlab="mu(t,c)",theta = -30,
phi = 30, expand = 0.5, col = "lightblue", ltheta = 120,
shade = 0.05, ticktype = "detailed")



#Examples for asyvarCS1D
asyvarCS1D(1.2,.8)

tseq<-seq(0.01,5,by=.05)
cseq<-seq(0.01,.99,by=.05)

ltseq<-length(tseq)
lcseq<-length(cseq)

var.grid<-matrix(0,nrow=ltseq,ncol=lcseq)
for (i in 1:ltseq)
  for (j in 1:lcseq)
  {
    var.grid[i,j]<-asyvarCS1D(tseq[i],cseq[j])
  }

persp(tseq,cseq,var.grid, xlab="t", ylab="c", zlab="var(t,c)", theta = -30,
phi = 30, expand = 0.5, col = "lightblue", ltheta = 120,
shade = 0.05, ticktype = "detailed")


#Examples for muCS1D
muCS1D(1.2,.4)
muCS1D(1.2,.6)

tseq<-seq(0.01,5,by=.05)
cseq<-seq(0.01,.99,by=.05)

ltseq<-length(tseq)
lcseq<-length(cseq)

mu.grid<-matrix(0,nrow=ltseq,ncol=lcseq)
for (i in 1:ltseq)
  for (j in 1:lcseq)
  {
    mu.grid[i,j]<-muCS1D(tseq[i],cseq[j])
  }

persp(tseq,cseq,mu.grid, xlab="t", ylab="c", zlab="mu(t,c)",theta = -30,
phi = 30, expand = 0.5, col = "lightblue", ltheta = 120,
shade = 0.05, ticktype = "detailed")



#Examples for asyvarCS1D
asyvarCS1D(1.2,.8)

tseq<-seq(0.01,5,by=.05)
cseq<-seq(0.01,.99,by=.05)

ltseq<-length(tseq)
lcseq<-length(cseq)

var.grid<-matrix(0,nrow=ltseq,ncol=lcseq)
for (i in 1:ltseq)
  for (j in 1:lcseq)
  {
    var.grid[i,j]<-asyvarCS1D(tseq[i],cseq[j])
  }

persp(tseq,cseq,var.grid, xlab="t", ylab="c", zlab="var(t,c)", theta = -30,
phi = 30, expand = 0.5, col = "lightblue", ltheta = 120,
shade = 0.05, ticktype = "detailed")

Returns the mean and (asymptotic) variance of arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D uniform data in one triangle

Description

Two functions: muCS2D and asyvarCS2D.

muCS2D returns the mean of the (arc) density of CS-PCD and asyvarCS2D returns the asymptotic variance of the arc density of CS-PCD with expansion parameter $t>0$ for 2D uniform data in a triangle.

CS proximity regions are defined with respect to the triangle and vertex regions are based on center of mass, $CM$ of the triangle.

See also (Ceyhan (2005); Ceyhan et al. (2007)).

Usage

muCS2D(t)

asyvarCS2D(t)
muCS2D(t)

asyvarCS2D(t)

Arguments

`t`	A positive real number which serves as the expansion parameter in CS proximity region.

Value

muCS2D returns the mean and asyvarCS2D returns the (asymptotic) variance of the arc density of CS-PCD for uniform data in any triangle

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
#Examples for muCS2D
muCS2D(.5)

tseq<-seq(0.01,5,by=.1)
ltseq<-length(tseq)

mu<-vector()
for (i in 1:ltseq)
{
  mu<-c(mu,muCS2D(tseq[i]))
}

plot(tseq, mu,type="l",xlab="t",ylab=expression(mu(t)),lty=1,xlim=range(tseq))

## End(Not run)

## Not run: 
#Examples for asyvarCS2D
asyvarCS2D(.5)

tseq<-seq(0.01,10,by=.1)
ltseq<-length(tseq)

asyvar<-vector()
for (i in 1:ltseq)
{
  asyvar<-c(asyvar,asyvarCS2D(tseq[i]))
}

par(mar=c(5,5,4,2))
plot(tseq, asyvar,type="l",xlab="t",ylab=expression(paste(sigma^2,"(t)")),lty=1,xlim=range(tseq))

## End(Not run)

## Not run: 
#Examples for muCS2D
muCS2D(.5)

tseq<-seq(0.01,5,by=.1)
ltseq<-length(tseq)

mu<-vector()
for (i in 1:ltseq)
{
  mu<-c(mu,muCS2D(tseq[i]))
}

plot(tseq, mu,type="l",xlab="t",ylab=expression(mu(t)),lty=1,xlim=range(tseq))

## End(Not run)

## Not run: 
#Examples for asyvarCS2D
asyvarCS2D(.5)

tseq<-seq(0.01,10,by=.1)
ltseq<-length(tseq)

asyvar<-vector()
for (i in 1:ltseq)
{
  asyvar<-c(asyvar,asyvarCS2D(tseq[i]))
}

par(mar=c(5,5,4,2))
plot(tseq, asyvar,type="l",xlab="t",ylab=expression(paste(sigma^2,"(t)")),lty=1,xlim=range(tseq))

## End(Not run)

Returns the mean and (asymptotic) variance of arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - end interval case

Description

Two functions: muCSend.int and asyvarCSend.int.

muCSend.int returns the mean of the arc density of CS-PCD and asyvarCSend.int returns the asymptotic variance of the arc density of CS-PCD for a given expansion parameter $t>0$ for 1D uniform data in the left and right end intervals for the interval $(a,b)$ .

See also (Ceyhan (2016)).

Usage

muCSend.int(t)

asyvarCSend.int(t)
muCSend.int(t)

asyvarCSend.int(t)

Arguments

`t`	A positive real number which serves as the expansion parameter in CS proximity region.

Details

funsMuVarCSend.int

Value

muCSend.int returns the mean and asyvarCSend.int returns the asymptotic variance of the arc density of CS-PCD for uniform data in end intervals

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

#Examples for muCSend.int
muCSend.int(1.2)

tseq<-seq(0.01,5,by=.05)
ltseq<-length(tseq)

mu.end<-vector()
for (i in 1:ltseq)
{
  mu.end<-c(mu.end,muCSend.int(tseq[i]))
}

oldpar <- par(no.readonly = TRUE)
par(mar = c(5,4,4,2) + 0.1)
plot(tseq, mu.end,type="l",
ylab=expression(paste(mu,"(t)")),xlab="t",lty=1,xlim=range(tseq),ylim=c(0,1))
par(oldpar)

#Examples for asyvarCSend.int
asyvarCSend.int(1.2)

tseq<-seq(.01,5,by=.05)
ltseq<-length(tseq)

var.end<-vector()
for (i in 1:ltseq)
{
  var.end<-c(var.end,asyvarCSend.int(tseq[i]))
}

oldpar <- par(no.readonly = TRUE)
par(mar=c(5,5,4,2))
plot(tseq, var.end,type="l",xlab="t",ylab=expression(paste(sigma^2,"(t)")),lty=1,xlim=range(tseq))
par(oldpar)

#Examples for muCSend.int
muCSend.int(1.2)

tseq<-seq(0.01,5,by=.05)
ltseq<-length(tseq)

mu.end<-vector()
for (i in 1:ltseq)
{
  mu.end<-c(mu.end,muCSend.int(tseq[i]))
}

oldpar <- par(no.readonly = TRUE)
par(mar = c(5,4,4,2) + 0.1)
plot(tseq, mu.end,type="l",
ylab=expression(paste(mu,"(t)")),xlab="t",lty=1,xlim=range(tseq),ylim=c(0,1))
par(oldpar)

#Examples for asyvarCSend.int
asyvarCSend.int(1.2)

tseq<-seq(.01,5,by=.05)
ltseq<-length(tseq)

var.end<-vector()
for (i in 1:ltseq)
{
  var.end<-c(var.end,asyvarCSend.int(tseq[i]))
}

oldpar <- par(no.readonly = TRUE)
par(mar=c(5,5,4,2))
plot(tseq, var.end,type="l",xlab="t",ylab=expression(paste(sigma^2,"(t)")),lty=1,xlim=range(tseq))
par(oldpar)

Returns the mean and (asymptotic) variance of arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - middle interval case

Description

The functions muPE1D and asyvarPE1D and their auxiliary functions.

muPE1D returns the mean of the (arc) density of PE-PCD and asyvarPE1D returns the (asymptotic) variance of the arc density of PE-PCD for a given centrality parameter $c \in (0,1)$ and an expansion parameter $r \ge 1$ and for 1D uniform data in a finite interval $(a,b)$ , i.e., data from $U(a,b)$ distribution.

muPE1D uses auxiliary (internal) function mu1PE1D which yields mean (i.e., expected value) of the arc density of PE-PCD for a given $c \in (0,1/2)$ and $r \ge 1$ .

asyvarPE1D uses auxiliary (internal) functions fvar1 which yields asymptotic variance of the arc density of PE-PCD for $c \in (1/4,1/2)$ and $r \ge 1$ ; and fvar2 which yields asymptotic variance of the arc density of PE-PCD for $c \in (0,1/4)$ and $r \ge 1$ .

See also (Ceyhan (2012)).

Usage

mu1PE1D(r, c)

muPE1D(r, c)

fvar1(r, c)

fvar2(r, c)

asyvarPE1D(r, c)
mu1PE1D(r, c)

muPE1D(r, c)

fvar1(r, c)

fvar2(r, c)

asyvarPE1D(r, c)

Arguments

`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ . For the interval, $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

muPE1D returns the mean and asyvarPE1D returns the asymptotic variance of the arc density of PE-PCD for $U(a,b)$ data

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
#Examples for muPE1D
muPE1D(1.2,.4)
muPE1D(1.2,.6)

rseq<-seq(1.01,5,by=.1)
cseq<-seq(0.01,.99,by=.1)

lrseq<-length(rseq)
lcseq<-length(cseq)

mu.grid<-matrix(0,nrow=lrseq,ncol=lcseq)
for (i in 1:lrseq)
  for (j in 1:lcseq)
  {
    mu.grid[i,j]<-muPE1D(rseq[i],cseq[j])
  }

persp(rseq,cseq,mu.grid, xlab="r", ylab="c", zlab="mu(r,c)", theta = -30, phi = 30,
expand = 0.5, col = "lightblue", ltheta = 120, shade = 0.05, ticktype = "detailed")

## End(Not run)

## Not run: 
#Examples for asyvarPE1D
asyvarPE1D(1.2,.8)

rseq<-seq(1.01,5,by=.1)
cseq<-seq(0.01,.99,by=.1)

lrseq<-length(rseq)
lcseq<-length(cseq)

var.grid<-matrix(0,nrow=lrseq,ncol=lcseq)
for (i in 1:lrseq)
  for (j in 1:lcseq)
  {
    var.grid[i,j]<-asyvarPE1D(rseq[i],cseq[j])
  }

persp(rseq,cseq,var.grid, xlab="r", ylab="c", zlab="var(r,c)", theta = -30, phi = 30,
expand = 0.5, col = "lightblue", ltheta = 120, shade = 0.05, ticktype = "detailed")

## End(Not run)

## Not run: 
#Examples for muPE1D
muPE1D(1.2,.4)
muPE1D(1.2,.6)

rseq<-seq(1.01,5,by=.1)
cseq<-seq(0.01,.99,by=.1)

lrseq<-length(rseq)
lcseq<-length(cseq)

mu.grid<-matrix(0,nrow=lrseq,ncol=lcseq)
for (i in 1:lrseq)
  for (j in 1:lcseq)
  {
    mu.grid[i,j]<-muPE1D(rseq[i],cseq[j])
  }

persp(rseq,cseq,mu.grid, xlab="r", ylab="c", zlab="mu(r,c)", theta = -30, phi = 30,
expand = 0.5, col = "lightblue", ltheta = 120, shade = 0.05, ticktype = "detailed")

## End(Not run)

## Not run: 
#Examples for asyvarPE1D
asyvarPE1D(1.2,.8)

rseq<-seq(1.01,5,by=.1)
cseq<-seq(0.01,.99,by=.1)

lrseq<-length(rseq)
lcseq<-length(cseq)

var.grid<-matrix(0,nrow=lrseq,ncol=lcseq)
for (i in 1:lrseq)
  for (j in 1:lcseq)
  {
    var.grid[i,j]<-asyvarPE1D(rseq[i],cseq[j])
  }

persp(rseq,cseq,var.grid, xlab="r", ylab="c", zlab="var(r,c)", theta = -30, phi = 30,
expand = 0.5, col = "lightblue", ltheta = 120, shade = 0.05, ticktype = "detailed")

## End(Not run)

Returns the mean and (asymptotic) variance of arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D uniform data in one triangle

Description

Two functions: muPE2D and asyvarPE2D.

muPE2D returns the mean of the (arc) density of PE-PCD and asyvarPE2D returns the asymptotic variance of the arc density of PE-PCD for 2D uniform data in a triangle.

PE proximity regions are defined with expansion parameter $r \ge 1$ with respect to the triangle in which the points reside and vertex regions are based on center of mass, $CM$ of the triangle.

Usage

muPE2D(r)

asyvarPE2D(r)
muPE2D(r)

asyvarPE2D(r)

Arguments

`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .

Value

muPE2D returns the mean and asyvarPE2D returns the (asymptotic) variance of the arc density of PE-PCD for uniform data in any triangle.

Author(s)

Elvan Ceyhan

References

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Examples

## Not run: 
#Examples for muPE2D
muPE2D(1.2)

rseq<-seq(1.01,5,by=.05)
lrseq<-length(rseq)

mu<-vector()
for (i in 1:lrseq)
{
  mu<-c(mu,muPE2D(rseq[i]))
}

plot(rseq, mu,type="l",xlab="r",ylab=expression(mu(r)),lty=1,
xlim=range(rseq),ylim=c(0,1))

## End(Not run)

## Not run: 
#Examples for asyvarPE2D
asyvarPE2D(1.2)

rseq<-seq(1.01,5,by=.05)
lrseq<-length(rseq)

avar<-vector()
for (i in 1:lrseq)
{
  avar<-c(avar,asyvarPE2D(rseq[i]))
}

par(mar=c(5,5,4,2))
plot(rseq, avar,type="l",xlab="r",
ylab=expression(paste(sigma^2,"(r)")),lty=1,xlim=range(rseq))

## End(Not run)

## Not run: 
#Examples for muPE2D
muPE2D(1.2)

rseq<-seq(1.01,5,by=.05)
lrseq<-length(rseq)

mu<-vector()
for (i in 1:lrseq)
{
  mu<-c(mu,muPE2D(rseq[i]))
}

plot(rseq, mu,type="l",xlab="r",ylab=expression(mu(r)),lty=1,
xlim=range(rseq),ylim=c(0,1))

## End(Not run)

## Not run: 
#Examples for asyvarPE2D
asyvarPE2D(1.2)

rseq<-seq(1.01,5,by=.05)
lrseq<-length(rseq)

avar<-vector()
for (i in 1:lrseq)
{
  avar<-c(avar,asyvarPE2D(rseq[i]))
}

par(mar=c(5,5,4,2))
plot(rseq, avar,type="l",xlab="r",
ylab=expression(paste(sigma^2,"(r)")),lty=1,xlim=range(rseq))

## End(Not run)

Returns the mean and (asymptotic) variance of arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - end interval case

Description

Two functions: muPEend.int and asyvarPEend.int.

muPEend.int returns the mean of the arc density of PE-PCD and asyvarPEend.int returns the asymptotic variance of the arc density of PE-PCD for a given expansion parameter $r \ge 1$ for 1D uniform data in the left and right end intervals for the interval $(a,b)$ .

See also (Ceyhan (2012)).

Usage

muPEend.int(r)

asyvarPEend.int(r)
muPEend.int(r)

asyvarPEend.int(r)

Arguments

`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .

Value

muPEend.int returns the mean and asyvarPEend.int returns the asymptotic variance of the arc density of PE-PCD for uniform data in end intervals

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
#Examples for muPEend.int
muPEend.int(1.2)

rseq<-seq(1.01,5,by=.1)
lrseq<-length(rseq)

mu.end<-vector()
for (i in 1:lrseq)
{
  mu.end<-c(mu.end,muPEend.int(rseq[i]))
}

plot(rseq, mu.end,type="l",
ylab=expression(paste(mu,"(r)")),xlab="r",lty=1,xlim=range(rseq),ylim=c(0,1))

## End(Not run)

## Not run: 
#Examples for asyvarPEend.int
asyvarPEend.int(1.2)

rseq<-seq(1.01,5,by=.1)
lrseq<-length(rseq)

var.end<-vector()
for (i in 1:lrseq)
{
  var.end<-c(var.end,asyvarPEend.int(rseq[i]))
}

par(mar=c(5,5,4,2))
plot(rseq, var.end,type="l",
xlab="r",ylab=expression(paste(sigma^2,"(r)")),lty=1,xlim=range(rseq))

## End(Not run)

## Not run: 
#Examples for muPEend.int
muPEend.int(1.2)

rseq<-seq(1.01,5,by=.1)
lrseq<-length(rseq)

mu.end<-vector()
for (i in 1:lrseq)
{
  mu.end<-c(mu.end,muPEend.int(rseq[i]))
}

plot(rseq, mu.end,type="l",
ylab=expression(paste(mu,"(r)")),xlab="r",lty=1,xlim=range(rseq),ylim=c(0,1))

## End(Not run)

## Not run: 
#Examples for asyvarPEend.int
asyvarPEend.int(1.2)

rseq<-seq(1.01,5,by=.1)
lrseq<-length(rseq)

var.end<-vector()
for (i in 1:lrseq)
{
  var.end<-c(var.end,asyvarPEend.int(rseq[i]))
}

par(mar=c(5,5,4,2))
plot(rseq, var.end,type="l",
xlab="r",ylab=expression(paste(sigma^2,"(r)")),lty=1,xlim=range(rseq))

## End(Not run)

The functions for probability of domination number $= 2$ for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case

Description

The function Pdom.num2PE1D and its auxiliary functions.

Returns $P(\gamma=2)$ for PE-PCD whose vertices are a uniform data set of size n in a finite interval $(a,b)$ where $\gamma$ stands for the domination number.

The PE proximity region $N_{PE}(x,r,c)$ is defined with respect to $(a,b)$ with centrality parameter $c \in (0,1)$ and expansion parameter $r \ge 1$ .

To compute the probability $P(\gamma=2)$ for PE-PCD in the 1D case, we partition the domain $(r,c)=(1,\infty) \times (0,1)$ , and compute the probability for each partition set. The sample size (i.e., number of vertices or data points) is a positive integer, n.

Usage

Pdom.num2AI(r, c, n)

Pdom.num2AII(r, c, n)

Pdom.num2AIII(r, c, n)

Pdom.num2AIV(r, c, n)

Pdom.num2A(r, c, n)

Pdom.num2Asym(r, c, n)

Pdom.num2BIII(r, c, n)

Pdom.num2B(r, c, n)

Pdom.num2Bsym(r, c, n)

Pdom.num2CIV(r, c, n)

Pdom.num2C(r, c, n)

Pdom.num2Csym(r, c, n)

Pdom.num2PE1D(r, c, n)
Pdom.num2AI(r, c, n)

Pdom.num2AII(r, c, n)

Pdom.num2AIII(r, c, n)

Pdom.num2AIV(r, c, n)

Pdom.num2A(r, c, n)

Pdom.num2Asym(r, c, n)

Pdom.num2BIII(r, c, n)

Pdom.num2B(r, c, n)

Pdom.num2Bsym(r, c, n)

Pdom.num2CIV(r, c, n)

Pdom.num2C(r, c, n)

Pdom.num2Csym(r, c, n)

Pdom.num2PE1D(r, c, n)

Arguments

`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ . For the interval, $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`n`	A positive integer representing the size of the uniform data set.

Value

$P($ domination number $\le 1)$ for PE-PCD whose vertices are a uniform data set of size n in a finite interval $(a,b)$

Auxiliary Functions for `Pdom.num2PE1D`

The auxiliary functions are Pdom.num2AI, Pdom.num2AII, Pdom.num2AIII, Pdom.num2AIV, Pdom.num2A, Pdom.num2Asym, Pdom.num2BIII, Pdom.num2B, Pdom.num2B, Pdom.num2Bsym, Pdom.num2CIV, Pdom.num2C, and Pdom.num2Csym, each corresponding to a partition of the domain of r and c. In particular, the domain partition is handled in 3 cases as

CASE A: $c \in ((3-\sqrt{5})/2, 1/2)$

CASE B: $c \in (1/4,(3-\sqrt{5})/2)$ and

CASE C: $c \in (0,1/4)$ .

Case A - $c \in ((3-\sqrt{5})/2, 1/2)$

In Case A, we compute $P(\gamma=2)$ with

Pdom.num2AIV(r,c,n) if $1 < r < (1-c)/c$ ;

Pdom.num2AIII(r,c,n) if $(1-c)/c< r < 1/(1-c)$ ;

Pdom.num2AII(r,c,n) if $1/(1-c)< r < 1/c$ ;

and Pdom.num2AI(r,c,n) otherwise.

Pdom.num2A(r,c,n) combines these functions in Case A: $c \in ((3-\sqrt{5})/2,1/2)$ . Due to the symmetry in the PE proximity regions, we use Pdom.num2Asym(r,c,n) for $c$ in $(1/2,(\sqrt{5}-1)/2)$ with the same auxiliary functions

Pdom.num2AIV(r,1-c,n) if $1 < r < c/(1-c)$ ;

Pdom.num2AIII(r,1-c,n) if $(c/(1-c) < r < 1/c$ ;

Pdom.num2AII(r,1-c,n) if $1/c < r < 1/(1-c)$ ;

and Pdom.num2AI(r,1-c,n) otherwise.

Case B - $c \in (1/4,(3-\sqrt{5})/2)$

In Case B, we compute $P(\gamma=2)$ with

Pdom.num2AIV(r,c,n) if $1 < r < 1/(1-c)$ ;

Pdom.num2BIII(r,c,n) if $1/(1-c) < r < (1-c)/c$ ;

Pdom.num2AII(r,c,n) if $(1-c)/c < r < 1/c$ ;

and Pdom.num2AI(r,c,n) otherwise.

Pdom.num2B(r,c,n) combines these functions in Case B: $c \in (1/4,(3-\sqrt{5})/2)$ . Due to the symmetry in the PE proximity regions, we use Pdom.num2Bsym(r,c,n) for c in $((\sqrt{5}-1)/2,3/4)$ with the same auxiliary functions

Pdom.num2AIV(r,1-c,n) if $1< r < 1/c$ ;

Pdom.num2BIII(r,1-c,n) if $1/c < r < c/(1-c)$ ;

Pdom.num2AII(r,1-c,n) if $c/(1-c) < r < 1/(1-c)$ ;

and Pdom.num2AI(r,1-c,n) otherwise.

Case C - $c \in (0,1/4)$

In Case C, we compute $P(\gamma=2)$ with

Pdom.num2AIV(r,c,n) if $1< r < 1/(1-c)$ ;

Pdom.num2BIII(r,c,n) if $1/(1-c) < r < (1-\sqrt{1-4 c})/(2 c)$ ;

Pdom.num2CIV(r,c,n) if $(1-\sqrt{1-4 c})/(2 c) < r < (1+\sqrt{1-4 c})/(2 c)$ ;

Pdom.num2BIII(r,c,n) if $(1+\sqrt{1-4 c})/(2 c) < r <1/(1-c)$ ;

Pdom.num2AII(r,c,n) if $1/(1-c) < r < 1/c$ ;

and Pdom.num2AI(r,c,n) otherwise.

Pdom.num2C(r,c,n) combines these functions in Case C: $c \in (0,1/4)$ . Due to the symmetry in the PE proximity regions, we use Pdom.num2Csym(r,c,n) for $c \in (3/4,1)$ with the same auxiliary functions

Pdom.num2AIV(r,1-c,n) if $1< r < 1/c$ ;

Pdom.num2BIII(r,1-c,n) if $1/c < r < (1-\sqrt{1-4(1-c)})/(2(1-c))$ ;

Pdom.num2CIV(r,1-c,n) if $(1-\sqrt{1-4(1-c)})/(2(1-c)) < r < (1+\sqrt{1-4(1-c)})/(2(1-c))$ ;

Pdom.num2BIII(r,1-c,n) if $(1+\sqrt{1-4(1-c)})/(2(1-c)) < r < c/(1-c)$ ;

Pdom.num2AII(r,1-c,n) if $c/(1-c)< r < 1/(1-c)$ ;

and Pdom.num2AI(r,1-c,n) otherwise.

Combining Cases A, B, and C, we get our main function Pdom.num2PE1D which computes $P(\gamma=2)$ for any (r,c) in its domain.

Author(s)

Elvan Ceyhan

Examples

#Examples for the main function Pdom.num2PE1D
r<-2
c<-.5

Pdom.num2PE1D(r,c,n=10)
Pdom.num2PE1D(r=1.5,c=1/1.5,n=100)

#Examples for the main function Pdom.num2PE1D
r<-2
c<-.5

Pdom.num2PE1D(r,c,n=10)
Pdom.num2PE1D(r=1.5,c=1/1.5,n=100)

Returns the ranks and orders of points in decreasing distance to the edges of the triangle

Description

Two functions: rank.dist2edges.std.tri and order.dist2edges.std.tri.

rank.dist2edges.std.tri finds the ranks of the distances of points in data, Xp, to the edges of the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$

dec is a logical argument, default is TRUE, so the ranks are for decreasing distances, if FALSE it will be in increasing distances.

order.dist2edges.std.tri finds the orders of the distances of points in data, Xp, to the edges of $T_e$ . The arguments are as in rank.dist2edges.std.tri.

Usage

rank.dist2edges.std.tri(Xp, dec = TRUE)

order.dist2edges.std.tri(Xp, dec = TRUE)
rank.dist2edges.std.tri(Xp, dec = TRUE)

order.dist2edges.std.tri(Xp, dec = TRUE)

Arguments

`Xp`	A set of 2D points representing the data set in which ranking in terms of the distance to the edges of $T_e$ is performed.
`dec`	A logical argument indicating the how the ranking will be performed. If `TRUE`, the ranks are for decreasing distances, and if `FALSE` they will be in increasing distances, default is `TRUE`.

Value

A list with two elements

`distances`	Distances from data points to the edges of $T_e$
`dist.rank`	The ranks of the data points in decreasing distances to the edges of $T_e$

Author(s)

Elvan Ceyhan

Examples

## Not run: 
#Examples for rank.dist2edges.std.tri
n<-10
set.seed(1)
Xp<-runif.std.tri(n)$gen.points

dec.dist<-rank.dist2edges.std.tri(Xp)
dec.dist
dec.dist.rank<-dec.dist[[2]]
#the rank of distances to the edges in decreasing order
dec.dist.rank

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.0,.01),
ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp,pch=".")
text(Xp,labels = factor(dec.dist.rank) )

inc.dist<-rank.dist2edges.std.tri(Xp,dec = FALSE)
inc.dist
inc.dist.rank<-inc.dist[[2]]
#the rank of distances to the edges in increasing order
inc.dist.rank
dist<-inc.dist[[1]] #distances to the edges of the std eq. triangle
dist

plot(A,pch=".",xlab="",ylab="",xlim=Xlim,ylim=Ylim)
polygon(Te)
points(Xp,pch=".",xlab="",ylab="", main="",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05))
text(Xp,labels = factor(inc.dist.rank))

## End(Not run)

## Not run: 
#Examples for order.dist2edges.std.tri
n<-10
set.seed(1)
Xp<-runif.std.tri(n)$gen.points  #try also Xp<-cbind(runif(n),runif(n))

dec.dist<-order.dist2edges.std.tri(Xp)
dec.dist
dec.dist.order<-dec.dist[[2]]
#the order of distances to the edges in decreasing order
dec.dist.order

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.01,.01),
ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp,pch=".")
text(Xp[dec.dist.order,],labels = factor(1:n) )

inc.dist<-order.dist2edges.std.tri(Xp,dec = FALSE)
inc.dist
inc.dist.order<-inc.dist[[2]]
#the order of distances to the edges in increasing order
inc.dist.order
dist<-inc.dist[[1]] #distances to the edges of the std eq. triangle
dist
dist[inc.dist.order] #distances in increasing order

plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,pch=".")
text(Xp[inc.dist.order,],labels = factor(1:n))

## End(Not run)

## Not run: 
#Examples for rank.dist2edges.std.tri
n<-10
set.seed(1)
Xp<-runif.std.tri(n)$gen.points

dec.dist<-rank.dist2edges.std.tri(Xp)
dec.dist
dec.dist.rank<-dec.dist[[2]]
#the rank of distances to the edges in decreasing order
dec.dist.rank

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.0,.01),
ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp,pch=".")
text(Xp,labels = factor(dec.dist.rank) )

inc.dist<-rank.dist2edges.std.tri(Xp,dec = FALSE)
inc.dist
inc.dist.rank<-inc.dist[[2]]
#the rank of distances to the edges in increasing order
inc.dist.rank
dist<-inc.dist[[1]] #distances to the edges of the std eq. triangle
dist

plot(A,pch=".",xlab="",ylab="",xlim=Xlim,ylim=Ylim)
polygon(Te)
points(Xp,pch=".",xlab="",ylab="", main="",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05))
text(Xp,labels = factor(inc.dist.rank))

## End(Not run)

## Not run: 
#Examples for order.dist2edges.std.tri
n<-10
set.seed(1)
Xp<-runif.std.tri(n)$gen.points  #try also Xp<-cbind(runif(n),runif(n))

dec.dist<-order.dist2edges.std.tri(Xp)
dec.dist
dec.dist.order<-dec.dist[[2]]
#the order of distances to the edges in decreasing order
dec.dist.order

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.01,.01),
ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp,pch=".")
text(Xp[dec.dist.order,],labels = factor(1:n) )

inc.dist<-order.dist2edges.std.tri(Xp,dec = FALSE)
inc.dist
inc.dist.order<-inc.dist[[2]]
#the order of distances to the edges in increasing order
inc.dist.order
dist<-inc.dist[[1]] #distances to the edges of the std eq. triangle
dist
dist[inc.dist.order] #distances in increasing order

plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,pch=".")
text(Xp[inc.dist.order,],labels = factor(1:n))

## End(Not run)

Two functions `lineD1CCinTb` and `lineD2CCinTb` which are of class `"TriLines"` — The lines joining the midpoints of edges to the circumcenter ( $CC$ ) in the standard basic triangle.

Description

Returns the equation, slope, intercept, and $y$ -coordinates of the lines joining $D_1$ and $CC$ and also $D_2$ and $CC$ , in the standard basic triangle $T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ and $D_1=(B+C)/2$ and $D_2=(A+C)/2$ are the midpoints of edges $BC$ and $AC$ .

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis. $x$ -coordinates are provided in vector x.

Usage

lineD1CCinTb(x, c1, c2)

lineD2CCinTb(x, c1, c2)
lineD1CCinTb(x, c1, c2)

lineD2CCinTb(x, c1, c2)

Arguments

`x`	A single scalar or a `vector` of scalars.
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Value

A list with the elements

`txt1`	Longer description of the line.
`txt2`	Shorter description of the line (to be inserted over the line in the plot).
`mtitle`	The `"main"` title for the plot of the line.
`cent`	The center chosen inside the standard equilateral triangle.
`cent.name`	The name of the center inside the standard basic triangle. It is `"CC"` for these two functions.
`tri`	The triangle (it is the standard basic triangle for this function).
`x`	The input vector, can be a scalar or a `vector` of scalars, which constitute the $x$ -coordinates of the point(s) of interest on the line.
`y`	The output vector, will be a scalar if `x` is a scalar or a `vector` of scalars if `x` is a `vector` of scalar, constitutes the $y$ -coordinates of the point(s) of interest on the line.
`slope`	Slope of the line.
`intercept`	Intercept of the line.
`equation`	Equation of the line.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
#Examples for lineD1CCinTb
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);  #the vertices of the standard basic triangle Tb

Tb<-rbind(A,B,C)

xfence<-abs(A[1]-B[1])*.25 #how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by=.1)  #try also by=.01

lnD1CC<-lineD1CCinTb(x,c1,c2)
lnD1CC
summary(lnD1CC)
plot(lnD1CC)

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

x1<-seq(0,1,by=.1)  #try also by=.01
y1<-lineD1CCinTb(x1,c1,c2)$y

Xlim<-range(Tb[,1],x1)
Ylim<-range(Tb[,2],y1)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.04,.03,.02,.09,-.08,0)
yc<-txt[,2]+c(.02,.02,.04,.08,.03,.03,-.05)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

lines(x1,y1,type="l",lty=2)
text(.8,.5,"lineD1CCinTb")

c1<-.4; c2<-.6;
x1<-seq(0,1,by=.1)  #try also by=.01
lineD1CCinTb(x1,c1,c2)

## End(Not run)

## Not run: 
#Examples for lineD2CCinTb
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);  #the vertices of the standard basic triangle Tb

Tb<-rbind(A,B,C)

xfence<-abs(A[1]-B[1])*.25 #how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by=.1)  #try also by=.01

lnD2CC<-lineD2CCinTb(x,c1,c2)
lnD2CC
summary(lnD2CC)
plot(lnD2CC)

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

x2<-seq(0,1,by=.1)  #try also by=.01
y2<-lineD2CCinTb(x2,c1,c2)$y

Xlim<-range(Tb[,1],x1)
Ylim<-range(Tb[,2],y2)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.04,.03,.02,.09,-.08,0)
yc<-txt[,2]+c(.02,.02,.04,.08,.03,.03,-.05)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

lines(x2,y2,type="l",lty=2)
text(0,.5,"lineD2CCinTb")

## End(Not run)

## Not run: 
#Examples for lineD1CCinTb
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);  #the vertices of the standard basic triangle Tb

Tb<-rbind(A,B,C)

xfence<-abs(A[1]-B[1])*.25 #how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by=.1)  #try also by=.01

lnD1CC<-lineD1CCinTb(x,c1,c2)
lnD1CC
summary(lnD1CC)
plot(lnD1CC)

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

x1<-seq(0,1,by=.1)  #try also by=.01
y1<-lineD1CCinTb(x1,c1,c2)$y

Xlim<-range(Tb[,1],x1)
Ylim<-range(Tb[,2],y1)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.04,.03,.02,.09,-.08,0)
yc<-txt[,2]+c(.02,.02,.04,.08,.03,.03,-.05)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

lines(x1,y1,type="l",lty=2)
text(.8,.5,"lineD1CCinTb")

c1<-.4; c2<-.6;
x1<-seq(0,1,by=.1)  #try also by=.01
lineD1CCinTb(x1,c1,c2)

## End(Not run)

## Not run: 
#Examples for lineD2CCinTb
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);  #the vertices of the standard basic triangle Tb

Tb<-rbind(A,B,C)

xfence<-abs(A[1]-B[1])*.25 #how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,by=.1)  #try also by=.01

lnD2CC<-lineD2CCinTb(x,c1,c2)
lnD2CC
summary(lnD2CC)
plot(lnD2CC)

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2; #midpoints of the edges
Ds<-rbind(D1,D2,D3)

x2<-seq(0,1,by=.1)  #try also by=.01
y2<-lineD2CCinTb(x2,c1,c2)$y

Xlim<-range(Tb[,1],x1)
Ylim<-range(Tb[,2],y2)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

txt<-rbind(Tb,CC,D1,D2,D3)
xc<-txt[,1]+c(-.03,.04,.03,.02,.09,-.08,0)
yc<-txt[,2]+c(.02,.02,.04,.08,.03,.03,-.05)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

lines(x2,y2,type="l",lty=2)
text(0,.5,"lineD2CCinTb")

## End(Not run)

The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case

Description

Returns $I(p2 \in N_{AS}(p1))$ for points p1 and p2, that is, returns 1 if $p2$ is in $N_{AS}(p1)$ , returns 0 otherwise, where $N_{AS}(x)$ is the AS proximity region for point $x$ .

AS proximity region is constructed in the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Vertex regions are based on the center M="CC" for circumcenter of $T_b$ ; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_b$ ; default is M="CC" i.e., circumcenter of $T_b$ . rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside $T_b$ , the function returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010)).

Usage

IarcASbasic.tri(p1, p2, c1, c2, M = "CC", rv = NULL)
IarcASbasic.tri(p1, p2, c1, c2, M = "CC", rv = NULL)

Arguments

`p1`	A 2D point whose AS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the AS proximity region of `p1` or not.
`c1`, `c2`	Positive real numbers representing the top vertex in standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ , $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	The center of the triangle. `"CC"` stands for circumcenter or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$ ; default is `M="CC"` i.e., the circumcenter of $T_b$ .
`rv`	The index of the `M`-vertex region in $T_b$ containing the point, either `1,2,3` or `NULL` (default is `NULL`).

Value

$I(p2 \in N_{AS}(p1))$ for points p1 and p2, that is, returns 1 if $p2$ is in $N_{AS}(p1)$ (i.e., if there is an arc from p1 to p2), returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcASbasic.tri(P1,P2,c1,c2,M)

P1<-c(.3,.2)
P2<-c(.6,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

#or try
Rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
IarcASbasic.tri(P1,P2,c1,c2,M,Rv)

P1<-c(.3,.2)
P2<-c(.8,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

P3<-c(.5,.4)
IarcASbasic.tri(P1,P3,c1,c2,M)

P4<-c(1.5,.4)
IarcASbasic.tri(P1,P4,c1,c2,M)
IarcASbasic.tri(P4,P4,c1,c2,M)

c1<-.4; c2<-.6;
P1<-c(.3,.2)
P2<-c(.6,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcASbasic.tri(P1,P2,c1,c2,M)

P1<-c(.3,.2)
P2<-c(.6,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

#or try
Rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
IarcASbasic.tri(P1,P2,c1,c2,M,Rv)

P1<-c(.3,.2)
P2<-c(.8,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

P3<-c(.5,.4)
IarcASbasic.tri(P1,P3,c1,c2,M)

P4<-c(1.5,.4)
IarcASbasic.tri(P1,P4,c1,c2,M)
IarcASbasic.tri(P4,P4,c1,c2,M)

c1<-.4; c2<-.6;
P1<-c(.3,.2)
P2<-c(.6,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

## End(Not run)

The indicator for the presence of an arc from a point in set `S` to the point `p` for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns I( $pt \in N_{AS}(x)$ for some $x \in S$ ), that is, returns 1 if $p$ is in $\cup_{x \in S}N_{AS}(x)$ , returns 0 otherwise, where $N_{AS}(x)$ is the AS proximity region for point $x$ .

AS proximity regions are constructed with respect to the triangle, tri $=T(A,B,C)=$ (rv=1,rv=2,rv=3), and vertices of tri are also labeled as 1,2, and 3, respectively.

If p is not in S and either p or all points in S are outside tri, it returns 0, but if p is in S, then it always returns 1 (i.e., loops are allowed).

See also (Ceyhan (2005, 2010)).

Usage

IarcASset2pnt.tri(S, p, tri, M = "CC")
IarcASset2pnt.tri(S, p, tri, M = "CC")

Arguments

`S`	A set of 2D points whose AS proximity regions are considered.
`p`	A 2D point. The function determines whether `p` is inside the union of AS proximity regions of points in `S` or not.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of `tri`; default is `M="CC"` i.e., the circumcenter of `tri`.

Value

$I(pt \in \cup_{x in S}N_{AS}(x,r))$ , that is, returns 1 if p is in S or inside $N_{AS}(x)$ for at least one $x$ in S, returns 0 otherwise, where AS proximity region is constructed in tri

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

IarcASset2pnt.tri(S,Xp[3,],Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcASset2pnt.tri(S,Xp[3,],Tr,M)

IarcASset2pnt.tri(S,Xp[6,],Tr,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcASset2pnt.tri(S,Xp[3,],Tr,M)

IarcASset2pnt.tri(c(.2,.5),Xp[2,],Tr,M)
IarcASset2pnt.tri(Xp,c(.2,.5),Tr,M)
IarcASset2pnt.tri(Xp,Xp[2,],Tr,M)
IarcASset2pnt.tri(c(.2,.5),c(.2,.5),Tr,M)
IarcASset2pnt.tri(Xp[5,],Xp[2,],Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
IarcASset2pnt.tri(S,Xp[3,],Tr,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcASset2pnt.tri(Xp,P,Tr,M)
IarcASset2pnt.tri(S,P,Tr,M)

IarcASset2pnt.tri(rbind(S,S),P,Tr,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

IarcASset2pnt.tri(S,Xp[3,],Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcASset2pnt.tri(S,Xp[3,],Tr,M)

IarcASset2pnt.tri(S,Xp[6,],Tr,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcASset2pnt.tri(S,Xp[3,],Tr,M)

IarcASset2pnt.tri(c(.2,.5),Xp[2,],Tr,M)
IarcASset2pnt.tri(Xp,c(.2,.5),Tr,M)
IarcASset2pnt.tri(Xp,Xp[2,],Tr,M)
IarcASset2pnt.tri(c(.2,.5),c(.2,.5),Tr,M)
IarcASset2pnt.tri(Xp[5,],Xp[2,],Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
IarcASset2pnt.tri(S,Xp[3,],Tr,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcASset2pnt.tri(Xp,P,Tr,M)
IarcASset2pnt.tri(S,P,Tr,M)

IarcASset2pnt.tri(rbind(S,S),P,Tr,M)

## End(Not run)

The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns $I(p2 \in N_{AS}(p1))$ for points p1 and p2, that is, returns 1 if $p2$ is in $N_{AS}(p1)$ , returns 0 otherwise, where $N_{AS}(x)$ is the AS proximity region for point $x$ .

AS proximity regions are constructed with respect to the triangle, tri $=T(A,B,C)=$ (rv=1,rv=2,rv=3), and vertex regions are based on the center M="CC" for circumcenter of tri; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is M="CC" i.e., circumcenter of tri. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside tri, the function returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010)).

Usage

IarcAStri(p1, p2, tri, M = "CC", rv = NULL)
IarcAStri(p1, p2, tri, M = "CC", rv = NULL)

Arguments

`p1`	A 2D point whose AS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the AS proximity region of `p1` or not.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of `tri`; default is `M="CC"` i.e., the circumcenter of `tri`.
`rv`	The index of the `M`-vertex region in `tri` containing the point, either `1,2,3` or `NULL` (default is `NULL`).

Value

$I(p2 \in N_{AS}(p1))$ for p1, that is, returns 1 if $p2$ is in $N_{AS}(p1)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcAStri(P1,P2,Tr,M)

P1<-c(1.3,1.2)
P2<-c(1.8,.5)
IarcAStri(P1,P2,Tr,M)
IarcAStri(P1,P1,Tr,M)

#or try
Rv<-rel.vert.triCC(P1,Tr)$rv
IarcAStri(P1,P2,Tr,M,Rv)

P3<-c(1.6,1.4)
IarcAStri(P1,P3,Tr,M)

P4<-c(1.5,1.0)
IarcAStri(P1,P4,Tr,M)

P5<-c(.5,1.0)
IarcAStri(P1,P5,Tr,M)
IarcAStri(P5,P5,Tr,M)

#or try
Rv<-rel.vert.triCC(P5,Tr)$rv
IarcAStri(P5,P5,Tr,M,Rv)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcAStri(P1,P2,Tr,M)

P1<-c(1.3,1.2)
P2<-c(1.8,.5)
IarcAStri(P1,P2,Tr,M)
IarcAStri(P1,P1,Tr,M)

#or try
Rv<-rel.vert.triCC(P1,Tr)$rv
IarcAStri(P1,P2,Tr,M,Rv)

P3<-c(1.6,1.4)
IarcAStri(P1,P3,Tr,M)

P4<-c(1.5,1.0)
IarcAStri(P1,P4,Tr,M)

P5<-c(.5,1.0)
IarcAStri(P1,P5,Tr,M)
IarcAStri(P5,P5,Tr,M)

#or try
Rv<-rel.vert.triCC(P5,Tr)$rv
IarcAStri(P5,P5,Tr,M,Rv)

## End(Not run)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - first one-sixth of the standard equilateral triangle case

Description

Returns $I($ p2 is in $N_{CS}(p1,t=1))$ for points p1 and p2, that is, returns 1 if p2 is in $N_{CS}(p1,t=1)$ , returns 0 otherwise, where $N_{CS}(x,t=1)$ is the CS proximity region for point $x$ with expansion parameter $t=1$ .

CS proximity region is defined with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ and edge regions are based on the center of mass $CM=(1/2,\sqrt{3}/6)$ . Here p1 must lie in the first one-sixth of $T_e$ , which is the triangle with vertices $T(A,D_3,CM)=T((0,0),(1/2,0),CM)$ . If p1 and p2 are distinct and p1 is outside of $T(A,D_3,CM)$ or p2 is outside $T_e$ , it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Usage

IarcCS.Te.onesixth(p1, p2)
IarcCS.Te.onesixth(p1, p2)

Arguments

`p1`	A 2D point whose CS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not.

Value

$I($ p2 is in $N_{CS}(p1,t=1))$ for p1 in the first one-sixth of $T_e$ , $T(A,D_3,CM)$ , that is, returns 1 if p2 is in $N_{CS}(p1,t=1)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard basic triangle case

Description

CS proximity region is defined with respect to the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Edge regions are based on the center, $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the standard basic triangle $T_b$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_b$ . re is the index of the edge region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside $T_b$ , it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation, and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

IarcCSbasic.tri(p1, p2, t, c1, c2, M = c(1, 1, 1), re = NULL)
IarcCSbasic.tri(p1, p2, t, c1, c2, M = c(1, 1, 1), re = NULL)

Arguments

`p1`	A 2D point whose CS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not.
`t`	A positive real number which serves as the expansion parameter in CS proximity region; must be $\ge 1$
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle or circumcenter of $T_b$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_b$ .
`re`	The index of the edge region in $T_b$ containing the point, either `1,2,3` or `NULL` (default is `NULL`).

Value

$I($ p2 is in $N_{CS}(p1,t))$ for points p1 and p2, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)

tau<-2

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcCSbasic.tri(P1,P2,tau,c1,c2,M)

P1<-c(.4,.2)
P2<-c(.5,.26)
IarcCSbasic.tri(P1,P2,tau,c1,c2,M)
IarcCSbasic.tri(P1,P1,tau,c1,c2,M)

#or try
Re<-rel.edge.basic.tri(P1,c1,c2,M)$re
IarcCSbasic.tri(P1,P2,tau,c1,c2,M,Re)
IarcCSbasic.tri(P1,P1,tau,c1,c2,M,Re)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)

tau<-2

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcCSbasic.tri(P1,P2,tau,c1,c2,M)

P1<-c(.4,.2)
P2<-c(.5,.26)
IarcCSbasic.tri(P1,P2,tau,c1,c2,M)
IarcCSbasic.tri(P1,P1,tau,c1,c2,M)

#or try
Re<-rel.edge.basic.tri(P1,c1,c2,M)$re
IarcCSbasic.tri(P1,P2,tau,c1,c2,M,Re)
IarcCSbasic.tri(P1,P1,tau,c1,c2,M,Re)

## End(Not run)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns $I($ p2 is in $N_{CS}(p1,t))$ for points p1 and p2, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ , returns 0 otherwise, where $N_{CS}(x,t)$ is the CS proximity region for point $x$ with expansion parameter $t>0$ . This function is equivalent to IarcCSstd.tri, except that it computes the indicator using the functions IarcCSstd.triRAB, IarcCSstd.triRBC and IarcCSstd.triRAC which are edge-region specific indicator functions. For example, IarcCSstd.triRAB computes $I($ p2 is in $N_{CS}(p1,t))$ for points p1 and p2 when p1 resides in the edge region of edge $AB$ .

CS proximity region is defined with respect to the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$ . re is the index of the edge region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside $T_e$ , it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

IarcCSedge.reg.std.tri(p1, p2, t, M = c(1, 1, 1), re = NULL)
IarcCSedge.reg.std.tri(p1, p2, t, M = c(1, 1, 1), re = NULL)

Arguments

`p1`	A 2D point whose CS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .
`re`	The index of the edge region in $T_e$ containing the point, either `1,2,3` or `NULL` (default is `NULL`).

Value

$I($ p2 is in $N_{CS}(p1,t))$ for p1, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M)
IarcCSstd.tri(Xp[1,],Xp[2,],t,M)

#or try
re<-rel.edge.std.triCM(Xp[1,])$re
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M,re=re)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M)
IarcCSstd.tri(Xp[1,],Xp[2,],t,M)

#or try
re<-rel.edge.std.triCM(Xp[1,])$re
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M,re=re)

## End(Not run)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - end interval case

Description

Returns $I(p_2$ in $N_{CS}(p_1,t))$ for points $p_1$ and $p_2$ , that is, returns 1 if $p_2$ is in $N_{CS}(p_1,t)$ , returns 0 otherwise, where $N_{CS}(x,t)$ is the CS proximity region for point $x$ with expansion parameter $t>0$ for the region outside the interval $(a,b)$ .

rv is the index of the end vertex region $p_1$ resides, with default=NULL, and rv=1 for left end interval and rv=2 for the right end interval. If $p_1$ and $p_2$ are distinct and either of them are inside interval int, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2016)).

Usage

IarcCSend.int(p1, p2, int, t, rv = NULL)
IarcCSend.int(p1, p2, int, t, rv = NULL)

Arguments

`p1`	A 1D point for which the CS proximity region is constructed.
`p2`	A 1D point to check whether it is inside the proximity region or not.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`rv`	Index of the end interval containing the point, either `1,2` or `NULL` (default=`NULL`).

Value

$I(p_2$ in $N_{CS}(p_1,t))$ for points $p_1$ and $p_2$ , that is, returns 1 if $p_2$ is in $N_{CS}(p_1,t)$ (i.e., if there is an arc from $p_1$ to $p_2$ ), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

a<-0; b<-10; int<-c(a,b)
t<-2

IarcCSend.int(15,17,int,t)
IarcCSend.int(15,15,int,t)

IarcCSend.int(1.5,17,int,t)
IarcCSend.int(1.5,1.5,int,t)

IarcCSend.int(-15,17,int,t)

IarcCSend.int(-15,-17,int,t)

a<-0; b<-10; int<-c(a,b)
t<-.5

IarcCSend.int(15,17,int,t)

a<-0; b<-10; int<-c(a,b)
t<-2

IarcCSend.int(15,17,int,t)
IarcCSend.int(15,15,int,t)

IarcCSend.int(1.5,17,int,t)
IarcCSend.int(1.5,1.5,int,t)

IarcCSend.int(-15,17,int,t)

IarcCSend.int(-15,-17,int,t)

a<-0; b<-10; int<-c(a,b)
t<-.5

IarcCSend.int(15,17,int,t)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one interval case

Description

Returns $I(p_2$ in $N_{CS}(p_1,t,c))$ for points $p_1$ and $p_2$ , that is, returns 1 if $p_2$ is in $N_{CS}(p_1,t,c)$ , returns 0 otherwise, where $N_{CS}(x,t,c)$ is the CS proximity region for point $x$ with expansion parameter $t>0$ and centrality parameter $c \in (0,1)$ .

CS proximity region is constructed with respect to the interval $(a,b)$ . This function works whether $p_1$ and $p_2$ are inside or outside the interval int.

Vertex regions for middle intervals are based on the center associated with the centrality parameter $c \in (0,1)$ . If $p_1$ and $p_2$ are identical, then it returns 1 regardless of their locations (i.e., loops are allowed in the digraph).

See also (Ceyhan (2016)).

Usage

IarcCSint(p1, p2, int, t, c = 0.5)
IarcCSint(p1, p2, int, t, c = 0.5)

Arguments

`p1`	A 1D point for which the proximity region is constructed.
`p2`	A 1D point for which it is checked whether it resides in the proximity region of $p_1$ or not.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

$I(p_2$ in $N_{CS}(p_1,t,c))$ for p2, that is, returns 1 if $p_2$ in $N_{CS}(p_1,t,c)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

IarcCSint(7,5,int,t,c)
IarcCSint(17,17,int,t,c)
IarcCSint(15,17,int,t,c)
IarcCSint(1,3,int,t,c)

IarcCSint(-17,17,int,t,c)

IarcCSint(3,5,int,t,c)
IarcCSint(3,3,int,t,c)
IarcCSint(4,5,int,t,c)
IarcCSint(a,5,int,t,c)

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

IarcCSint(7,5,int,t,c)

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

IarcCSint(7,5,int,t,c)
IarcCSint(17,17,int,t,c)
IarcCSint(15,17,int,t,c)
IarcCSint(1,3,int,t,c)

IarcCSint(-17,17,int,t,c)

IarcCSint(3,5,int,t,c)
IarcCSint(3,3,int,t,c)
IarcCSint(4,5,int,t,c)
IarcCSint(a,5,int,t,c)

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

IarcCSint(7,5,int,t,c)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - middle interval case

Description

Returns $I(p_2$ in $N_{CS}(p_1,t,c))$ for points $p_1$ and $p_2$ , that is, returns 1 if $p_2$ is in $N_{CS}(p_1,t,c)$ , returns 0 otherwise, where $N_{CS}(x,t,c)$ is the CS proximity region for point $x$ and is constructed with expansion parameter $t>0$ and centrality parameter $c \in (0,1)$ for the interval $(a,b)$ .

CS proximity regions are defined with respect to the middle interval int and vertex regions are based on the center associated with the centrality parameter $c \in (0,1)$ . For the interval, int $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ . rv is the index of the vertex region $p_1$ resides, with default=NULL.

If $p_1$ and $p_2$ are distinct and either of them are outside interval int, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., loops are allowed in the digraph).

See also (Ceyhan (2016)).

Usage

IarcCSmid.int(p1, p2, int, t, c = 0.5, rv = NULL)
IarcCSmid.int(p1, p2, int, t, c = 0.5, rv = NULL)

Arguments

`p1`, `p2`	1D points; $p_1$ is the point for which the proximity region, $N_{CS}(p_1,t,c)$ is constructed and $p_2$ is the point which the function is checking whether its inside $N_{CS}(p_1,t,c)$ or not.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`rv`	Index of the end interval containing the point, either `1,2` or `NULL` (default is `NULL`).

Value

$I(p_2$ in $N_{CS}(p_1,t,c))$ for points $p_1$ and $p_2$ that is, returns 1 if $p_2$ is in $N_{CS}(p_1,t,c)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

c<-.5
t<-2
a<-0; b<-10; int<-c(a,b)

IarcCSmid.int(7,5,int,t,c)
IarcCSmid.int(7,7,int,t,c)
IarcCSmid.int(7,5,int,t,c=.4)

IarcCSmid.int(1,3,int,t,c)

IarcCSmid.int(9,11,int,t,c)

IarcCSmid.int(19,1,int,t,c)
IarcCSmid.int(19,19,int,t,c)

IarcCSmid.int(3,5,int,t,c)

#or try
Rv<-rel.vert.mid.int(3,int,c)$rv
IarcCSmid.int(3,5,int,t,c,rv=Rv)

IarcCSmid.int(7,5,int,t,c)

c<-.5
t<-2
a<-0; b<-10; int<-c(a,b)

IarcCSmid.int(7,5,int,t,c)
IarcCSmid.int(7,7,int,t,c)
IarcCSmid.int(7,5,int,t,c=.4)

IarcCSmid.int(1,3,int,t,c)

IarcCSmid.int(9,11,int,t,c)

IarcCSmid.int(19,1,int,t,c)
IarcCSmid.int(19,19,int,t,c)

IarcCSmid.int(3,5,int,t,c)

#or try
Rv<-rel.vert.mid.int(3,int,c)$rv
IarcCSmid.int(3,5,int,t,c,rv=Rv)

IarcCSmid.int(7,5,int,t,c)

The indicator for the presence of an arc from a point in set `S` to the point `p` for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns $I($ p in $N_{CS}(x,t)$ for some $x$ in S $)$ , that is, returns 1 if p is in $\cup_{x in S} N_{CS}(x,t)$ , returns 0 otherwise, CS proximity region is constructed with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with the expansion parameter $t>0$ and edge regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$ (which is equivalent to circumcenter of $T_e$ ).

Edges of $T_e$ , $AB$ , $BC$ , $AC$ , are also labeled as edges 3, 1, and 2, respectively. If p is not in S and either p or all points in S are outside $T_e$ , it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

See also (Ceyhan (2012)).

Usage

IarcCSset2pnt.std.tri(S, p, t, M = c(1, 1, 1))
IarcCSset2pnt.std.tri(S, p, t, M = c(1, 1, 1))

Arguments

`S`	A set of 2D points. Presence of an arc from a point in `S` to point `p` is checked by the function.
`p`	A 2D point. Presence of an arc from a point in `S` to point `p` is checked by the function.
`t`	A positive real number which serves as the expansion parameter in CS proximity region in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$ .

Value

$I($ p is in $\cup_{x in S} N_{CS}(x,t))$ , that is, returns 1 if p is in S or inside $N_{CS}(x,t)$ for at least one $x$ in S, returns 0 otherwise. CS proximity region is constructed with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with M-edge regions.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(.5,.5)
IarcCSset2pnt.std.tri(S,Xp[3,],t,M)
IarcCSset2pnt.std.tri(S,Xp[3,],t=1,M)
IarcCSset2pnt.std.tri(S,Xp[3,],t=1.5,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcCSset2pnt.std.tri(S,Xp[3,],t,M)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(.5,.5)
IarcCSset2pnt.std.tri(S,Xp[3,],t,M)
IarcCSset2pnt.std.tri(S,Xp[3,],t=1,M)
IarcCSset2pnt.std.tri(S,Xp[3,],t=1.5,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcCSset2pnt.std.tri(S,Xp[3,],t,M)

## End(Not run)

The indicator for the presence of an arc from a point in set `S` to the point `p` for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case

Description

Returns I(p in $N_{CS}(x,t)$ for some $x$ in S), that is, returns 1 if p in $\cup_{x in S} N_{CS}(x,t)$ , returns 0 otherwise.

CS proximity region is constructed with respect to the triangle tri with the expansion parameter $t>0$ and edge regions are based on the center, $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is $M=(1,1,1)$ i.e., the center of mass of tri.

Edges of tri $=T(A,B,C)$ , $AB$ , $BC$ , $AC$ , are also labeled as edges 3, 1, and 2, respectively. If p is not in S and either p or all points in S are outside tri, it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

Usage

IarcCSset2pnt.tri(S, p, tri, t, M = c(1, 1, 1))
IarcCSset2pnt.tri(S, p, tri, t, M = c(1, 1, 1))

Arguments

`S`	A set of 2D points. Presence of an arc from a point in `S` to point `p` is checked by the function.
`p`	A 2D point. Presence of an arc from a point in `S` to point `p` is checked by the function.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region constructed in the triangle `tri`.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`.

Value

I(p is in $\cup_{x in S} N_{CS}(x,t)$ ), that is, returns 1 if p is in S or inside $N_{CS}(x,t)$ for at least one $x$ in S, returns 0 otherwise where CS proximity region is constructed with respect to the triangle tri

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

tau<-.5

IarcCSset2pnt.tri(S,Xp[3,],Tr,tau,M)
IarcCSset2pnt.tri(S,Xp[3,],Tr,t=1,M)
IarcCSset2pnt.tri(S,Xp[3,],Tr,t=1.5,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcCSset2pnt.tri(S,Xp[3,],Tr,tau,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

tau<-.5

IarcCSset2pnt.tri(S,Xp[3,],Tr,tau,M)
IarcCSset2pnt.tri(S,Xp[3,],Tr,t=1,M)
IarcCSset2pnt.tri(S,Xp[3,],Tr,t=1.5,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcCSset2pnt.tri(S,Xp[3,],Tr,tau,M)

## End(Not run)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

CS proximity region is defined with respect to the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ and vertex regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$ . rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside $T_e$ , it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

IarcCSstd.tri(p1, p2, t, M = c(1, 1, 1), re = NULL)
IarcCSstd.tri(p1, p2, t, M = c(1, 1, 1), re = NULL)

Arguments

`p1`	A 2D point whose CS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .
`re`	The index of the edge region in $T_e$ containing the point, either `1,2,3` or `NULL` (default is `NULL`).

Value

$I($ p2 is in $N_{CS}(p1,t))$ for points p1 and p2, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2) or M=(A+B+C)/3

IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M)
IarcCSstd.tri(c(0,1),Xp[3,],t=2,M)

#or try
Re<-rel.edge.tri(Xp[1,],Te,M) $re
IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M,Re)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2) or M=(A+B+C)/3

IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M)
IarcCSstd.tri(c(0,1),Xp[3,],t=2,M)

#or try
Re<-rel.edge.tri(Xp[1,],Te,M) $re
IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M,Re)

## End(Not run)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case with $t=1$

Description

If p1 and p2 are distinct and either are outside $T_e$ , it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Usage

IarcCSt1.std.tri(p1, p2)
IarcCSt1.std.tri(p1, p2)

Arguments

`p1`	A 2D point whose CS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not.

Value

$I($ p2 is in $N_{CS}(p1,t=1))$ for p1 in $T_e$ that is, returns 1 if p2 is in $N_{CS}(p1,t=1)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

IarcCSt1.std.tri(Xp[1,],Xp[2,])
IarcCSt1.std.tri(c(.2,.5),Xp[2,])

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

IarcCSt1.std.tri(Xp[1,],Xp[2,])
IarcCSt1.std.tri(c(.2,.5),Xp[2,])

## End(Not run)

The indicator for the presence of an arc from one point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs)

Description

Returns $I($ p2 is in $N_{CS}(p1,t))$ for points p1 and p2, that is, returns 1 if p2 is in $NCS(p1,t)$ , returns 0 otherwise, where $N_{CS}(x,t)$ is the CS proximity region for point $x$ with the expansion parameter $t>0$ .

CS proximity region is constructed with respect to the triangle tri and edge regions are based on the center, $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of tri or based on the circumcenter of tri. re is the index of the edge region p resides, with default=NULL

If p1 and p2 are distinct and either of them are outside tri, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

IarcCStri(p1, p2, tri, t, M, re = NULL)
IarcCStri(p1, p2, tri, t, M, re = NULL)

Arguments

`p1`	A 2D point whose CS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`.
`re`	Index of the `M`-edge region containing the point `p`, either `1,2,3` or `NULL` (default is `NULL`).

Value

I(p2 is in $NCS(p1,t)$ ) for p1, that is, returns 1 if p2 is in $NCS(p1,t)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
tau<-1.5

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

n<-10
set.seed(1)
Xp<-runif.tri(n,Tr)$g

IarcCStri(Xp[1,],Xp[2,],Tr,tau,M)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcCStri(P1,P2,Tr,tau,M)

#or try
re<-rel.edges.tri(P1,Tr,M)$re
IarcCStri(P1,P2,Tr,tau,M,re)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
tau<-1.5

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

n<-10
set.seed(1)
Xp<-runif.tri(n,Tr)$g

IarcCStri(Xp[1,],Xp[2,],Tr,tau,M)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcCStri(P1,P2,Tr,tau,M)

#or try
re<-rel.edges.tri(P1,Tr,M)$re
IarcCStri(P1,P2,Tr,tau,M,re)

## End(Not run)

An alternative to the function `IarcCStri` which yields the indicator for the presence of an arc from one point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs)

Description

CS proximity region is constructed with respect to the triangle tri and edge regions are based on the center of mass, $CM$ . re is the index of the $CM$ -edge region p resides, with default=NULL but must be provided as vertices $(y_1,y_2,y_3)$ for $re=3$ as rbind(y2,y3,y1) for $re=1$ and as rbind(y1,y3,y2) for $re=2$ for triangle $T(y_1,y_2,y_3)$ .

If p1 and p2 are distinct and either of them are outside tri, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

IarcCStri.alt(p1, p2, tri, t, re = NULL)
IarcCStri.alt(p1, p2, tri, t, re = NULL)

Arguments

`p1`	A 2D point whose CS proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the CS proximity region of `p1` or not.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`re`	Index of the $CM$ -edge region containing the point `p`, either `1,2,3` or `NULL`, default=`NULL` but must be provided (row-wise) as vertices $(y_1,y_2,y_3)$ for `re=3` as $(y_2,y_3,y_1)$ for `re=1` and as $(y_1,y_3,y_2)$ for `re=2` for triangle $T(y_1,y_2,y_3)$ .

Value

$I($ p2 is in $N_{CS}(p1,t))$ for p1, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ , returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.6,2);
Tr<-rbind(A,B,C);
t<-1.5

P1<-c(.4,.2)
P2<-c(1.8,.5)
IarcCStri(P1,P2,Tr,t,M=c(1,1,1))
IarcCStri.alt(P1,P2,Tr,t)

IarcCStri(P2,P1,Tr,t,M=c(1,1,1))
IarcCStri.alt(P2,P1,Tr,t)

#or try
re<-rel.edges.triCM(P1,Tr)$re
IarcCStri(P1,P2,Tr,t,M=c(1,1,1),re)
IarcCStri.alt(P1,P2,Tr,t,re)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.6,2);
Tr<-rbind(A,B,C);
t<-1.5

P1<-c(.4,.2)
P2<-c(1.8,.5)
IarcCStri(P1,P2,Tr,t,M=c(1,1,1))
IarcCStri.alt(P1,P2,Tr,t)

IarcCStri(P2,P1,Tr,t,M=c(1,1,1))
IarcCStri.alt(P2,P1,Tr,t)

#or try
re<-rel.edges.triCM(P1,Tr)$re
IarcCStri(P1,P2,Tr,t,M=c(1,1,1),re)
IarcCStri.alt(P1,P2,Tr,t,re)

## End(Not run)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic triangle case

Description

Returns $I($ p2 is in $N_{PE}(p1,r))$ for points p1 and p2 in the standard basic triangle, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , and returns 0 otherwise, where $N_{PE}(x,r)$ is the PE proximity region for point $x$ with expansion parameter $r \ge 1$ .

PE proximity region is defined with respect to the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Vertex regions are based on the center, $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the standard basic triangle $T_b$ or based on circumcenter of $T_b$ ; default is $M=(1,1,1)$ , i.e., the center of mass of $T_b$ . rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside $T_b$ , it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010); Ceyhan et al. (2006)).

Usage

IarcPEbasic.tri(p1, p2, r, c1, c2, M = c(1, 1, 1), rv = NULL)
IarcPEbasic.tri(p1, p2, r, c1, c2, M = c(1, 1, 1), rv = NULL)

Arguments

`p1`	A 2D point whose PE proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the PE proximity region of `p1` or not.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle or circumcenter of $T_b$ which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of $T_b$ .
`rv`	The index of the vertex region in $T_b$ containing the point, either `1,2,3` or `NULL` (default is `NULL`).

Value

$I($ p2 is in $N_{PE}(p1,r))$ for points p1 and p2 in the standard basic triangle, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , and returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)

r<-2

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcPEbasic.tri(P1,P2,r,c1,c2,M)

P1<-c(.4,.2)
P2<-c(.5,.26)
IarcPEbasic.tri(P1,P2,r,c1,c2,M)
IarcPEbasic.tri(P2,P1,r,c1,c2,M)

#or try
Rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
IarcPEbasic.tri(P1,P2,r,c1,c2,M,Rv)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)

r<-2

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcPEbasic.tri(P1,P2,r,c1,c2,M)

P1<-c(.4,.2)
P2<-c(.5,.26)
IarcPEbasic.tri(P1,P2,r,c1,c2,M)
IarcPEbasic.tri(P2,P1,r,c1,c2,M)

#or try
Rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
IarcPEbasic.tri(P1,P2,r,c1,c2,M,Rv)

## End(Not run)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - end interval case

Description

Returns $I(p_2 \in N_{PE}(p_1,r))$ for points $p_1$ and $p_2$ , that is, returns 1 if $p_2$ is in $N_{PE}(p_1,r)$ , returns 0 otherwise, where $N_{PE}(x,r)$ is the PE proximity region for point $x$ with expansion parameter $r \ge 1$ for the region outside the interval $(a,b)$ .

See also (Ceyhan (2012)).

Usage

IarcPEend.int(p1, p2, int, r, rv = NULL)
IarcPEend.int(p1, p2, int, r, rv = NULL)

Arguments

`p1`	A 1D point whose PE proximity region is constructed.
`p2`	A 1D point. The function determines whether $p_2$ is inside the PE proximity region of $p_1$ or not.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`rv`	Index of the end interval containing the point, either `1,2` or `NULL` (default is `NULL`).

Value

$I(p_2 \in N_{PE}(p_1,r))$ for points $p_1$ and $p_2$ , that is, returns 1 if $p_2$ is in $N_{PE}(p_1,r)$ (i.e., if there is an arc from $p_1$ to $p_2$ ), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

a<-0; b<-10; int<-c(a,b)
r<-2

IarcPEend.int(15,17,int,r)
IarcPEend.int(1.5,17,int,r)
IarcPEend.int(-15,17,int,r)

a<-0; b<-10; int<-c(a,b)
r<-2

IarcPEend.int(15,17,int,r)
IarcPEend.int(1.5,17,int,r)
IarcPEend.int(-15,17,int,r)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one interval case

Description

Returns $I(p_2 \in N_{PE}(p_1,r,c))$ for points $p_1$ and $p_2$ , that is, returns 1 if $p_2$ is in $N_{PE}(p_1,r,c)$ , returns 0 otherwise, where $N_{PE}(x,r,c)$ is the PE proximity region for point $x$ with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ .

PE proximity region is constructed with respect to the interval $(a,b)$ . This function works whether $p_1$ and $p_2$ are inside or outside the interval int.

See also (Ceyhan (2012)).

Usage

IarcPEint(p1, p2, int, r, c = 0.5)
IarcPEint(p1, p2, int, r, c = 0.5)

Arguments

`p1`	A 1D point for which the proximity region is constructed.
`p2`	A 1D point for which it is checked whether it resides in the proximity region of $p_1$ or not.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

$I(p_2 \in N_{PE}(p_1,r,c))$ , that is, returns 1 if $p_2$ in $N_{PE}(p_1,r,c)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

IarcPEint(7,5,int,r,c)
IarcPEint(15,17,int,r,c)
IarcPEint(1,3,int,r,c)

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

IarcPEint(7,5,int,r,c)
IarcPEint(15,17,int,r,c)
IarcPEint(1,3,int,r,c)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case

Description

Returns $I(p_2 \in N_{PE}(p_1,r,c))$ for points $p_1$ and $p_2$ , that is, returns 1 if $p_2$ is in $N_{PE}(p_1,r,c)$ , returns 0 otherwise, where $N_{PE}(x,r,c)$ is the PE proximity region for point $x$ and is constructed with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ for the interval $(a,b)$ .

PE proximity regions are defined with respect to the middle interval int and vertex regions are based on the center associated with the centrality parameter $c \in (0,1)$ . For the interval, int $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ . rv is the index of the vertex region $p_1$ resides, with default=NULL. If $p_1$ and $p_2$ are distinct and either of them are outside interval int, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., loops are allowed in the digraph).

See also (Ceyhan (2012, 2016)).

Usage

IarcPEmid.int(p1, x2, int, r, c = 0.5, rv = NULL)
IarcPEmid.int(p1, x2, int, r, c = 0.5, rv = NULL)

Arguments

`p1`, `x2`	1D points; $p_1$ is the point for which the proximity region, $N_{PE}(p_1,r,c)$ is constructed and $p_2$ is the point which the function is checking whether its inside $N_{PE}(p_1,r,c)$ or not.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`rv`	The index of the vertex region $p_1$ resides, with default=`NULL`.

Value

$I(p_2 \in N_{PE}(p_1,r,c))$ for points $p_1$ and $p_2$ that is, returns 1 if $p_2$ is in $N_{PE}(p_1,r,c)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

IarcPEmid.int(7,5,int,r,c)
IarcPEmid.int(1,3,int,r,c)

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

IarcPEmid.int(7,5,int,r,c)
IarcPEmid.int(1,3,int,r,c)

The indicator for the presence of an arc from a point in set `S` to the point `p` or Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns $I($ p in $N_{PE}(x,r)$ for some $x$ in S $)$ for S, in the standard equilateral triangle, that is, returns 1 if p is in $\cup_{x in S}N_{PE}(x,r)$ , and returns 0 otherwise.

PE proximity region is constructed with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with the expansion parameter $r \ge 1$ and vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ , i.e., the center of mass of $T_e$ (which is equivalent to the circumcenter for $T_e$ ).

Vertices of $T_e$ are also labeled as 1, 2, and 3, respectively. If p is not in S and either p or all points in S are outside $T_e$ , it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

Usage

IarcPEset2pnt.std.tri(S, p, r, M = c(1, 1, 1))
IarcPEset2pnt.std.tri(S, p, r, M = c(1, 1, 1))

Arguments

`S`	A set of 2D points. Presence of an arc from a point in `S` to point `p` is checked by the function.
`p`	A 2D point. Presence of an arc from a point in `S` to point `p` is checked by the function.
`r`	A positive real number which serves as the expansion parameter in PE proximity region in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ ; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$ .

Value

$I($ p is in U_x in S $N_{PE}(x,r))$ for S in the standard equilateral triangle, that is, returns 1 if p is in S or inside $N_{PE}(x,r)$ for at least one $x$ in S, and returns 0 otherwise. PE proximity region is constructed with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with M-vertex regions

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(.5,.5)
IarcPEset2pnt.std.tri(S,Xp[3,],r,M)
IarcPEset2pnt.std.tri(S,Xp[3,],r=1,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcPEset2pnt.std.tri(S,Xp[3,],r,M)

IarcPEset2pnt.std.tri(S,Xp[6,],r,M)
IarcPEset2pnt.std.tri(S,Xp[6,],r=1.25,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcPEset2pnt.std.tri(Xp,P,r,M)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(.5,.5)
IarcPEset2pnt.std.tri(S,Xp[3,],r,M)
IarcPEset2pnt.std.tri(S,Xp[3,],r=1,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcPEset2pnt.std.tri(S,Xp[3,],r,M)

IarcPEset2pnt.std.tri(S,Xp[6,],r,M)
IarcPEset2pnt.std.tri(S,Xp[6,],r=1.25,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcPEset2pnt.std.tri(Xp,P,r,M)

## End(Not run)

The indicator for the presence of an arc from a point in set `S` to the point `p` for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns $I($ p in $N_{PE}(x,r)$ for some $x$ in S $)$ , that is, returns 1 if p is in $\cup_{x in S}N_{PE}(x,r)$ , and returns 0 otherwise.

PE proximity region is constructed with respect to the triangle tri with the expansion parameter $r \ge 1$ and vertex regions are based on the center, $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri or based on the circumcenter of tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri. Vertices of tri are also labeled as 1, 2, and 3, respectively.

If p is not in S and either p or all points in S are outside tri, it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

Usage

IarcPEset2pnt.tri(S, p, tri, r, M = c(1, 1, 1))
IarcPEset2pnt.tri(S, p, tri, r, M = c(1, 1, 1))

Arguments

`S`	A set of 2D points. Presence of an arc from a point in `S` to point `p` is checked by the function.
`p`	A 2D point. Presence of an arc from a point in `S` to point `p` is checked by the function.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region constructed in the triangle `tri`; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.

Value

I(p is in U_x in S N_PE(x,r)), that is, returns 1 if p is in S or inside $N_{PE}(x,r)$ for at least one $x$ in S, and returns 0 otherwise, where PE proximity region is constructed with respect to the triangle tri

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)
IarcPEset2pnt.tri(S,Xp[3,],r=1,Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcPEset2pnt.tri(Xp,P,Tr,r,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)
IarcPEset2pnt.tri(S,Xp[3,],r=1,Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcPEset2pnt.tri(Xp,P,Tr,r,M)

## End(Not run)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case

Description

Returns $I($ p2 is in $N_{PE}(p1,r))$ for points p1 and p2, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , returns 0 otherwise, where $N_{PE}(x,r)$ is the PE proximity region for point $x$ with expansion parameter $r \ge 1$ .

PE proximity region is defined with respect to the standard regular tetrahedron $T_h=T(v=1,v=2,v=3,v=4)=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))$ and vertex regions are based on the circumcenter (which is equivalent to the center of mass for standard regular tetrahedron) of $T_h$ . rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside $T_h$ , it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010)).

Usage

IarcPEstd.tetra(p1, p2, r, rv = NULL)
IarcPEstd.tetra(p1, p2, r, rv = NULL)

Arguments

`p1`	A 3D point whose PE proximity region is constructed.
`p2`	A 3D point. The function determines whether `p2` is inside the PE proximity region of `p1` or not.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`rv`	Index of the vertex region containing the point, either `1,2,3,4` (default is `NULL`).

Value

$I($ p2 is in $N_{PE}(p1,r))$ for points p1 and p2, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-3  #try also n<-20
Xp<-runif.std.tetra(n)$g
r<-1.5
IarcPEstd.tetra(Xp[1,],Xp[3,],r)
IarcPEstd.tetra(c(.4,.4,.4),c(.5,.5,.5),r)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
IarcPEstd.tetra(Xp[1,],Xp[3,],r,rv=RV)

P1<-c(.1,.1,.1)
P2<-c(.5,.5,.5)
IarcPEstd.tetra(P1,P2,r)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-3  #try also n<-20
Xp<-runif.std.tetra(n)$g
r<-1.5
IarcPEstd.tetra(Xp[1,],Xp[3,],r)
IarcPEstd.tetra(c(.4,.4,.4),c(.5,.5,.5),r)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
IarcPEstd.tetra(Xp[1,],Xp[3,],r,rv=RV)

P1<-c(.1,.1,.1)
P2<-c(.5,.5,.5)
IarcPEstd.tetra(P1,P2,r)

## End(Not run)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns $I($ p2 is in $N_{PE}(p1,r))$ for points p1 and p2 in the standard equilateral triangle, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , and returns 0 otherwise, where $N_{PE}(x,r)$ is the PE proximity region for point $x$ with expansion parameter $r \ge 1$ .

PE proximity region is defined with respect to the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ and vertex regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ , i.e., the center of mass of $T_e$ . rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside $T_e$ , it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

IarcPEstd.tri(p1, p2, r, M = c(1, 1, 1), rv = NULL)
IarcPEstd.tri(p1, p2, r, M = c(1, 1, 1), rv = NULL)

Arguments

`p1`	A 2D point whose PE proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the PE proximity region of `p1` or not.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .
`rv`	The index of the vertex region in $T_e$ containing the point, either `1,2,3` or `NULL` (default is `NULL`).

Value

$I($ p2 is in $N_{PE}(p1,r))$ for points p1 and p2 in the standard equilateral triangle, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , and returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

IarcPEstd.tri(Xp[1,],Xp[3,],r=1.5,M)
IarcPEstd.tri(Xp[1,],Xp[3,],r=2,M)

#or try
Rv<-rel.vert.std.triCM(Xp[1,])$rv
IarcPEstd.tri(Xp[1,],Xp[3,],r=2,rv=Rv)

P1<-c(.4,.2)
P2<-c(.5,.26)
r<-2
IarcPEstd.tri(P1,P2,r,M)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

IarcPEstd.tri(Xp[1,],Xp[3,],r=1.5,M)
IarcPEstd.tri(Xp[1,],Xp[3,],r=2,M)

#or try
Rv<-rel.vert.std.triCM(Xp[1,])$rv
IarcPEstd.tri(Xp[1,],Xp[3,],r=2,rv=Rv)

P1<-c(.4,.2)
P2<-c(.5,.26)
r<-2
IarcPEstd.tri(P1,P2,r,M)

## End(Not run)

The indicator for the presence of an arc from one 3D point to another 3D point for Proportional Edge Proximity Catch Digraphs (PE-PCDs)

Description

Returns $I($ p2 is in $N_{PE}(p1,r))$ for 3D points p1 and p2, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , returns 0 otherwise, where N_PE(x,r) is the PE proximity region for point $x$ with the expansion parameter $r \ge 1$ .

PE proximity region is constructed with respect to the tetrahedron th and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM". rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside th, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010)).

Usage

IarcPEtetra(p1, p2, th, r, M = "CM", rv = NULL)
IarcPEtetra(p1, p2, th, r, M = "CM", rv = NULL)

Arguments

`p1`	A 3D point whose PE proximity region is constructed.
`p2`	A 3D point. The function determines whether `p2` is inside the PE proximity region of `p1` or not.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.
`rv`	Index of the `M`-vertex region containing the point, either `1,2,3,4` (default is `NULL`).

Value

$I($ p2 is in $N_{PE}(p1,r))$ for p1, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-3  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.5

IarcPEtetra(Xp[1,],Xp[2,],tetra,r)  #uses the default M="CM"
IarcPEtetra(Xp[1,],Xp[2,],tetra,r,M)

IarcPEtetra(c(.4,.4,.4),c(.5,.5,.5),tetra,r,M)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
IarcPEtetra(Xp[1,],Xp[3,],tetra,r,M,rv=RV)

P1<-c(.1,.1,.1)
P2<-c(.5,.5,.5)
IarcPEtetra(P1,P2,tetra,r,M)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-3  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.5

IarcPEtetra(Xp[1,],Xp[2,],tetra,r)  #uses the default M="CM"
IarcPEtetra(Xp[1,],Xp[2,],tetra,r,M)

IarcPEtetra(c(.4,.4,.4),c(.5,.5,.5),tetra,r,M)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
IarcPEtetra(Xp[1,],Xp[3,],tetra,r,M,rv=RV)

P1<-c(.1,.1,.1)
P2<-c(.5,.5,.5)
IarcPEtetra(P1,P2,tetra,r,M)

## End(Not run)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns $I($ p2 is in $N_{PE}(p1,r))$ for points p1 and p2, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , and returns 0 otherwise, where $N_{PE}(x,r)$ is the PE proximity region for point $x$ with the expansion parameter $r \ge 1$ .

PE proximity region is constructed with respect to the triangle tri and vertex regions are based on the center, $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of tri or based on the circumcenter of tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside tri, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)).

Usage

IarcPEtri(p1, p2, tri, r, M = c(1, 1, 1), rv = NULL)
IarcPEtri(p1, p2, tri, r, M = c(1, 1, 1), rv = NULL)

Arguments

`p1`	A 2D point whose PE proximity region is constructed.
`p2`	A 2D point. The function determines whether `p2` is inside the PE proximity region of `p1` or not.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.
`rv`	Index of the `M`-vertex region containing the point, either `1,2,3` or `NULL` (default is `NULL`).

Value

$I($ p2 is in $N_{PE}(p1,r))$ for points p1 and p2, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ , and returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0);

r<-1.5

n<-3
set.seed(1)
Xp<-runif.tri(n,Tr)$g

IarcPEtri(Xp[1,],Xp[2,],Tr,r,M)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcPEtri(P1,P2,Tr,r,M)

P1<-c(.4,.2)
P2<-c(1.8,.5)
IarcPEtri(P1,P2,Tr,r,M)
IarcPEtri(P2,P1,Tr,r,M)

M<-c(1.3,1.3)
r<-2

#or try
Rv<-rel.vert.tri(P1,Tr,M)$rv
IarcPEtri(P1,P2,Tr,r,M,Rv)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0);

r<-1.5

n<-3
set.seed(1)
Xp<-runif.tri(n,Tr)$g

IarcPEtri(Xp[1,],Xp[2,],Tr,r,M)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcPEtri(P1,P2,Tr,r,M)

P1<-c(.4,.2)
P2<-c(1.8,.5)
IarcPEtri(P1,P2,Tr,r,M)
IarcPEtri(P2,P1,Tr,r,M)

M<-c(1.3,1.3)
r<-2

#or try
Rv<-rel.vert.tri(P1,Tr,M)$rv
IarcPEtri(P1,P2,Tr,r,M,Rv)

## End(Not run)

Indicator for an upper bound for the domination number by the exact algorithm

Description

Returns 1 if the domination number is less than or equal to the prespecified value k and also the indices (i.e., row numbers) of a dominating set of size k based on the incidence matrix Inc.Mat of a graph or a digraph. Here the row number in the incidence matrix corresponds to the index of the vertex (i.e., index of the data point). The function works whether loops are allowed or not (i.e., whether the first diagonal is all 1 or all 0). It takes a rather long time for large number of vertices (i.e., large number of row numbers).

Usage

Idom.num.up.bnd(Inc.Mat, k)
Idom.num.up.bnd(Inc.Mat, k)

Arguments

`Inc.Mat`	A square matrix consisting of 0's and 1's which represents the incidence matrix of a graph or digraph.
`k`	A positive integer for the upper bound (to be checked) for the domination number.

Value

A list with two elements

`dom.up.bnd`	The upper bound (to be checked) for the domination number. It is prespecified as `k` in the function arguments.
`Idom.num.up.bnd`	The indicator for the upper bound for domination number of the graph or digraph being the specified value `k` or not. It returns 1 if the upper bound is `k`, and 0 otherwise based on the incidence matrix `Inc.Mat` of the graph or digraph.
`ind.dom.set`	Indices of the rows in the incidence matrix `Inc.Mat` that correspond to the vertices in the dominating set of size `k` if it exists, otherwise it yields `NULL`.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
n<-10
M<-matrix(sample(c(0,1),n^2,replace=TRUE),nrow=n)
diag(M)<-1

dom.num.greedy(M)
Idom.num.up.bnd(M,2)

for (k in 1:n)
print(c(k,Idom.num.up.bnd(M,k)))

## End(Not run)

## Not run: 
n<-10
M<-matrix(sample(c(0,1),n^2,replace=TRUE),nrow=n)
diag(M)<-1

dom.num.greedy(M)
Idom.num.up.bnd(M,2)

for (k in 1:n)
print(c(k,Idom.num.up.bnd(M,k)))

## End(Not run)

The indicator for a point being a dominating point for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case

Description

Returns I(p is a dominating point of the AS-PCD) where the vertices of the AS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point of AS-PCD, returns 0 otherwise. AS proximity regions are defined with respect to the standard basic triangle, $T_b$ , $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

Vertex regions are based on the center M="CC" for circumcenter of $T_b$ ; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_b$ ; default is M="CC". Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as $1,2,3$ in the order they are stacked row-wise.

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1ASbasic.tri(p, Xp, c1, c2, M = "CC", rv = NULL, ch.data.pnt = FALSE)
Idom.num1ASbasic.tri(p, Xp, c1, c2, M = "CC", rv = NULL, ch.data.pnt = FALSE)

Arguments

`p`	A 2D point that is to be tested for being a dominating point or not of the AS-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the AS-PCD.
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle $T_b$ or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$ ; default is `M="CC"` i.e., the circumcenter of $T_b$ .
`rv`	Index of the vertex whose region contains point `p`, `rv` takes the vertex labels as $1,2,3$ as in the row order of the vertices in $T_b$ .
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

I(p is a dominating point of the AS-PCD) where the vertices of the AS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Idom.num1ASbasic.tri(Xp[1,],Xp,c1,c2,M)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1ASbasic.tri(Xp[i,],Xp,c1,c2,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
Idom.num1ASbasic.tri(Xp[1,],Xp,c1,c2,M,Rv)

Idom.num1ASbasic.tri(c(.2,.4),Xp,c1,c2,M)
Idom.num1ASbasic.tri(c(.2,.4),c(.2,.4),c1,c2,M)

Xp2<-rbind(Xp,c(.2,.4))
Idom.num1ASbasic.tri(Xp[1,],Xp2,c1,c2,M)

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges.basic.tri(c1,c2,M)
}

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)

txt<-rbind(Tb,cent,Ds)
xc<-txt[,1]+c(-.03,.03,.02,.06,.06,-0.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.0,.03,.03,-.03)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1ASbasic.tri(c(.4,.2),Xp,c1,c2,M)

Idom.num1ASbasic.tri(c(.5,.11),Xp,c1,c2,M)

Idom.num1ASbasic.tri(c(.5,.11),Xp,c1,c2,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since the point is not in the standard basic triangle

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Idom.num1ASbasic.tri(Xp[1,],Xp,c1,c2,M)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1ASbasic.tri(Xp[i,],Xp,c1,c2,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
Idom.num1ASbasic.tri(Xp[1,],Xp,c1,c2,M,Rv)

Idom.num1ASbasic.tri(c(.2,.4),Xp,c1,c2,M)
Idom.num1ASbasic.tri(c(.2,.4),c(.2,.4),c1,c2,M)

Xp2<-rbind(Xp,c(.2,.4))
Idom.num1ASbasic.tri(Xp[1,],Xp2,c1,c2,M)

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges.basic.tri(c1,c2,M)
}

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)

txt<-rbind(Tb,cent,Ds)
xc<-txt[,1]+c(-.03,.03,.02,.06,.06,-0.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.0,.03,.03,-.03)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1ASbasic.tri(c(.4,.2),Xp,c1,c2,M)

Idom.num1ASbasic.tri(c(.5,.11),Xp,c1,c2,M)

Idom.num1ASbasic.tri(c(.5,.11),Xp,c1,c2,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since the point is not in the standard basic triangle

## End(Not run)

The indicator for a point being a dominating point for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns I(p is a dominating point of the AS-PCD whose vertices are the 2D data set Xp), that is, returns 1 if p is a dominating point of AS-PCD, returns 0 otherwise. Point, p, is in the region of vertex rv (default is NULL); vertices are labeled as $1,2,3$ in the order they are stacked row-wise in tri.

AS proximity regions are defined with respect to the triangle tri and vertex regions are based on the center M="CC" for circumcenter of tri; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is M="CC" the circumcenter of tri.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1AStri(p, Xp, tri, M = "CC", rv = NULL, ch.data.pnt = FALSE)
Idom.num1AStri(p, Xp, tri, M = "CC", rv = NULL, ch.data.pnt = FALSE)

Arguments

`p`	A 2D point that is to be tested for being a dominating point or not of the AS-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the AS-PCD.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$ ; default is `M="CC"` i.e., the circumcenter of `tri`.
`rv`	Index of the vertex whose region contains point `p`, `rv` takes the vertex labels as $1,2,3$ as in the row order of the vertices in `tri`.
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

I(p is a dominating point of the AS-PCD whose vertices are the 2D data set Xp), that is, returns 1 if p is a dominating point of the AS-PCD, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Idom.num1AStri(Xp[1,],Xp,Tr,M)
Idom.num1AStri(Xp[1,],Xp[1,],Tr,M)
Idom.num1AStri(c(1.5,1.5),c(1.6,1),Tr,M)
Idom.num1AStri(c(1.6,1),c(1.5,1.5),Tr,M)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1AStri(Xp[i,],Xp,Tr,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.triCC(Xp[1,],Tr)$rv
Idom.num1AStri(Xp[1,],Xp,Tr,M,Rv)

Idom.num1AStri(c(.2,.4),Xp,Tr,M)
Idom.num1AStri(c(.2,.4),c(.2,.4),Tr,M)

Xp2<-rbind(Xp,c(.2,.4))
Idom.num1AStri(Xp[1,],Xp2,Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

CC<-circumcenter.tri(Tr)  #the circumcenter

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)

txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1AStri(c(1.5,1.1),Xp,Tr,M)

Idom.num1AStri(c(1.5,1.1),Xp,Tr,M)

Idom.num1AStri(c(1.5,1.1),Xp,Tr,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since point p is not a data point in Xp

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Idom.num1AStri(Xp[1,],Xp,Tr,M)
Idom.num1AStri(Xp[1,],Xp[1,],Tr,M)
Idom.num1AStri(c(1.5,1.5),c(1.6,1),Tr,M)
Idom.num1AStri(c(1.6,1),c(1.5,1.5),Tr,M)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1AStri(Xp[i,],Xp,Tr,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.triCC(Xp[1,],Tr)$rv
Idom.num1AStri(Xp[1,],Xp,Tr,M,Rv)

Idom.num1AStri(c(.2,.4),Xp,Tr,M)
Idom.num1AStri(c(.2,.4),c(.2,.4),Tr,M)

Xp2<-rbind(Xp,c(.2,.4))
Idom.num1AStri(Xp[1,],Xp2,Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

CC<-circumcenter.tri(Tr)  #the circumcenter

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)

txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1AStri(c(1.5,1.1),Xp,Tr,M)

Idom.num1AStri(c(1.5,1.1),Xp,Tr,M)

Idom.num1AStri(c(1.5,1.1),Xp,Tr,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since point p is not a data point in Xp

## End(Not run)

The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - first one-sixth of the standard equilateral triangle case

Description

Returns $I($ p is a dominating point of the 2D data set Xp of CS-PCD $)$ in the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ , that is, returns 1 if p is a dominating point of CS-PCD, returns 0 otherwise.

Point, p, must lie in the first one-sixth of $T_e$ , which is the triangle with vertices $T(A,D_3,CM)=T((0,0),(1/2,0),CM)$ .

CS proximity region is constructed with respect to $T_e$ with expansion parameter $t=1$ .

Usage

Idom.num1CS.Te.onesixth(p, Xp, ch.data.pnt = FALSE)
Idom.num1CS.Te.onesixth(p, Xp, ch.data.pnt = FALSE)

Arguments

`p`	A 2D point that is to be tested for being a dominating point or not of the CS-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the CS-PCD.
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

$I($ p is a dominating point of the CS-PCD $)$ where the vertices of the CS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) for an interval

Description

Returns $I($ p is a dominating point of CS-PCD $)$ where the vertices of the CS-PCD are the 1D data set Xp).

CS proximity region is defined with respect to the interval int with an expansion parameter, $t>0$ , and a centrality parameter, $c \in (0,1)$ , so arcs may exist for Xp points inside the interval int $=(a,b)$ .

Vertex regions are based on the center associated with the centrality parameter $c \in (0,1)$ . rv is the index of the vertex region p resides, with default=NULL.

Usage

Idom.num1CSint(p, Xp, int, t, c = 0.5, rv = NULL, ch.data.pnt = FALSE)
Idom.num1CSint(p, Xp, int, t, c = 0.5, rv = NULL, ch.data.pnt = FALSE)

Arguments

`p`	A 1D point that is to be tested for being a dominating point or not of the CS-PCD.
`Xp`	A set of 1D points which constitutes the vertices of the CS-PCD.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`rv`	Index of the vertex region in which the point resides, either `1,2` or `NULL` (default is `NULL`).
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

$I($ p is a dominating point of CS-PCD $)$ where the vertices of the CS-PCD are the 1D data set Xp), that is, returns 1 if p is a dominating point, returns 0 otherwise

Author(s)

Elvan Ceyhan

Examples

t<-2
c<-.4
a<-0; b<-10; int<-c(a,b)

Mc<-centerMc(int,c)
n<-10

set.seed(1)
Xp<-runif(n,a,b)

Idom.num1CSint(Xp[5],Xp,int,t,c)

Idom.num1CSint(2,Xp,int,t,c,ch.data.pnt = FALSE)
#gives an error if ch.data.pnt = TRUE since p is not a data point in Xp

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSint(Xp[i],Xp,int,t,c))}

ind.gam1<-which(gam.vec==1)
ind.gam1

domset<-Xp[ind.gam1]
if (length(ind.gam1)==0)
{domset<-NA}

#or try
Rv<-rel.vert.mid.int(Xp[5],int,c)$rv
Idom.num1CSint(Xp[5],Xp,int,t,c,Rv)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",xlim=Xlim+xd*c(-.05,.05))
abline(h=0)
abline(v=c(a,b,Mc),col=c(1,1,2),lty=2)
points(cbind(Xp,0))
points(cbind(domset,0),pch=4,col=2)
text(cbind(c(a,b,Mc),-0.1),c("a","b","Mc"))

Idom.num1CSint(Xp[5],Xp,int,t,c)

n<-10
Xp2<-runif(n,a+b,b+10)
Idom.num1CSint(5,Xp2,int,t,c)

t<-2
c<-.4
a<-0; b<-10; int<-c(a,b)

Mc<-centerMc(int,c)
n<-10

set.seed(1)
Xp<-runif(n,a,b)

Idom.num1CSint(Xp[5],Xp,int,t,c)

Idom.num1CSint(2,Xp,int,t,c,ch.data.pnt = FALSE)
#gives an error if ch.data.pnt = TRUE since p is not a data point in Xp

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSint(Xp[i],Xp,int,t,c))}

ind.gam1<-which(gam.vec==1)
ind.gam1

domset<-Xp[ind.gam1]
if (length(ind.gam1)==0)
{domset<-NA}

#or try
Rv<-rel.vert.mid.int(Xp[5],int,c)$rv
Idom.num1CSint(Xp[5],Xp,int,t,c,Rv)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",xlim=Xlim+xd*c(-.05,.05))
abline(h=0)
abline(v=c(a,b,Mc),col=c(1,1,2),lty=2)
points(cbind(Xp,0))
points(cbind(domset,0),pch=4,col=2)
text(cbind(c(a,b,Mc),-0.1),c("a","b","Mc"))

Idom.num1CSint(Xp[5],Xp,int,t,c)

n<-10
Xp2<-runif(n,a+b,b+10)
Idom.num1CSint(5,Xp2,int,t,c)

The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns $I($ p is a dominating point of the CS-PCD $)$ where the vertices of the CS-PCD are the 2D data set Xp in the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ , that is, returns 1 if p is a dominating point of CS-PCD, returns 0 otherwise.

CS proximity region is constructed with respect to $T_e$ with expansion parameter $t>0$ and edge regions are based on center of mass $CM=(1/2,\sqrt{3}/6)$ .

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1CSstd.tri(p, Xp, t, ch.data.pnt = FALSE)
Idom.num1CSstd.tri(p, Xp, t, ch.data.pnt = FALSE)

Arguments

`p`	A 2D point that is to be tested for being a dominating point or not of the CS-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the CS-PCD.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

$I($ p is a dominating point of the CS-PCD $)$ where the vertices of the CS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
Te<-rbind(A,B,C);
t<-1.5
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num1CSstd.tri(Xp[3,],Xp,t)
Idom.num1CSstd.tri(c(1,2),c(1,2),t)
Idom.num1CSstd.tri(c(1,2),c(1,2),t,ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSstd.tri(Xp[i,],Xp,t))}

ind.gam1<-which(gam.vec==1)
ind.gam1

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE);
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.02,.01,.05)
yc<-txt[,2]+c(.02,.02,.03,.02)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

Idom.num1CSstd.tri(c(1,2),Xp,t,ch.data.pnt = FALSE)
#gives an error if ch.data.pnt = TRUE message since p is not a data point

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
Te<-rbind(A,B,C);
t<-1.5
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num1CSstd.tri(Xp[3,],Xp,t)
Idom.num1CSstd.tri(c(1,2),c(1,2),t)
Idom.num1CSstd.tri(c(1,2),c(1,2),t,ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSstd.tri(Xp[i,],Xp,t))}

ind.gam1<-which(gam.vec==1)
ind.gam1

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE);
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.02,.01,.05)
yc<-txt[,2]+c(.02,.02,.03,.02)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

Idom.num1CSstd.tri(c(1,2),Xp,t,ch.data.pnt = FALSE)
#gives an error if ch.data.pnt = TRUE message since p is not a data point

## End(Not run)

The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case with $t=1$

Description

Point, p, is in the edge region of edge re (default is NULL); vertices are labeled as $1,2,3$ in the order they are stacked row-wise in $T_e$ , and the opposite edges are labeled with label of the vertices (that is, edge numbering is 1,2, and 3 for edges $AB$ , $BC$ , and $AC$ ).

CS proximity region is constructed with respect to $T_e$ with expansion parameter $t=1$ and edge regions are based on center of mass $CM=(1/2,\sqrt{3}/6)$ .

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1CSt1std.tri(p, Xp, re = NULL, ch.data.pnt = FALSE)
Idom.num1CSt1std.tri(p, Xp, re = NULL, ch.data.pnt = FALSE)

Arguments

`p`	A 2D point that is to be tested for being a dominating point or not of the CS-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the CS-PCD.
`re`	The index of the edge region in $T_e$ containing the point, either `1,2,3` or `NULL` (default is `NULL`).
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

$I($ p is a dominating point of the CS-PCD $)$ where the vertices of the CS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num1CSt1std.tri(Xp[3,],Xp)

Idom.num1CSt1std.tri(c(1,2),c(1,2))
Idom.num1CSt1std.tri(c(1,2),c(1,2),ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSt1std.tri(Xp[i,],Xp))}

ind.gam1<-which(gam.vec==1)
ind.gam1

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE);
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.02,.01,.05)
yc<-txt[,2]+c(.02,.02,.03,.02)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num1CSt1std.tri(Xp[3,],Xp)

Idom.num1CSt1std.tri(c(1,2),c(1,2))
Idom.num1CSt1std.tri(c(1,2),c(1,2),ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSt1std.tri(Xp[i,],Xp))}

ind.gam1<-which(gam.vec==1)
ind.gam1

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE);
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.02,.01,.05)
yc<-txt[,2]+c(.02,.02,.03,.02)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

## End(Not run)

The indicator for a point being a dominating point or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic triangle case

Description

Returns $I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp for data in the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ , that is, returns 1 if p is a dominating point of PE-PCD, and returns 0 otherwise.

PE proximity regions are defined with respect to the standard basic triangle $T_b$ . In the standard basic triangle, $T_b$ , $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of a standard basic triangle to the edges on the extension of the lines joining M to the vertices or based on the circumcenter of $T_b$ ; default is $M=(1,1,1)$ , i.e., the center of mass of $T_b$ . Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as $1,2,3$ in the order they are stacked row-wise.

See also (Ceyhan (2005, 2011)).

Usage

Idom.num1PEbasic.tri(
  p,
  Xp,
  r,
  c1,
  c2,
  M = c(1, 1, 1),
  rv = NULL,
  ch.data.pnt = FALSE
)
Idom.num1PEbasic.tri(
  p,
  Xp,
  r,
  c1,
  c2,
  M = c(1, 1, 1),
  rv = NULL,
  ch.data.pnt = FALSE
)

Arguments

`p`	A 2D point that is to be tested for being a dominating point or not of the PE-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the PE-PCD.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle $T_b$ or the circumcenter of $T_b$ which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of $T_b$ .
`rv`	Index of the vertex whose region contains point `p`, `rv` takes the vertex labels as $1,2,3$ as in the row order of the vertices in $T_b$ .
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

$I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, and returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)
r<-2

P<-c(.4,.2)
Idom.num1PEbasic.tri(P,Xp,r,c1,c2,M)
Idom.num1PEbasic.tri(Xp[1,],Xp,r,c1,c2,M)

Idom.num1PEbasic.tri(c(1,1),Xp,r,c1,c2,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since point p=c(1,1) is not a data point in Xp

#or try
Rv<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv
Idom.num1PEbasic.tri(Xp[1,],Xp,r,c1,c2,M,Rv)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEbasic.tri(Xp[i,],Xp,r,c1,c2,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

if (identical(M,circumcenter.tri(Tb)))
{
  plot(Tb,pch=".",asp=1,xlab="",ylab="",axes=TRUE,
  xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
  polygon(Tb)
  points(Xp,pch=1,col=1)
  Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2)
} else
{plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
  polygon(Tb)
  points(Xp,pch=1,col=1)
  Ds<-prj.cent2edges.basic.tri(c1,c2,M)}
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)

txt<-rbind(Tb,M,Ds)
xc<-txt[,1]+c(-.02,.02,.02,-.02,.03,-.03,.01)
yc<-txt[,2]+c(.02,.02,.02,-.02,.02,.02,-.03)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1PEbasic.tri(c(.2,.1),Xp,r,c1,c2,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since point p is not a data point in Xp

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)
r<-2

P<-c(.4,.2)
Idom.num1PEbasic.tri(P,Xp,r,c1,c2,M)
Idom.num1PEbasic.tri(Xp[1,],Xp,r,c1,c2,M)

Idom.num1PEbasic.tri(c(1,1),Xp,r,c1,c2,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since point p=c(1,1) is not a data point in Xp

#or try
Rv<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv
Idom.num1PEbasic.tri(Xp[1,],Xp,r,c1,c2,M,Rv)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEbasic.tri(Xp[i,],Xp,r,c1,c2,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

if (identical(M,circumcenter.tri(Tb)))
{
  plot(Tb,pch=".",asp=1,xlab="",ylab="",axes=TRUE,
  xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
  polygon(Tb)
  points(Xp,pch=1,col=1)
  Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2)
} else
{plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
  polygon(Tb)
  points(Xp,pch=1,col=1)
  Ds<-prj.cent2edges.basic.tri(c1,c2,M)}
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)

txt<-rbind(Tb,M,Ds)
xc<-txt[,1]+c(-.02,.02,.02,-.02,.03,-.03,.01)
yc<-txt[,2]+c(.02,.02,.02,-.02,.02,.02,-.03)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1PEbasic.tri(c(.2,.1),Xp,r,c1,c2,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since point p is not a data point in Xp

## End(Not run)

The indicator for a point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) for an interval

Description

Returns $I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 1D data set Xp.

PE proximity region is defined with respect to the interval int with an expansion parameter, $r \ge 1$ , and a centrality parameter, $c \in (0,1)$ , so arcs may exist for Xp points inside the interval int $=(a,b)$ .

Vertex regions are based on the center associated with the centrality parameter $c \in (0,1)$ . rv is the index of the vertex region p resides, with default=NULL.

Usage

Idom.num1PEint(p, Xp, int, r, c = 0.5, rv = NULL, ch.data.pnt = FALSE)
Idom.num1PEint(p, Xp, int, r, c = 0.5, rv = NULL, ch.data.pnt = FALSE)

Arguments

`p`	A 1D point that is to be tested for being a dominating point or not of the PE-PCD.
`Xp`	A set of 1D points which constitutes the vertices of the PE-PCD.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ . For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ ; default `c=.5`.
`rv`	Index of the vertex region in which the point resides, either `1,2` or `NULL` (default is `NULL`).
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

$I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 1D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

Author(s)

Elvan Ceyhan

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10
int=c(a,b)

Mc<-centerMc(int,c)

n<-10

set.seed(1)
Xp<-runif(n,a,b)

Idom.num1PEint(Xp[5],Xp,int,r,c)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEint(Xp[i],Xp,int,r,c))}

ind.gam1<-which(gam.vec==1)
ind.gam1

domset<-Xp[ind.gam1]
if (length(ind.gam1)==0)
{domset<-NA}

#or try
Rv<-rel.vert.mid.int(Xp[5],int,c)$rv
Idom.num1PEint(Xp[5],Xp,int,r,c,Rv)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",xlim=Xlim+xd*c(-.05,.05))
abline(h=0)
points(cbind(Xp,0))
abline(v=c(a,b,Mc),col=c(1,1,2),lty=2)
points(cbind(domset,0),pch=4,col=2)
text(cbind(c(a,b,Mc),-0.1),c("a","b","Mc"))

Idom.num1PEint(2,Xp,int,r,c,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since point p is not a data point in Xp

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10
int=c(a,b)

Mc<-centerMc(int,c)

n<-10

set.seed(1)
Xp<-runif(n,a,b)

Idom.num1PEint(Xp[5],Xp,int,r,c)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEint(Xp[i],Xp,int,r,c))}

ind.gam1<-which(gam.vec==1)
ind.gam1

domset<-Xp[ind.gam1]
if (length(ind.gam1)==0)
{domset<-NA}

#or try
Rv<-rel.vert.mid.int(Xp[5],int,c)$rv
Idom.num1PEint(Xp[5],Xp,int,r,c,Rv)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",xlim=Xlim+xd*c(-.05,.05))
abline(h=0)
points(cbind(Xp,0))
abline(v=c(a,b,Mc),col=c(1,1,2),lty=2)
points(cbind(domset,0),pch=4,col=2)
text(cbind(c(a,b,Mc),-0.1),c("a","b","Mc"))

Idom.num1PEint(2,Xp,int,r,c,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since point p is not a data point in Xp

## End(Not run)

The indicator for a 3D point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case

Description

Returns $I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 3D data set Xp in the standard regular tetrahedron $T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))$ , that is, returns 1 if p is a dominating point of PE-PCD, returns 0 otherwise.

Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as 1,2,3,4 in the order they are stacked row-wise in $T_h$ .

PE proximity region is constructed with respect to the tetrahedron $T_h$ with expansion parameter $r \ge 1$ and vertex regions are based on center of mass $CM$ (equivalent to circumcenter in this case).

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1PEstd.tetra(p, Xp, r, rv = NULL, ch.data.pnt = FALSE)
Idom.num1PEstd.tetra(p, Xp, r, rv = NULL, ch.data.pnt = FALSE)

Arguments

`p`	A 3D point that is to be tested for being a dominating point or not of the PE-PCD.
`Xp`	A set of 3D points which constitutes the vertices of the PE-PCD.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`rv`	Index of the vertex whose region contains point `p`, `rv` takes the vertex labels as `1,2,3,4` as in the row order of the vertices in standard regular tetrahedron, default is `NULL`.
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

$I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 3D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
set.seed(123)
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-5 #try also n<-20
Xp<-runif.std.tetra(n)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))
r<-1.5

P<-c(.4,.1,.2)
Idom.num1PEstd.tetra(Xp[1,],Xp,r)
Idom.num1PEstd.tetra(P,Xp,r)

Idom.num1PEstd.tetra(Xp[1,],Xp,r)
Idom.num1PEstd.tetra(Xp[1,],Xp[1,],r)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
Idom.num1PEstd.tetra(Xp[1,],Xp,r,rv=RV)

Idom.num1PEstd.tetra(c(-1,-1,-1),Xp,r)
Idom.num1PEstd.tetra(c(-1,-1,-1),c(-1,-1,-1),r)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEstd.tetra(Xp[i,],Xp,r))}

ind.gam1<-which(gam.vec==1)
ind.gam1
g1.pts<-Xp[ind.gam1,]

Xlim<-range(tetra[,1],Xp[,1])
Ylim<-range(tetra[,2],Xp[,2])
Zlim<-range(tetra[,3],Xp[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi =0,theta=40, bty = "g",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
         pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
if (length(g1.pts)!=0)
{
  if (length(g1.pts)==3) g1.pts<-matrix(g1.pts,nrow=1)
  plot3D::points3D(g1.pts[,1],g1.pts[,2],g1.pts[,3], pch=4,col="red", add=TRUE)}

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), add=TRUE)

CM<-apply(tetra,2,mean)
D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CM,6),ncol=3,byrow=TRUE)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2)

P<-c(.4,.1,.2)
Idom.num1PEstd.tetra(P,Xp,r)

Idom.num1PEstd.tetra(c(-1,-1,-1),Xp,r,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE

## End(Not run)

## Not run: 
set.seed(123)
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-5 #try also n<-20
Xp<-runif.std.tetra(n)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))
r<-1.5

P<-c(.4,.1,.2)
Idom.num1PEstd.tetra(Xp[1,],Xp,r)
Idom.num1PEstd.tetra(P,Xp,r)

Idom.num1PEstd.tetra(Xp[1,],Xp,r)
Idom.num1PEstd.tetra(Xp[1,],Xp[1,],r)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
Idom.num1PEstd.tetra(Xp[1,],Xp,r,rv=RV)

Idom.num1PEstd.tetra(c(-1,-1,-1),Xp,r)
Idom.num1PEstd.tetra(c(-1,-1,-1),c(-1,-1,-1),r)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEstd.tetra(Xp[i,],Xp,r))}

ind.gam1<-which(gam.vec==1)
ind.gam1
g1.pts<-Xp[ind.gam1,]

Xlim<-range(tetra[,1],Xp[,1])
Ylim<-range(tetra[,2],Xp[,2])
Zlim<-range(tetra[,3],Xp[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi =0,theta=40, bty = "g",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
         pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
if (length(g1.pts)!=0)
{
  if (length(g1.pts)==3) g1.pts<-matrix(g1.pts,nrow=1)
  plot3D::points3D(g1.pts[,1],g1.pts[,2],g1.pts[,3], pch=4,col="red", add=TRUE)}

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), add=TRUE)

CM<-apply(tetra,2,mean)
D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CM,6),ncol=3,byrow=TRUE)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2)

P<-c(.4,.1,.2)
Idom.num1PEstd.tetra(P,Xp,r)

Idom.num1PEstd.tetra(c(-1,-1,-1),Xp,r,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE

## End(Not run)

The indicator for a 3D point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns $I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp in the tetrahedron th, that is, returns 1 if p is a dominating point of PE-PCD, returns 0 otherwise.

Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as 1,2,3,4 in the order they are stacked row-wise in th.

PE proximity region is constructed with respect to the tetrahedron th with expansion parameter $r \ge 1$ and vertex regions are based on center of mass (M="CM") or circumcenter (M="CC") only. and vertex regions are based on center of mass $CM$ (equivalent to circumcenter in this case).

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1PEtetra(p, Xp, th, r, M = "CM", rv = NULL, ch.data.pnt = FALSE)
Idom.num1PEtetra(p, Xp, th, r, M = "CM", rv = NULL, ch.data.pnt = FALSE)

Arguments

`p`	A 3D point that is to be tested for being a dominating point or not of the PE-PCD.
`Xp`	A set of 3D points which constitutes the vertices of the PE-PCD.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.
`rv`	Index of the vertex whose region contains point `p`, `rv` takes the vertex labels as `1,2,3,4` as in the row order of the vertices in standard tetrahedron, default is `NULL`.
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

$I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5 #try also n<-20

Xp<-runif.tetra(n,tetra)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))

M<-"CM"; cent<-apply(tetra,2,mean)  #center of mass
#try also M<-"CC"; cent<-circumcenter.tetra(tetra)  #circumcenter

r<-2

P<-c(.4,.1,.2)
Idom.num1PEtetra(Xp[1,],Xp,tetra,r,M)
Idom.num1PEtetra(P,Xp,tetra,r,M)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
Idom.num1PEtetra(Xp[1,],Xp,tetra,r,M,rv=RV)

Idom.num1PEtetra(c(-1,-1,-1),Xp,tetra,r,M)
Idom.num1PEtetra(c(-1,-1,-1),c(-1,-1,-1),tetra,r,M)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEtetra(Xp[i,],Xp,tetra,r,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1
g1.pts<-Xp[ind.gam1,]

Xlim<-range(tetra[,1],Xp[,1],cent[1])
Ylim<-range(tetra[,2],Xp[,2],cent[2])
Zlim<-range(tetra[,3],Xp[,3],cent[3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi =0,theta=40, bty = "g",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
         pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
if (length(g1.pts)!=0)
{plot3D::points3D(g1.pts[,1],g1.pts[,2],g1.pts[,3], pch=4,col="red", add=TRUE)}

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), add=TRUE)

D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-rbind(cent,cent,cent,cent,cent,cent)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2)

P<-c(.4,.1,.2)
Idom.num1PEtetra(P,Xp,tetra,r,M)

Idom.num1PEtetra(c(-1,-1,-1),Xp,tetra,r,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since p is not a data point

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5 #try also n<-20

Xp<-runif.tetra(n,tetra)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))

M<-"CM"; cent<-apply(tetra,2,mean)  #center of mass
#try also M<-"CC"; cent<-circumcenter.tetra(tetra)  #circumcenter

r<-2

P<-c(.4,.1,.2)
Idom.num1PEtetra(Xp[1,],Xp,tetra,r,M)
Idom.num1PEtetra(P,Xp,tetra,r,M)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
Idom.num1PEtetra(Xp[1,],Xp,tetra,r,M,rv=RV)

Idom.num1PEtetra(c(-1,-1,-1),Xp,tetra,r,M)
Idom.num1PEtetra(c(-1,-1,-1),c(-1,-1,-1),tetra,r,M)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEtetra(Xp[i,],Xp,tetra,r,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1
g1.pts<-Xp[ind.gam1,]

Xlim<-range(tetra[,1],Xp[,1],cent[1])
Ylim<-range(tetra[,2],Xp[,2],cent[2])
Zlim<-range(tetra[,3],Xp[,3],cent[3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi =0,theta=40, bty = "g",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
         pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
if (length(g1.pts)!=0)
{plot3D::points3D(g1.pts[,1],g1.pts[,2],g1.pts[,3], pch=4,col="red", add=TRUE)}

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), add=TRUE)

D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-rbind(cent,cent,cent,cent,cent,cent)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2)

P<-c(.4,.1,.2)
Idom.num1PEtetra(P,Xp,tetra,r,M)

Idom.num1PEtetra(c(-1,-1,-1),Xp,tetra,r,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since p is not a data point

## End(Not run)

The indicator for a point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns $I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp in the triangle tri, that is, returns 1 if p is a dominating point of PE-PCD, and returns 0 otherwise.

Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as $1,2,3$ in the order they are stacked row-wise in tri.

PE proximity region is constructed with respect to the triangle tri with expansion parameter $r \ge 1$ and vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri or based on the circumcenter of tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

Idom.num1PEtri(p, Xp, tri, r, M = c(1, 1, 1), rv = NULL, ch.data.pnt = FALSE)
Idom.num1PEtri(p, Xp, tri, r, M = c(1, 1, 1), rv = NULL, ch.data.pnt = FALSE)

Arguments

`p`	A 2D point that is to be tested for being a dominating point or not of the PE-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the PE-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.
`rv`	Index of the vertex whose region contains point `p`, `rv` takes the vertex labels as $1,2,3$ as in the row order of the vertices in `tri`.
`ch.data.pnt`	A logical argument for checking whether point `p` is a data point in `Xp` or not (default is `FALSE`).

Value

$I($ p is a dominating point of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, and returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5  #try also r<-2

Idom.num1PEtri(Xp[1,],Xp,Tr,r,M)
Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M)
Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M,ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEtri(Xp[i,],Xp,Tr,r,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.tri(Xp[1,],Tr,M)$rv
Idom.num1PEtri(Xp[1,],Xp,Tr,r,M,Rv)

Ds<-prj.cent2edges(Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

Xlim<-range(Tr[,1],Xp[,1],M[1])
Ylim<-range(Tr[,2],Xp[,2],M[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Tr,M,Ds)
xc<-txt[,1]+c(-.02,.03,.02,-.02,.04,-.03,.0)
yc<-txt[,2]+c(.02,.02,.05,-.03,.04,.06,-.07)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

P<-c(1.4,1)
Idom.num1PEtri(P,P,Tr,r,M)
Idom.num1PEtri(Xp[1,],Xp,Tr,r,M)

Idom.num1PEtri(c(1,2),Xp,Tr,r,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since p is not a data point

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5  #try also r<-2

Idom.num1PEtri(Xp[1,],Xp,Tr,r,M)
Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M)
Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M,ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEtri(Xp[i,],Xp,Tr,r,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.tri(Xp[1,],Tr,M)$rv
Idom.num1PEtri(Xp[1,],Xp,Tr,r,M,Rv)

Ds<-prj.cent2edges(Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

Xlim<-range(Tr[,1],Xp[,1],M[1])
Ylim<-range(Tr[,2],Xp[,2],M[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Tr,M,Ds)
xc<-txt[,1]+c(-.02,.03,.02,-.02,.04,-.03,.0)
yc<-txt[,2]+c(.02,.02,.05,-.03,.04,.06,-.07)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

P<-c(1.4,1)
Idom.num1PEtri(P,P,Tr,r,M)
Idom.num1PEtri(Xp[1,],Xp,Tr,r,M)

Idom.num1PEtri(c(1,2),Xp,Tr,r,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since p is not a data point

## End(Not run)

The indicator for two points being a dominating set for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case

Description

Returns $I($ {p1,p2} is a dominating set of AS-PCD $)$ where vertices of AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise.

AS proximity regions are defined with respect to the standard basic triangle $T_b=T(c(0,0),c(1,0),c(c1,c2))$ , In the standard basic triangle, $T_b$ , $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

Point, p1, is in the vertex region of vertex rv1 (default is NULL) and point, p2, is in the vertex region of vertex rv2 (default is NULL); vertices are labeled as $1,2,3$ in the order they are stacked row-wise.

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would be a dominating set if they actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2ASbasic.tri(
  p1,
  p2,
  Xp,
  c1,
  c2,
  M = "CC",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)
Idom.num2ASbasic.tri(
  p1,
  p2,
  Xp,
  c1,
  c2,
  M = "CC",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

`p1`, `p2`	Two 2D points to be tested for constituting a dominating set of the AS-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the AS-PCD.
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle $T_b$ or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$ ; default is `M="CC"` i.e., the circumcenter of $T_b$ .
`rv1`, `rv2`	The indices of the vertices whose regions contains `p1` and `p2`, respectively. They take the vertex labels as $1,2,3$ as in the row order of the vertices in $T_b$ (default is `NULL` for both).
`ch.data.pnts`	A logical argument for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`).

Value

$I($ {p1,p2} is a dominating set of the AS-PCD $)$ where the vertices of AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M)
Idom.num2ASbasic.tri(Xp[1,],Xp[1,],Xp,c1,c2,M)  #one point can not a dominating set of size two

Idom.num2ASbasic.tri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),c1,c2,M)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2ASbasic.tri(Xp[i,],Xp[j,],Xp,c1,c2,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
rv2<-rel.vert.basic.triCC(Xp[2,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv1,rv2)
Idom.num2ASbasic.tri(c(.2,.4),Xp[2,],Xp,c1,c2,M,rv1,rv2)

#or try
rv1<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv1)

#or try
Rv2<-rel.vert.basic.triCC(Xp[2,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv2=Rv2)

Idom.num2ASbasic.tri(c(.3,.2),c(.35,.25),Xp,c1,c2,M)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M)
Idom.num2ASbasic.tri(Xp[1,],Xp[1,],Xp,c1,c2,M)  #one point can not a dominating set of size two

Idom.num2ASbasic.tri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),c1,c2,M)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2ASbasic.tri(Xp[i,],Xp[j,],Xp,c1,c2,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
rv2<-rel.vert.basic.triCC(Xp[2,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv1,rv2)
Idom.num2ASbasic.tri(c(.2,.4),Xp[2,],Xp,c1,c2,M,rv1,rv2)

#or try
rv1<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv1)

#or try
Rv2<-rel.vert.basic.triCC(Xp[2,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv2=Rv2)

Idom.num2ASbasic.tri(c(.3,.2),c(.35,.25),Xp,c1,c2,M)

## End(Not run)

The indicator for two points constituting a dominating set for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns $I($ {p1,p2} is a dominating set of the AS-PCD $)$ where vertices of the AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise.

AS proximity regions are defined with respect to the triangle tri. Point, p1, is in the region of vertex rv1 (default is NULL) and point, p2, is in the region of vertex rv2 (default is NULL); vertices (and hence rv1 and rv2) are labeled as $1,2,3$ in the order they are stacked row-wise in tri.

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would constitute dominating set if they actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2AStri(
  p1,
  p2,
  Xp,
  tri,
  M = "CC",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)
Idom.num2AStri(
  p1,
  p2,
  Xp,
  tri,
  M = "CC",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

`p1`, `p2`	Two 2D points to be tested for constituting a dominating set of the AS-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the AS-PCD.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$ ; default is `M="CC"` i.e., the circumcenter of `tri`.
`rv1`, `rv2`	The indices of the vertices whose regions contains `p1` and `p2`, respectively. They take the vertex labels as $1,2,3$ as in the row order of the vertices in `tri` (default is `NULL` for both).
`ch.data.pnts`	A logical argument for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`).

Value

$I($ {p1,p2} is a dominating set of the AS-PCD $)$ where vertices of the AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M)
Idom.num2AStri(Xp[1,],Xp[1,],Xp,Tr,M)  #same two points cannot be a dominating set of size 2

Idom.num2AStri(c(.2,.4),Xp[2,],Xp,Tr,M)
Idom.num2AStri(c(.2,.4),c(.2,.5),Xp,Tr,M)
Idom.num2AStri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),Tr,M)

#or try
rv1<-rel.vert.triCC(c(.2,.4),Tr)$rv
rv2<-rel.vert.triCC(c(.2,.5),Tr)$rv
Idom.num2AStri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),Tr,M,rv1,rv2)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2AStri(Xp[i,],Xp[j,],Xp,Tr,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.triCC(Xp[1,],Tr)$rv
rv2<-rel.vert.triCC(Xp[2,],Tr)$rv
Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv1,rv2)

#or try
rv1<-rel.vert.triCC(Xp[1,],Tr)$rv
Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv1)

#or try
Rv2<-rel.vert.triCC(Xp[2,],Tr)$rv
Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv2=Rv2)

Idom.num2AStri(c(1.3,1.2),c(1.35,1.25),Xp,Tr,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M)
Idom.num2AStri(Xp[1,],Xp[1,],Xp,Tr,M)  #same two points cannot be a dominating set of size 2

Idom.num2AStri(c(.2,.4),Xp[2,],Xp,Tr,M)
Idom.num2AStri(c(.2,.4),c(.2,.5),Xp,Tr,M)
Idom.num2AStri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),Tr,M)

#or try
rv1<-rel.vert.triCC(c(.2,.4),Tr)$rv
rv2<-rel.vert.triCC(c(.2,.5),Tr)$rv
Idom.num2AStri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),Tr,M,rv1,rv2)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2AStri(Xp[i,],Xp[j,],Xp,Tr,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.triCC(Xp[1,],Tr)$rv
rv2<-rel.vert.triCC(Xp[2,],Tr)$rv
Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv1,rv2)

#or try
rv1<-rel.vert.triCC(Xp[1,],Tr)$rv
Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv1)

#or try
Rv2<-rel.vert.triCC(Xp[2,],Tr)$rv
Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv2=Rv2)

Idom.num2AStri(c(1.3,1.2),c(1.35,1.25),Xp,Tr,M)

## End(Not run)

The indicator for two points constituting a dominating set for Central Similarity Proximity Catch Digraphs (CS-PCDs) - first one-sixth of the standard equilateral triangle case

Description

Returns $I($ {p1,p2} is a dominating set of the CS-PCD $)$ where the vertices of the CS-PCD are the 2D data set Xp), that is, returns 1 if p is a dominating point of CS-PCD, returns 0 otherwise.

CS proximity region is constructed with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ and with expansion parameter $t=1$ . Point, p1, must lie in the first one-sixth of $T_e$ , which is the triangle with vertices $T(A,D_3,CM)=T((0,0),(1/2,0),CM)$ .

Usage

Idom.num2CS.Te.onesixth(p1, p2, Xp, ch.data.pnts = FALSE)
Idom.num2CS.Te.onesixth(p1, p2, Xp, ch.data.pnts = FALSE)

Arguments

`p1`, `p2`	Two 2D points to be tested for constituting a dominating set of the CS-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the CS-PCD.
`ch.data.pnts`	A logical argument for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`).

Value

$I($ {p1,p2} is a dominating set of the CS-PCD $)$ where the vertices of the CS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of CS-PCD, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

The indicator for two points being a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic triangle case

Description

Returns $I($ {p1,p2} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp in the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ , that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, and returns 0 otherwise.

PE proximity regions are defined with respect to $T_b$ . In the standard basic triangle, $T_b$ , $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of a standard basic triangle $T_b$ ; default is $M=(1,1,1)$ , i.e., the center of mass of $T_b$ . Point, p1, is in the vertex region of vertex rv1 (default is NULL); and point, p2, is in the vertex region of vertex rv2 (default is NULL); vertices are labeled as $1,2,3$ in the order they are stacked row-wise.

ch.data.pnts is for checking whether points p1 and p2 are both data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would constitute a dominating set if they both were actually in the data set.

See also (Ceyhan (2005, 2011)).

Usage

Idom.num2PEbasic.tri(
  p1,
  p2,
  Xp,
  r,
  c1,
  c2,
  M = c(1, 1, 1),
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)
Idom.num2PEbasic.tri(
  p1,
  p2,
  Xp,
  r,
  c1,
  c2,
  M = c(1, 1, 1),
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

`p1`, `p2`	Two 2D points to be tested for constituting a dominating set of the PE-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the PE-PCD.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle. adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle $T_b$ or the circumcenter of $T_b$ which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of $T_b$ .
`rv1`, `rv2`	The indices of the vertices whose regions contains `p1` and `p2`, respectively. They take the vertex labels as $1,2,3$ as in the row order of the vertices in $T_b$ (default is `NULL` for both).
`ch.data.pnts`	A logical argument for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`).

Value

$I($ {p1,p2} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, and returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)

r<-2

Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M)

Idom.num2PEbasic.tri(c(1,2),c(1,3),rbind(c(1,2),c(1,3)),r,c1,c2,M)
Idom.num2PEbasic.tri(c(1,2),c(1,3),rbind(c(1,2),c(1,3)),r,c1,c2,M,
ch.data.pnts = TRUE)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2PEbasic.tri(Xp[i,],Xp[j,],Xp,r,c1,c2,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv;
rv2<-rel.vert.basic.tri(Xp[2,],c1,c2,M)$rv;
Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M,rv1,rv2)

#or try
rv1<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv;
Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M,rv1)

#or try
rv2<-rel.vert.basic.tri(Xp[2,],c1,c2,M)$rv;
Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M,rv2=rv2)

Idom.num2PEbasic.tri(c(1,2),Xp[2,],Xp,r,c1,c2,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not both points are data points in Xp

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)

r<-2

Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M)

Idom.num2PEbasic.tri(c(1,2),c(1,3),rbind(c(1,2),c(1,3)),r,c1,c2,M)
Idom.num2PEbasic.tri(c(1,2),c(1,3),rbind(c(1,2),c(1,3)),r,c1,c2,M,
ch.data.pnts = TRUE)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2PEbasic.tri(Xp[i,],Xp[j,],Xp,r,c1,c2,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv;
rv2<-rel.vert.basic.tri(Xp[2,],c1,c2,M)$rv;
Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M,rv1,rv2)

#or try
rv1<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv;
Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M,rv1)

#or try
rv2<-rel.vert.basic.tri(Xp[2,],c1,c2,M)$rv;
Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M,rv2=rv2)

Idom.num2PEbasic.tri(c(1,2),Xp[2,],Xp,r,c1,c2,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not both points are data points in Xp

## End(Not run)

The indicator for two 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case

Description

Returns $I($ {p1,p2} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 3D data set Xp in the standard regular tetrahedron $T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))$ , that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL) and point, p2, is in the region of vertex rv2 (default is NULL); vertices (and hence rv1 and rv2) are labeled as 1,2,3,4 in the order they are stacked row-wise in $T_h$ .

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would constitute a dominating set if they actually were both in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2PEstd.tetra(
  p1,
  p2,
  Xp,
  r,
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)
Idom.num2PEstd.tetra(
  p1,
  p2,
  Xp,
  r,
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

`p1`, `p2`	Two 3D points to be tested for constituting a dominating set of the PE-PCD.
`Xp`	A set of 3D points which constitutes the vertices of the PE-PCD.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`rv1`, `rv2`	The indices of the vertices whose regions contains `p1` and `p2`, respectively. They take the vertex labels as `1,2,3,4` as in the row order of the vertices in $T_h$ (default is `NULL` for both).
`ch.data.pnts`	A logical argument for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`).

Value

$I($ {p1,p2} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 3D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-5 #try also n<-20
Xp<-runif.std.tetra(n)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))
r<-1.5

Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r)

ind.gam2<-vector()
for (i in 1:(n-1))
 for (j in (i+1):n)
 {if (Idom.num2PEstd.tetra(Xp[i,],Xp[j,],Xp,r)==1)
  ind.gam2<-rbind(ind.gam2,c(i,j))}

ind.gam2

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r,rv1,rv2)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.4,.1,.2)
Idom.num2PEstd.tetra(P1,P2,Xp,r)

Idom.num2PEstd.tetra(c(-1,-1,-1),Xp[2,],Xp,r,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-5 #try also n<-20
Xp<-runif.std.tetra(n)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))
r<-1.5

Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r)

ind.gam2<-vector()
for (i in 1:(n-1))
 for (j in (i+1):n)
 {if (Idom.num2PEstd.tetra(Xp[i,],Xp[j,],Xp,r)==1)
  ind.gam2<-rbind(ind.gam2,c(i,j))}

ind.gam2

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r,rv1,rv2)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.4,.1,.2)
Idom.num2PEstd.tetra(P1,P2,Xp,r)

Idom.num2PEstd.tetra(c(-1,-1,-1),Xp[2,],Xp,r,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp

## End(Not run)

The indicator for two 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns $I($ {p1,p2} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 3D data set Xp in the tetrahedron th, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise.

PE proximity region is constructed with respect to the tetrahedron th with expansion parameter $r \ge 1$ and vertex regions are based on center of mass (M="CM") or circumcenter (M="CC") only.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2PEtetra(
  p1,
  p2,
  Xp,
  th,
  r,
  M = "CM",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)
Idom.num2PEtetra(
  p1,
  p2,
  Xp,
  th,
  r,
  M = "CM",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

`p1`, `p2`	Two 3D points to be tested for constituting a dominating set of the PE-PCD.
`Xp`	A set of 3D points which constitutes the vertices of the PE-PCD.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.
`rv1`, `rv2`	The indices of the vertices whose regions contains `p1` and `p2`, respectively. They take the vertex labels as `1,2,3,4` as in the row order of the vertices in `th` (default is `NULL` for both).
`ch.data.pnts`	A logical argument for checking whether both points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`).

Value

$I($ {p1,p2} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 3D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5

set.seed(1)
Xp<-runif.tetra(n,tetra)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))

M<-"CM";  #try also M<-"CC";
r<-1.5

Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M)
Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M)

ind.gam2<-ind.gamn2<-vector()
for (i in 1:(n-1))
 for (j in (i+1):n)
 {if (Idom.num2PEtetra(Xp[i,],Xp[j,],Xp,tetra,r,M)==1)
 {ind.gam2<-rbind(ind.gam2,c(i,j))
 }
 }
ind.gam2

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1,rv2)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.4,.1,.2)
Idom.num2PEtetra(P1,P2,Xp,tetra,r,M)

Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5

set.seed(1)
Xp<-runif.tetra(n,tetra)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))

M<-"CM";  #try also M<-"CC";
r<-1.5

Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M)
Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M)

ind.gam2<-ind.gamn2<-vector()
for (i in 1:(n-1))
 for (j in (i+1):n)
 {if (Idom.num2PEtetra(Xp[i,],Xp[j,],Xp,tetra,r,M)==1)
 {ind.gam2<-rbind(ind.gam2,c(i,j))
 }
 }
ind.gam2

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1,rv2)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.4,.1,.2)
Idom.num2PEtetra(P1,P2,Xp,tetra,r,M)

Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp

## End(Not run)

The indicator for two points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns $I($ {p1,p2} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, and returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL) and point, p2, is in the region of vertex rv2 (default is NULL); vertices (and hence rv1 and rv2) are labeled as $1,2,3$ in the order they are stacked row-wise in tri.

PE proximity regions are defined with respect to the triangle tri and vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri or circumcenter of tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

Idom.num2PEtri(
  p1,
  p2,
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)
Idom.num2PEtri(
  p1,
  p2,
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

`p1`, `p2`	Two 2D points to be tested for constituting a dominating set of the PE-PCD.
`Xp`	A set of 2D points which constitutes the vertices of the PE-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.
`rv1`, `rv2`	The indices of the vertices whose regions contains `p1` and `p2`, respectively. They take the vertex labels as $1,2,3$ as in the row order of the vertices in `tri` (default is `NULL` for both).
`ch.data.pnts`	A logical argument for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`).

Value

$I($ {p1,p2} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, and returns 0 otherwise.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5  #try also r<-2

Idom.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2PEtri(Xp[i,],Xp[j,],Xp,Tr,r,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.tri(Xp[1,],Tr,M)$rv;
rv2<-rel.vert.tri(Xp[2,],Tr,M)$rv
Idom.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M,rv1,rv2)

Idom.num2PEtri(Xp[1,],c(1,2),Xp,Tr,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5  #try also r<-2

Idom.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2PEtri(Xp[i,],Xp[j,],Xp,Tr,r,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.tri(Xp[1,],Tr,M)$rv;
rv2<-rel.vert.tri(Xp[2,],Tr,M)$rv
Idom.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M,rv1,rv2)

Idom.num2PEtri(Xp[1,],c(1,2),Xp,Tr,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp

## End(Not run)

The indicator for three 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case

Description

Returns $I(\{$ p1,p2,pt3} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 3D data set Xp in the standard regular tetrahedron $T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))$ , that is, returns 1 if {p1,p2,pt3} is a dominating set of PE-PCD, returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL), point, p2, is in the region of vertex rv2 (default is NULL); point, pt3), is in the region of vertex rv3) (default is NULL); vertices (and hence rv1, rv2 and rv3) are labeled as 1,2,3,4 in the order they are stacked row-wise in $T_h$ .

ch.data.pnts is for checking whether points p1, p2 and pt3 are all data points in Xp or not (default is FALSE), so by default this function checks whether the points p1, p2 and pt3 would constitute a dominating set if they actually were all in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num3PEstd.tetra(
  p1,
  p2,
  pt3,
  Xp,
  r,
  rv1 = NULL,
  rv2 = NULL,
  rv3 = NULL,
  ch.data.pnts = FALSE
)
Idom.num3PEstd.tetra(
  p1,
  p2,
  pt3,
  Xp,
  r,
  rv1 = NULL,
  rv2 = NULL,
  rv3 = NULL,
  ch.data.pnts = FALSE
)

Arguments

`p1`, `p2`, `pt3`	Three 3D points to be tested for constituting a dominating set of the PE-PCD.
`Xp`	A set of 3D points which constitutes the vertices of the PE-PCD.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`rv1`, `rv2`, `rv3`	The indices of the vertices whose regions contains `p1`, `p2` and `pt3`, respectively. They take the vertex labels as `1,2,3,4` as in the row order of the vertices in $T_h$ (default is `NULL` for all).
`ch.data.pnts`	A logical argument for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`).

Value

$I(\{$ p1,p2,pt3} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 3D data set Xp), that is, returns 1 if {p1,p2,pt3} is a dominating set of PE-PCD, returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
set.seed(123)
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5 #try 20, 40, 100 (larger n may take a long time)
Xp<-runif.std.tetra(n)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))
r<-1.25

Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r)

ind.gam3<-vector()
for (i in 1:(n-2))
 for (j in (i+1):(n-1))
   for (k in (j+1):n)
 {if (Idom.num3PEstd.tetra(Xp[i,],Xp[j,],Xp[k,],Xp,r)==1)
  ind.gam3<-rbind(ind.gam3,c(i,j,k))}

ind.gam3

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv; rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv;
rv3<-rel.vert.tetraCC(Xp[3,],tetra)$rv
Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv1,rv2,rv3)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.3,.3,.3)
P3<-c(.4,.1,.2)
Idom.num3PEstd.tetra(P1,P2,P3,Xp,r)

Idom.num3PEstd.tetra(Xp[1,],c(1,1,1),Xp[3,],Xp,r,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

## Not run: 
set.seed(123)
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5 #try 20, 40, 100 (larger n may take a long time)
Xp<-runif.std.tetra(n)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))
r<-1.25

Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r)

ind.gam3<-vector()
for (i in 1:(n-2))
 for (j in (i+1):(n-1))
   for (k in (j+1):n)
 {if (Idom.num3PEstd.tetra(Xp[i,],Xp[j,],Xp[k,],Xp,r)==1)
  ind.gam3<-rbind(ind.gam3,c(i,j,k))}

ind.gam3

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv; rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv;
rv3<-rel.vert.tetraCC(Xp[3,],tetra)$rv
Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv1,rv2,rv3)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.3,.3,.3)
P3<-c(.4,.1,.2)
Idom.num3PEstd.tetra(P1,P2,P3,Xp,r)

Idom.num3PEstd.tetra(Xp[1,],c(1,1,1),Xp[3,],Xp,r,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

The indicator for three 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns $I(\{$ p1,p2,pt3} is a dominating set of the PE-PCD $)$ where the vertices of the PE-PCD are the 3D data set Xp in the tetrahedron th, that is, returns 1 if {p1,p2,pt3} is a dominating set of PE-PCD, returns 0 otherwise.

PE proximity region is constructed with respect to the tetrahedron th with expansion parameter $r \ge 1$ and vertex regions are based on center of mass $CM$ (equivalent to circumcenter in this case).

See also (Ceyhan (2005, 2010)).

Usage

Idom.num3PEtetra(
  p1,
  p2,
  pt3,
  Xp,
  th,
  r,
  M = "CM",
  rv1 = NULL,
  rv2 = NULL,
  rv3 = NULL,
  ch.data.pnts = FALSE
)
Idom.num3PEtetra(
  p1,
  p2,
  pt3,
  Xp,
  th,
  r,
  M = "CM",
  rv1 = NULL,
  rv2 = NULL,
  rv3 = NULL,
  ch.data.pnts = FALSE
)

Arguments

`p1`, `p2`, `pt3`	Three 3D points to be tested for constituting a dominating set of the PE-PCD.
`Xp`	A set of 3D points which constitutes the vertices of the PE-PCD.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.
`rv1`, `rv2`, `rv3`	The indices of the vertices whose regions contains `p1`, `p2` and `pt3`, respectively. They take the vertex labels as `1,2,3,4` as in the row order of the vertices in `th` ( default is `NULL` for all).
`ch.data.pnts`	A logical argument for checking whether points `p1` and `p2` are data points in `Xp` or not (default is `FALSE`).

Value

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
set.seed(123)
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5 #try 20, 40, 100 (larger n may take a long time)
Xp<-runif.tetra(n,tetra)$g

M<-"CM";  #try also M<-"CC";
r<-1.25

Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M)

ind.gam3<-vector()
for (i in 1:(n-2))
 for (j in (i+1):(n-1))
   for (k in (j+1):n)
   {if (Idom.num3PEtetra(Xp[i,],Xp[j,],Xp[k,],Xp,tetra,r,M)==1)
    ind.gam3<-rbind(ind.gam3,c(i,j,k))}

ind.gam3

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv; rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv;
rv3<-rel.vert.tetraCC(Xp[3,],tetra)$rv
Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M,rv1,rv2,rv3)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.3,.3,.3)
P3<-c(.4,.1,.2)
Idom.num3PEtetra(P1,P2,P3,Xp,tetra,r,M)

Idom.num3PEtetra(Xp[1,],c(1,1,1),Xp[3,],Xp,tetra,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

## Not run: 
set.seed(123)
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5 #try 20, 40, 100 (larger n may take a long time)
Xp<-runif.tetra(n,tetra)$g

M<-"CM";  #try also M<-"CC";
r<-1.25

Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M)

ind.gam3<-vector()
for (i in 1:(n-2))
 for (j in (i+1):(n-1))
   for (k in (j+1):n)
   {if (Idom.num3PEtetra(Xp[i,],Xp[j,],Xp[k,],Xp,tetra,r,M)==1)
    ind.gam3<-rbind(ind.gam3,c(i,j,k))}

ind.gam3

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv; rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv;
rv3<-rel.vert.tetraCC(Xp[3,],tetra)$rv
Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M,rv1,rv2,rv3)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.3,.3,.3)
P3<-c(.4,.1,.2)
Idom.num3PEtetra(P1,P2,P3,Xp,tetra,r,M)

Idom.num3PEtetra(Xp[1,],c(1,1,1),Xp[3,],Xp,tetra,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

## End(Not run)

Indicator for an upper bound for the domination number of Arc Slice Proximity Catch Digraph (AS-PCD) by the exact algorithm - one triangle case

Description

Returns $I($ domination number of AS-PCD whose vertices are the data points Xp is less than or equal to k $)$ , that is, returns 1 if the domination number of AS-PCD is less than the prespecified value k, returns 0 otherwise. It also provides the vertices (i.e., data points) in a dominating set of size k of AS-PCD.

AS proximity regions are constructed with respect to the triangle tri and vertex regions are based on the center M="CC" for circumcenter of tri; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is M="CC" i.e., circumcenter of tri.

The vertices of triangle, tri, are labeled as $1,2,3$ according to the row number the vertex is recorded in tri. Loops are allowed in the digraph. It takes a long time for large number of vertices (i.e., large number of row numbers).

Usage

Idom.numASup.bnd.tri(Xp, k, tri, M = "CC")
Idom.numASup.bnd.tri(Xp, k, tri, M = "CC")

Arguments

`Xp`	A set of 2D points which constitute the vertices of the AS-PCD.
`k`	A positive integer to be tested for an upper bound for the domination number of AS-PCDs.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of `tri`; default is `M="CC"` i.e., the circumcenter of `tri`.

Value

A list with the elements

`domUB`	The suggested upper bound (to be checked) for the domination number of AS-PCD. It is prespecified as `k` in the function arguments.
`Idom.num.up.bnd`	The indicator for the upper bound for domination number of AS-PCD being the specified value `k` or not. It returns 1 if the upper bound is `k`, and 0 otherwise.
`ind.dom.set`	The vertices (i.e., data points) in the dominating set of size `k` if it exists, otherwise it yields `NULL`.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Idom.numASup.bnd.tri(Xp,1,Tr)

for (k in 1:n)
  print(c(k,Idom.numASup.bnd.tri(Xp,k,Tr,M)))

Idom.numASup.bnd.tri(Xp,k=4,Tr,M)

P<-c(.4,.2)
Idom.numASup.bnd.tri(P,1,Tr,M)

Idom.numASup.bnd.tri(rbind(Xp,Xp),k=2,Tr,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Idom.numASup.bnd.tri(Xp,1,Tr)

for (k in 1:n)
  print(c(k,Idom.numASup.bnd.tri(Xp,k,Tr,M)))

Idom.numASup.bnd.tri(Xp,k=4,Tr,M)

P<-c(.4,.2)
Idom.numASup.bnd.tri(P,1,Tr,M)

Idom.numASup.bnd.tri(rbind(Xp,Xp),k=2,Tr,M)

## End(Not run)

The indicator for `k` being an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - standard equilateral triangle case

Description

Returns $I($ domination number of CS-PCD is less than or equal to k $)$ where the vertices of the CS-PCD are the data points Xp, that is, returns 1 if the domination number of CS-PCD is less than the prespecified value k, returns 0 otherwise. It also provides the vertices (i.e., data points) in a dominating set of size k of CS-PCD.

CS proximity region is constructed with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with expansion parameter $t>0$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$ (which is equivalent to the circumcenter of $T_e$ ).

Edges of $T_e$ , $AB$ , $BC$ , $AC$ , are also labeled as 3, 1, and 2, respectively. Loops are allowed in the digraph. It takes a long time for large number of vertices (i.e., large number of row numbers).

See also (Ceyhan (2012)).

Usage

Idom.numCSup.bnd.std.tri(Xp, k, t, M = c(1, 1, 1))
Idom.numCSup.bnd.std.tri(Xp, k, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of CS-PCD.
`k`	A positive integer representing an upper bound for the domination number of CS-PCD.
`t`	A positive real number which serves as the expansion parameter in CS proximity region in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .

Value

A list with two elements

`domUB`	The upper bound `k` (to be checked) for the domination number of CS-PCD. It is prespecified as `k` in the function arguments.
`Idom.num.up.bnd`	The indicator for the upper bound for domination number of CS-PCD being the specified value `k` or not. It returns 1 if the upper bound is `k`, and 0 otherwise.
`ind.domset`	The vertices (i.e., data points) in the dominating set of size `k` if it exists, otherwise it is `NULL`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

Idom.numCSup.bnd.std.tri(Xp,1,t,M)

for (k in 1:n)
  print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$Idom.num.up.bnd))
  print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$domUB))

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

Idom.numCSup.bnd.std.tri(Xp,1,t,M)

for (k in 1:n)
  print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$Idom.num.up.bnd))
  print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$domUB))

## End(Not run)

Indicator for an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - one triangle case

Description

CS proximity region is constructed with respect to the triangle tri $=T(A,B,C)$ with expansion parameter $t>0$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of tri; default is $M=(1,1,1)$ i.e., the center of mass of tri.

Edges of tri, $AB$ , $BC$ , $AC$ , are also labeled as 3, 1, and 2, respectively. Loops are allowed in the digraph.

See also (Ceyhan (2012)).

Caveat: It takes a long time for large number of vertices (i.e., large number of row numbers).

Usage

Idom.numCSup.bnd.tri(Xp, k, tri, t, M = c(1, 1, 1))
Idom.numCSup.bnd.tri(Xp, k, tri, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of CS-PCD.
`k`	A positive integer to be tested for an upper bound for the domination number of CS-PCDs.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region in the triangle `tri`.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is $M=(1,1,1)$ , i.e. the center of mass of `tri`.

Value

A list with two elements

`domUB`	The upper bound `k` (to be checked) for the domination number of CS-PCD. It is prespecified as `k` in the function arguments.
`Idom.num.up.bnd`	The indicator for the upper bound for domination number of CS-PCD being the specified value `k` or not. It returns 1 if the upper bound is `k`, and 0 otherwise.
`ind.domset`	The vertices (i.e., data points) in the dominating set of size `k` if it exists, otherwise it is `NULL`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-.5

Idom.numCSup.bnd.tri(Xp,1,Tr,t,M)

for (k in 1:n)
  print(c(k,Idom.numCSup.bnd.tri(Xp,k,Tr,t,M)))

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-.5

Idom.numCSup.bnd.tri(Xp,1,Tr,t,M)

for (k in 1:n)
  print(c(k,Idom.numCSup.bnd.tri(Xp,k,Tr,t,M)))

## End(Not run)

The indicator for the set of points `S` being a dominating set or not for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns $I($ S a dominating set of AS-PCD $)$ , that is, returns 1 if S is a dominating set of AS-PCD, returns 0 otherwise.

AS-PCD has vertex set Xp and AS proximity region is constructed with vertex regions based on the center M="CC" for circumcenter of tri; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is M="CC" i.e., circumcenter of tri whose vertices are also labeled as edges 1, 2, and 3, respectively.

See also (Ceyhan (2005, 2010)).

Usage

Idom.setAStri(S, Xp, tri, M = "CC")
Idom.setAStri(S, Xp, tri, M = "CC")

Arguments

`S`	A set of 2D points which is to be tested for being a dominating set for the AS-PCDs.
`Xp`	A set of 2D points which constitute the vertices of the AS-PCD.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of `tri`; default is `M="CC"` i.e., the circumcenter of `tri`.

Value

$I($ S a dominating set of AS-PCD $)$ , that is, returns 1 if S is a dominating set of AS-PCD whose vertices are the data points in Xp; returns 0 otherwise, where AS proximity region is constructed in the triangle tri.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

S<-rbind(Xp[1,],Xp[2,])
Idom.setAStri(S,Xp,Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setAStri(S,Xp,Tr,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
Idom.setAStri(S,Xp,Tr,M)

Idom.setAStri(c(.2,.5),Xp,Tr,M)
Idom.setAStri(c(.2,.5),c(.2,.5),Tr,M)
Idom.setAStri(Xp[5,],Xp[2,],Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
Idom.setAStri(S,Xp[3,],Tr,M)

Idom.setAStri(Xp,Xp,Tr,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
Idom.setAStri(Xp,P,Tr,M)
Idom.setAStri(S,P,Tr,M)
Idom.setAStri(S,Xp,Tr,M)

Idom.setAStri(rbind(S,S),Xp,Tr,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

S<-rbind(Xp[1,],Xp[2,])
Idom.setAStri(S,Xp,Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setAStri(S,Xp,Tr,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
Idom.setAStri(S,Xp,Tr,M)

Idom.setAStri(c(.2,.5),Xp,Tr,M)
Idom.setAStri(c(.2,.5),c(.2,.5),Tr,M)
Idom.setAStri(Xp[5,],Xp[2,],Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
Idom.setAStri(S,Xp[3,],Tr,M)

Idom.setAStri(Xp,Xp,Tr,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
Idom.setAStri(Xp,P,Tr,M)
Idom.setAStri(S,P,Tr,M)
Idom.setAStri(S,Xp,Tr,M)

Idom.setAStri(rbind(S,S),Xp,Tr,M)

## End(Not run)

The indicator for the set of points `S` being a dominating set or not for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns $I($ S a dominating set of the CS-PCD $)$ where the vertices of the CS-PCD are the data set Xp), that is, returns 1 if S is a dominating set of CS-PCD, returns 0 otherwise.

Edges of $T_e$ , $AB$ , $BC$ , $AC$ , are also labeled as 3, 1, and 2, respectively.

See also (Ceyhan (2012)).

Usage

Idom.setCSstd.tri(S, Xp, t, M = c(1, 1, 1))
Idom.setCSstd.tri(S, Xp, t, M = c(1, 1, 1))

Arguments

`S`	A set of 2D points which is to be tested for being a dominating set for the CS-PCDs.
`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`t`	A positive real number which serves as the expansion parameter in CS proximity region in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .

Value

$I($ S a dominating set of the CS-PCD $)$ , that is, returns 1 if S is a dominating set of CS-PCD, returns 0 otherwise, where CS proximity region is constructed in the standard equilateral triangle $T_e$

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setCSstd.tri(S,Xp,t,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setCSstd.tri(S,Xp,t,M)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setCSstd.tri(S,Xp,t,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setCSstd.tri(S,Xp,t,M)

## End(Not run)

The indicator for the set of points `S` being a dominating set or not for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case

Description

Returns $I($ S a dominating set of CS-PCD whose vertices are the data set Xp $)$ , that is, returns 1 if S is a dominating set of CS-PCD, returns 0 otherwise.

CS proximity region is constructed with respect to the triangle tri with the expansion parameter $t>0$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is $M=(1,1,1)$ i.e., the center of mass of tri.

The triangle tri $=T(A,B,C)$ has edges $AB$ , $BC$ , $AC$ which are also labeled as edges 3, 1, and 2, respectively.

See also (Ceyhan (2012)).

Usage

Idom.setCStri(S, Xp, tri, t, M = c(1, 1, 1))
Idom.setCStri(S, Xp, tri, t, M = c(1, 1, 1))

Arguments

`S`	A set of 2D points which is to be tested for being a dominating set for the CS-PCDs.
`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region constructed in the triangle `tri`.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`.

Value

$I($ S a dominating set of the CS-PCD $)$ , that is, returns 1 if S is a dominating set of CS-PCD whose vertices are the data points in Xp; returns 0 otherwise, where CS proximity region is constructed in the triangle tri

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

tau<-.5
S<-rbind(Xp[1,],Xp[2,])
Idom.setCStri(S,Xp,Tr,tau,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setCStri(S,Xp,Tr,tau,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

tau<-.5
S<-rbind(Xp[1,],Xp[2,])
Idom.setCStri(S,Xp,Tr,tau,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setCStri(S,Xp,Tr,tau,M)

## End(Not run)

The indicator for the set of points `S` being a dominating set or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns $I($ S a dominating set of PE-PCD whose vertices are the data points Xp $)$ for S in the standard equilateral triangle, that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise.

PE proximity region is constructed with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with expansion parameter $r \ge 1$ and vertex regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ , i.e., the center of mass of $T_e$ (which is also equivalent to the circumcenter of $T_e$ ). Vertices of $T_e$ are also labeled as 1, 2, and 3, respectively.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

Idom.setPEstd.tri(S, Xp, r, M = c(1, 1, 1))
Idom.setPEstd.tri(S, Xp, r, M = c(1, 1, 1))

Arguments

`S`	A set of 2D points whose PE proximity regions are considered.
`Xp`	A set of 2D points which constitutes the vertices of the PE-PCD.
`r`	A positive real number which serves as the expansion parameter in PE proximity region in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ ; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .

Value

$I($ S a dominating set of PE-PCD $)$ for S in the standard equilateral triangle, that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise, where PE proximity region is constructed in the standard equilateral triangle $T_e$ .

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setPEstd.tri(S,Xp,r,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
Idom.setPEstd.tri(S,Xp[3,],r,M)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setPEstd.tri(S,Xp,r,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
Idom.setPEstd.tri(S,Xp[3,],r,M)

## End(Not run)

The indicator for the set of points `S` being a dominating set or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns $I($ S a dominating set of PE-PCD whose vertices are the data set Xp $)$ , that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise.

PE proximity region is constructed with respect to the triangle tri with the expansion parameter $r \ge 1$ and vertex regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri or based on the circumcenter of tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri. The triangle tri $=T(A,B,C)$ has edges $AB$ , $BC$ , $AC$ which are also labeled as edges 3, 1, and 2, respectively.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

Idom.setPEtri(S, Xp, tri, r, M = c(1, 1, 1))
Idom.setPEtri(S, Xp, tri, r, M = c(1, 1, 1))

Arguments

`S`	A set of 2D points which is to be tested for being a dominating set for the PE-PCDs.
`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region constructed in the triangle `tri`; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.

Value

$I($ S a dominating set of PE-PCD $)$ , that is, returns 1 if S is a dominating set of PE-PCD whose vertices are the data points in Xp; and returns 0 otherwise, where PE proximity region is constructed in the triangle tri.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setPEtri(S,Xp,Tr,r,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setPEtri(S,Xp,Tr,r,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
Idom.setPEtri(S,Xp,Tr,r,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setPEtri(S,Xp,Tr,r,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setPEtri(S,Xp,Tr,r,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
Idom.setPEtri(S,Xp,Tr,r,M)

## End(Not run)

Check whether a point is inside a circle

Description

Checks if the point p lies in the circle with center cent and radius rad, denoted as C(cent,rad). So, it returns 1 or TRUE if p is inside the circle, and 0 otherwise.

boundary is a logical argument (default=FALSE) to include boundary or not, so if it is TRUE, the function checks if the point, p, lies in the closure of the circle (i.e., interior and boundary combined) else it checks if p lies in the interior of the circle.

Usage

in.circle(p, cent, rad, boundary = TRUE)
in.circle(p, cent, rad, boundary = TRUE)

Arguments

`p`	A 2D point to be checked whether it is inside the circle or not.
`cent`	A 2D point in Cartesian coordinates which serves as the center of the circle.
`rad`	A positive real number which serves as the radius of the circle.
`boundary`	A logical parameter (default=`TRUE`) to include boundary or not, so if it is `TRUE`, the function checks if the point, `p`, lies in the closure of the circle (i.e., interior and boundary combined); else, it checks if `p` lies in the interior of the circle.

Value

Indicator for the point p being inside the circle or not, i.e., returns 1 or TRUE if p is inside the circle, and 0 otherwise.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
cent<-c(1,1); rad<-1; p<-c(1.4,1.2)
#try also cent<-runif(2); rad<-runif(1); p<-runif(2);

in.circle(p,cent,rad)

p<-c(.4,-.2)
in.circle(p,cent,rad)

p<-c(1,0)
in.circle(p,cent,rad)
in.circle(p,cent,rad,boundary=FALSE)

## End(Not run)

## Not run: 
cent<-c(1,1); rad<-1; p<-c(1.4,1.2)
#try also cent<-runif(2); rad<-runif(1); p<-runif(2);

in.circle(p,cent,rad)

p<-c(.4,-.2)
in.circle(p,cent,rad)

p<-c(1,0)
in.circle(p,cent,rad)
in.circle(p,cent,rad,boundary=FALSE)

## End(Not run)

Check whether a point is inside a tetrahedron

Description

Checks if the point p lies in the tetrahedron, th, using the barycentric coordinates, generally denoted as $(\alpha,\beta,\gamma)$ . If all (normalized or non-normalized) barycentric coordinates are positive then the point p is inside the tetrahedron, if all are nonnegative with one or more are zero, then p falls on the boundary. If some of the barycentric coordinates are negative, then p falls outside the tetrahedron.

boundary is a logical argument (default=FALSE) to include boundary or not, so if it is TRUE, the function checks if the point, p, lies in the closure of the tetrahedron (i.e., interior and boundary combined) else it checks if p lies in the interior of the tetrahedron.

Usage

in.tetrahedron(p, th, boundary = TRUE)
in.tetrahedron(p, th, boundary = TRUE)

Arguments

`p`	A 3D point to be checked whether it is inside the tetrahedron or not.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`boundary`	A logical parameter (default=`TRUE`) to include boundary or not, so if it is `TRUE`, the function checks if the point, `p`, lies in the closure of the tetrahedron (i.e., interior and boundary combined); else, it checks if `p` lies in the interior of the tetrahedron.

Value

A list with two elements

`in.tetra`	A logical output, if the point, `p`, is inside the tetrahedron, `th`, it is `TRUE`, else it is `FALSE`.
`barycentric`	The barycentric coordinates of the point `p` with respect to the tetrahedron, `th`.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3); P<-c(.1,.1,.1)
tetra<-rbind(A,B,C,D)

in.tetrahedron(P,tetra,boundary = FALSE)

in.tetrahedron(C,tetra)
in.tetrahedron(C,tetra,boundary = FALSE)

n1<-5; n2<-5; n<-n1+n2
Xp<-rbind(cbind(runif(n1),runif(n1,0,sqrt(3)/2),runif(n1,0,sqrt(6)/3)),
          runif.tetra(n2,tetra)$g)

in.tetra<-vector()
for (i in 1:n)
{in.tetra<-c(in.tetra,in.tetrahedron(Xp[i,],tetra,boundary = TRUE)$in.tetra) }

in.tetra
dat.tet<-Xp[in.tetra,]
if (is.vector(dat.tet)) {dat.tet<-matrix(dat.tet,nrow=1)}

Xlim<-range(tetra[,1],Xp[,1])
Ylim<-range(tetra[,2],Xp[,2])
Zlim<-range(tetra[,3],Xp[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi=40,theta=40,
bty = "g", pch = 20, cex = 1,
ticktype="detailed",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05),zlim=Zlim+zd*c(-.05,.05))
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
plot3D::points3D(dat.tet[,1],dat.tet[,2],dat.tet[,3],pch=4, add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

in.tetrahedron(P,tetra) #this works fine

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3); P<-c(.1,.1,.1)
tetra<-rbind(A,B,C,D)

in.tetrahedron(P,tetra,boundary = FALSE)

in.tetrahedron(C,tetra)
in.tetrahedron(C,tetra,boundary = FALSE)

n1<-5; n2<-5; n<-n1+n2
Xp<-rbind(cbind(runif(n1),runif(n1,0,sqrt(3)/2),runif(n1,0,sqrt(6)/3)),
          runif.tetra(n2,tetra)$g)

in.tetra<-vector()
for (i in 1:n)
{in.tetra<-c(in.tetra,in.tetrahedron(Xp[i,],tetra,boundary = TRUE)$in.tetra) }

in.tetra
dat.tet<-Xp[in.tetra,]
if (is.vector(dat.tet)) {dat.tet<-matrix(dat.tet,nrow=1)}

Xlim<-range(tetra[,1],Xp[,1])
Ylim<-range(tetra[,2],Xp[,2])
Zlim<-range(tetra[,3],Xp[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi=40,theta=40,
bty = "g", pch = 20, cex = 1,
ticktype="detailed",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05),zlim=Zlim+zd*c(-.05,.05))
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
plot3D::points3D(dat.tet[,1],dat.tet[,2],dat.tet[,3],pch=4, add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

in.tetrahedron(P,tetra) #this works fine

## End(Not run)

Check whether all points in a data set are inside the triangle

Description

Checks if all the data points in the 2D data set, Xp, lie in the triangle, tri, using the barycentric coordinates, generally denoted as $(\alpha,\beta,\gamma)$ .

If all (normalized or non-normalized) barycentric coordinates of a point are positive then the point is inside the triangle, if all are nonnegative with one or more are zero, then the point falls in the boundary. If some of the barycentric coordinates are negative, then the point falls outside the triangle.

boundary is a logical argument (default=TRUE) to include boundary or not, so if it is TRUE, the function checks if a point lies in the closure of the triangle (i.e., interior and boundary combined); else, it checks if the point lies in the interior of the triangle.

Usage

in.tri.all(Xp, tri, boundary = TRUE)
in.tri.all(Xp, tri, boundary = TRUE)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`boundary`	A logical parameter (default=`FALSE`) to include boundary or not, so if it is `TRUE`, the function checks if a point lies in the closure of the triangle (i.e., interior and boundary combined) else it checks if the point lies in the interior of the triangle.

Value

A logical output, if all data points in Xp are inside the triangle, tri, the output is TRUE, else it is FALSE.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2); p<-c(1.4,1.2)

Tr<-rbind(A,B,C)

in.tri.all(p,Tr)

#for the vertex A
in.tri.all(A,Tr)
in.tri.all(A,Tr,boundary = FALSE)

#for a point on the edge AB
D3<-(A+B)/2
in.tri.all(D3,Tr)
in.tri.all(D3,Tr,boundary = FALSE)

#data set
n<-10
Xp<-cbind(runif(n),runif(n))
in.tri.all(Xp,Tr,boundary = TRUE)

Xp<-runif.std.tri(n)$gen.points
in.tri.all(Xp,Tr)
in.tri.all(Xp,Tr,boundary = FALSE)

Xp<-runif.tri(n,Tr)$g
in.tri.all(Xp,Tr)
in.tri.all(Xp,Tr,boundary = FALSE)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2); p<-c(1.4,1.2)

Tr<-rbind(A,B,C)

in.tri.all(p,Tr)

#for the vertex A
in.tri.all(A,Tr)
in.tri.all(A,Tr,boundary = FALSE)

#for a point on the edge AB
D3<-(A+B)/2
in.tri.all(D3,Tr)
in.tri.all(D3,Tr,boundary = FALSE)

#data set
n<-10
Xp<-cbind(runif(n),runif(n))
in.tri.all(Xp,Tr,boundary = TRUE)

Xp<-runif.std.tri(n)$gen.points
in.tri.all(Xp,Tr)
in.tri.all(Xp,Tr,boundary = FALSE)

Xp<-runif.tri(n,Tr)$g
in.tri.all(Xp,Tr)
in.tri.all(Xp,Tr,boundary = FALSE)

## End(Not run)

Check whether a point is inside a triangle

Description

Checks if the point p lies in the triangle, tri, using the barycentric coordinates, generally denoted as $(\alpha,\beta,\gamma)$ .

If all (normalized or non-normalized) barycentric coordinates are positive then the point p is inside the triangle, if all are nonnegative with one or more are zero, then p falls in the boundary. If some of the barycentric coordinates are negative, then p falls outside the triangle.

boundary is a logical argument (default=TRUE) to include boundary or not, so if it is TRUE, the function checks if the point, p, lies in the closure of the triangle (i.e., interior and boundary combined); else, it checks if p lies in the interior of the triangle.

Usage

in.triangle(p, tri, boundary = TRUE)
in.triangle(p, tri, boundary = TRUE)

Arguments

`p`	A 2D point to be checked whether it is inside the triangle or not.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`boundary`	A logical parameter (default=`TRUE`) to include boundary or not, so if it is `TRUE`, the function checks if the point, `p`, lies in the closure of the triangle (i.e., interior and boundary combined); else, it checks if `p` lies in the interior of the triangle.

Value

A list with two elements

`in.tri`	A logical output, it is `TRUE`, if the point, `p`, is inside the triangle, `tri`, else it is `FALSE`.
`barycentric`	The barycentric coordinates $(\alpha,\beta,\gamma)$ of the point `p` with respect to the triangle, `tri`.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2); p<-c(1.4,1.2)
Tr<-rbind(A,B,C)
in.triangle(p,Tr)

p<-c(.4,-.2)
in.triangle(p,Tr)

#for the vertex A
in.triangle(A,Tr)
in.triangle(A,Tr,boundary = FALSE)

#for a point on the edge AB
D3<-(A+B)/2
in.triangle(D3,Tr)
in.triangle(D3,Tr,boundary = FALSE)

#for a NA entry point
p<-c(NA,.2)
in.triangle(p,Tr)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2); p<-c(1.4,1.2)
Tr<-rbind(A,B,C)
in.triangle(p,Tr)

p<-c(.4,-.2)
in.triangle(p,Tr)

#for the vertex A
in.triangle(A,Tr)
in.triangle(A,Tr,boundary = FALSE)

#for a point on the edge AB
D3<-(A+B)/2
in.triangle(D3,Tr)
in.triangle(D3,Tr,boundary = FALSE)

#for a NA entry point
p<-c(NA,.2)
in.triangle(p,Tr)

## End(Not run)

Incidence matrix for Arc Slice Proximity Catch Digraphs (AS-PCDs) - multiple triangle case

Description

Returns the incidence matrix for the AS-PCD whose vertices are a given 2D numerical data set, Xp, in the convex hull of Yp which is partitioned by the Delaunay triangles based on Yp points.

AS proximity regions are defined with respect to the Delaunay triangles based on Yp points and vertex regions are based on the center M="CC" for circumcenter of each Delaunay triangle or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle; default is M="CC" i.e., circumcenter of each triangle. Loops are allowed, so the diagonal entries are all equal to 1.

See (Ceyhan (2005, 2010)) for more on AS-PCDs. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

inci.matAS(Xp, Yp, M = "CC")
inci.matAS(Xp, Yp, M = "CC")

Arguments

`Xp`	A set of 2D points which constitute the vertices of the AS-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`M`	The center of the triangle. `"CC"` stands for circumcenter of each Delaunay triangle or 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle; default is `M="CC"` i.e., the circumcenter of each triangle.

Value

Incidence matrix for the AS-PCD whose vertices are the 2D data set, Xp, and AS proximity regions are defined in the Delaunay triangles based on Yp points.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-"CC"  #try also M<-c(1,1,1)

IM<-inci.matAS(Xp,Yp,M)
IM
dom.num.greedy(IM)  #try also dom.num.exact(IM)  #this might take a long time for large  nx

IM<-inci.matAS(Xp,Yp[1:3,],M)

inci.matAS(Xp,rbind(Yp,Yp))

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-"CC"  #try also M<-c(1,1,1)

IM<-inci.matAS(Xp,Yp,M)
IM
dom.num.greedy(IM)  #try also dom.num.exact(IM)  #this might take a long time for large  nx

IM<-inci.matAS(Xp,Yp[1:3,],M)

inci.matAS(Xp,rbind(Yp,Yp))

## End(Not run)

Incidence matrix for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns the incidence matrix for the AS-PCD whose vertices are the given 2D numerical data set, Xp.

AS proximity regions are defined with respect to the triangle tri $=T(v=1,v=2,v=3)$ and vertex regions based on the center M="CC" for circumcenter of tri; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is M="CC" i.e., circumcenter of tri. Loops are allowed, so the diagonal entries are all equal to 1.

See also (Ceyhan (2005, 2010)).

Usage

inci.matAStri(Xp, tri, M = "CC")
inci.matAStri(Xp, tri, M = "CC")

Arguments

`Xp`	A set of 2D points which constitute the vertices of AS-PCD.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of `tri`; default is `M="CC"` i.e., the circumcenter of `tri`.

Value

Incidence matrix for the AS-PCD whose vertices are 2D data set, Xp, and AS proximity regions are defined with respect to the triangle tri and vertex regions based on circumcenter.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

IM<-IncMatAStri(Xp,Tr,M)
IM

dom.num.greedy(IM)
dom.num.exact(IM)

## End(Not run)

## Not run: 

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

IM<-IncMatAStri(Xp,Tr,M)
IM

dom.num.greedy(IM)
dom.num.exact(IM)

## End(Not run)

Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) - multiple triangle case

Description

Returns the incidence matrix of Central Similarity Proximity Catch Digraph (CS-PCD) whose vertices are the data points in Xp in the multiple triangle case.

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points). For the incidence matrix loops are allowed, so the diagonal entries are all equal to 1.

Usage

inci.matCS(Xp, Yp, t, M = c(1, 1, 1))
inci.matCS(Xp, Yp, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle, default for $M=(1,1,1)$ which is the center of mass of each triangle.

Value

Incidence matrix for the CS-PCD with vertices being 2D data set, Xp. CS proximity regions are constructed with respect to the Delaunay triangles and M-edge regions.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2014). “Comparison of Relative Density of Two Random Geometric Digraph Families in Testing Spatial Clustering.” TEST, 23(1), 100-134.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

t<-1.5  #try also t<-2

IM<-inci.matCS(Xp,Yp,t,M)
IM
dom.num.greedy(IM) #try also dom.num.exact(IM)  #takes a very long time for large nx, try smaller nx
Idom.num.up.bnd(IM,3)  #takes a very long time for large nx, try smaller nx

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

t<-1.5  #try also t<-2

IM<-inci.matCS(Xp,Yp,t,M)
IM
dom.num.greedy(IM) #try also dom.num.exact(IM)  #takes a very long time for large nx, try smaller nx
Idom.num.up.bnd(IM,3)  #takes a very long time for large nx, try smaller nx

## End(Not run)

Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) for 1D data - multiple interval case

Description

Returns the incidence matrix for the CS-PCD for a given 1D numerical data set, Xp, as the vertices of the digraph and Yp determines the end points of the intervals (in the multi-interval case). Loops are allowed, so the diagonal entries are all equal to 1.

CS proximity region is constructed with an expansion parameter $t>0$ and a centrality parameter $c \in (0,1)$ .

See also (Ceyhan (2016)).

Usage

inci.matCS1D(Xp, Yp, t, c = 0.5)
inci.matCS1D(Xp, Yp, t, c = 0.5)

Arguments

`Xp`	a set of 1D points which constitutes the vertices of the digraph.
`Yp`	a set of 1D points which constitutes the end points of the intervals that partition the real line.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

Incidence matrix for the CS-PCD with vertices being 1D data set, Xp, and Yp determines the end points of the intervals (the multi-interval case)

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

t<-2
c<-.4
a<-0; b<-10;
nx<-10; ny<-4

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

IM<-inci.matCS1D(Xp,Yp,t,c)
IM
dom.num.greedy(IM)

dom.num.exact(IM)  #might take a long time depending on nx

Idom.num.up.bnd(IM,5)

Arcs<-arcsCS1D(Xp,Yp,t,c)
Arcs
summary(Arcs)
plot(Arcs)

inci.matCS1D(Xp,Yp+10,t,c)

t<-2
c<-.4
a<-0; b<-10;
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

inci.matCS1D(Xp,Yp,t,c)

t<-2
c<-.4
a<-0; b<-10;
nx<-10; ny<-4

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

IM<-inci.matCS1D(Xp,Yp,t,c)
IM
dom.num.greedy(IM)

dom.num.exact(IM)  #might take a long time depending on nx

Idom.num.up.bnd(IM,5)

Arcs<-arcsCS1D(Xp,Yp,t,c)
Arcs
summary(Arcs)
plot(Arcs)

inci.matCS1D(Xp,Yp+10,t,c)

t<-2
c<-.4
a<-0; b<-10;
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

inci.matCS1D(Xp,Yp,t,c)

Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) for 1D data - one interval case

Description

Returns the incidence matrix for the CS-PCD for a given 1D numerical data set, Xp, as the vertices of the digraph and int determines the end points of the interval (in the one interval case). Loops are allowed, so the diagonal entries are all equal to 1.

CS proximity region is constructed with an expansion parameter $t > 0$ and a centrality parameter $c \in (0,1)$ .

See also (Ceyhan (2016)).

Usage

inci.matCSint(Xp, int, t, c = 0.5)
inci.matCSint(Xp, int, t, c = 0.5)

Arguments

`Xp`	a set of 1D points which constitutes the vertices of the digraph.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

Incidence matrix for the CS-PCD with vertices being 1D data set, Xp, and int determines the end points of the intervals (in the one interval case)

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

## Not run: 
c<-.4
t<-1
a<-0; b<-10; int<-c(a,b)

xf<-(int[2]-int[1])*.1

set.seed(123)

n<-10
Xp<-runif(n,a-xf,b+xf)

IM<-inci.matCSint(Xp,int,t,c)
IM

dom.num.greedy(IM)
Idom.num.up.bnd(IM,3)
dom.num.exact(IM)

inci.matCSint(Xp,int+10,t,c)

## End(Not run)

## Not run: 
c<-.4
t<-1
a<-0; b<-10; int<-c(a,b)

xf<-(int[2]-int[1])*.1

set.seed(123)

n<-10
Xp<-runif(n,a-xf,b+xf)

IM<-inci.matCSint(Xp,int,t,c)
IM

dom.num.greedy(IM)
Idom.num.up.bnd(IM,3)
dom.num.exact(IM)

inci.matCSint(Xp,int+10,t,c)

## End(Not run)

Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns the incidence matrix for the CS-PCD whose vertices are the given 2D numerical data set, Xp, in the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ .

CS proximity region is defined with respect to the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$ . Loops are allowed, so the diagonal entries are all equal to 1.

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

inci.matCSstd.tri(Xp, t, M = c(1, 1, 1))
inci.matCSstd.tri(Xp, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates. which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .

Value

Incidence matrix for the CS-PCD with vertices being 2D data set, Xp and CS proximity regions are defined in the standard equilateral triangle $T_e$ with M-edge regions.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

inc.mat<-inci.matCSstd.tri(Xp,t=1.25,M)
inc.mat
sum(inc.mat)-n
num.arcsCSstd.tri(Xp,t=1.25)

dom.num.greedy(inc.mat) #try also dom.num.exact(inc.mat)  #might take a long time for large n
Idom.num.up.bnd(inc.mat,1)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

inc.mat<-inci.matCSstd.tri(Xp,t=1.25,M)
inc.mat
sum(inc.mat)-n
num.arcsCSstd.tri(Xp,t=1.25)

dom.num.greedy(inc.mat) #try also dom.num.exact(inc.mat)  #might take a long time for large n
Idom.num.up.bnd(inc.mat,1)

## End(Not run)

Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case

Description

Returns the incidence matrix for the CS-PCD whose vertices are the given 2D numerical data set, Xp, in the triangle tri $=T(v=1,v=2,v=3)$ .

CS proximity regions are constructed with respect to triangle tri with expansion parameter $t>0$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is $M=(1,1,1)$ i.e., the center of mass of tri. Loops are allowed, so the diagonal entries are all equal to 1.

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

inci.matCStri(Xp, tri, t, M = c(1, 1, 1))
inci.matCStri(Xp, tri, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of CS-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`.

Value

Incidence matrix for the CS-PCD with vertices being 2D data set, Xp, in the triangle tri with edge regions based on center M

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

IM<-inci.matCStri(Xp,Tr,t=1.25,M)
IM
dom.num.greedy(IM) #try also dom.num.exact(IM)
Idom.num.up.bnd(IM,3)

inci.matCStri(Xp,Tr,t=1.5,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

IM<-inci.matCStri(Xp,Tr,t=1.25,M)
IM
dom.num.greedy(IM) #try also dom.num.exact(IM)
Idom.num.up.bnd(IM,3)

inci.matCStri(Xp,Tr,t=1.5,M)

## End(Not run)

Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - multiple triangle case

Description

Returns the incidence matrix of Proportional Edge Proximity Catch Digraph (PE-PCD) whose vertices are the data points in Xp in the multiple triangle case.

Each Delaunay triangle is first converted to an (nonscaled) basic triangle so that M will be the same type of center for each Delaunay triangle (this conversion is not necessary when M is $CM$ ).

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points). For the incidence matrix loops are allowed, so the diagonal entries are all equal to 1.

Usage

inci.matPE(Xp, Yp, r, M = c(1, 1, 1))
inci.matPE(Xp, Yp, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as `M="CC"`), default for $M=(1,1,1)$ which is the center of mass of each triangle.

Value

Incidence matrix for the PE-PCD with vertices being 2D data set, Xp. PE proximity regions are constructed with respect to the Delaunay triangles and M-vertex regions.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

r<-1.5  #try also r<-2

IM<-inci.matPE(Xp,Yp,r,M)
IM
dom.num.greedy(IM)
#try also dom.num.exact(IM)
#might take a long time in this brute-force fashion ignoring the
#disconnected nature of the digraph inherent by the geometric construction of it

## End(Not run)

## Not run: 
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

r<-1.5  #try also r<-2

IM<-inci.matPE(Xp,Yp,r,M)
IM
dom.num.greedy(IM)
#try also dom.num.exact(IM)
#might take a long time in this brute-force fashion ignoring the
#disconnected nature of the digraph inherent by the geometric construction of it

## End(Not run)

Incidence matrix for Proportional-Edge Proximity Catch Digraphs (PE-PCDs) for 1D data - multiple interval case

Description

Returns the incidence matrix for the PE-PCD for a given 1D numerical data set, Xp, as the vertices of the digraph and Yp determines the end points of the intervals (in the multi-interval case). Loops are allowed, so the diagonal entries are all equal to 1.

PE proximity region is constructed with an expansion parameter $r \ge 1$ and a centrality parameter $c \in (0,1)$ .

See also (Ceyhan (2012)).

Usage

inci.matPE1D(Xp, Yp, r, c = 0.5)
inci.matPE1D(Xp, Yp, r, c = 0.5)

Arguments

`Xp`	a set of 1D points which constitutes the vertices of the digraph.
`Yp`	a set of 1D points which constitutes the end points of the intervals that partition the real line.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

Incidence matrix for the PE-PCD with vertices being 1D data set, Xp, and Yp determines the end points of the intervals (in the multi-interval case)

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10;
nx<-10; ny<-4

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

IM<-inci.matPE1D(Xp,Yp,r,c)
IM

dom.num.greedy(IM)
Idom.num.up.bnd(IM,6)
dom.num.exact(IM)

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10;
nx<-10; ny<-4

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

IM<-inci.matPE1D(Xp,Yp,r,c)
IM

dom.num.greedy(IM)
Idom.num.up.bnd(IM,6)
dom.num.exact(IM)

## End(Not run)

Incidence matrix for Proportional-Edge Proximity Catch Digraphs (PE-PCDs) for 1D data - one interval case

Description

Returns the incidence matrix for the PE-PCD for a given 1D numerical data set, Xp, as the vertices of the digraph and int determines the end points of the interval (in the one interval case). Loops are allowed, so the diagonal entries are all equal to 1.

PE proximity region is constructed with an expansion parameter $r \ge 1$ and a centrality parameter $c \in (0,1)$ .

See also (Ceyhan (2012)).

Usage

inci.matPEint(Xp, int, r, c = 0.5)
inci.matPEint(Xp, int, r, c = 0.5)

Arguments

`Xp`	a set of 1D points which constitutes the vertices of the digraph.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

Incidence matrix for the PE-PCD with vertices being 1D data set, Xp, and int determines the end points of the intervals (in the one interval case)

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

xf<-(int[2]-int[1])*.1

set.seed(123)

n<-10
Xp<-runif(n,a-xf,b+xf)

IM<-inci.matPEint(Xp,int,r,c)
IM

dom.num.greedy(IM)
Idom.num.up.bnd(IM,6)
dom.num.exact(IM)

inci.matPEint(Xp,int+10,r,c)

## End(Not run)

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

xf<-(int[2]-int[1])*.1

set.seed(123)

n<-10
Xp<-runif(n,a-xf,b+xf)

IM<-inci.matPEint(Xp,int,r,c)
IM

dom.num.greedy(IM)
Idom.num.up.bnd(IM,6)
dom.num.exact(IM)

inci.matPEint(Xp,int+10,r,c)

## End(Not run)

Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns the incidence matrix for the PE-PCD whose vertices are the given 2D numerical data set, Xp, in the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ .

PE proximity region is constructed with respect to the standard equilateral triangle $T_e$ with expansion parameter $r \ge 1$ and vertex regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ , i.e., the center of mass of $T_e$ . Loops are allowed, so the diagonal entries are all equal to 1.

See also (Ceyhan (2005, 2010)).

Usage

inci.matPEstd.tri(Xp, r, M = c(1, 1, 1))
inci.matPEstd.tri(Xp, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .

Value

Incidence matrix for the PE-PCD with vertices being 2D data set, Xp in the standard equilateral triangle where PE proximity regions are defined with M-vertex regions.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

inc.mat<-inci.matPEstd.tri(Xp,r=1.25,M)
inc.mat
sum(inc.mat)-n
num.arcsPEstd.tri(Xp,r=1.25)

dom.num.greedy(inc.mat)
Idom.num.up.bnd(inc.mat,2) #try also dom.num.exact(inc.mat)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

inc.mat<-inci.matPEstd.tri(Xp,r=1.25,M)
inc.mat
sum(inc.mat)-n
num.arcsPEstd.tri(Xp,r=1.25)

dom.num.greedy(inc.mat)
Idom.num.up.bnd(inc.mat,2) #try also dom.num.exact(inc.mat)

## End(Not run)

Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns the incidence matrix for the PE-PCD whose vertices are the given 3D numerical data set, Xp, in the tetrahedron $th=T(v=1,v=2,v=3,v=4)$ .

PE proximity regions are constructed with respect to tetrahedron th with expansion parameter $r \ge 1$ and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM". Loops are allowed, so the diagonal entries are all equal to 1.

See also (Ceyhan (2005, 2010)).

Usage

inci.matPEtetra(Xp, th, r, M = "CM")
inci.matPEtetra(Xp, th, r, M = "CM")

Arguments

`Xp`	A set of 3D points which constitute the vertices of PE-PCD.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.

Value

Incidence matrix for the PE-PCD with vertices being 3D data set, Xp, in the tetrahedron th with vertex regions based on circumcenter or center of mass

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5

Xp<-runif.tetra(n,tetra)$g  #try also Xp<-c(.5,.5,.5)

M<-"CM"  #try also M<-"CC"
r<-1.5

IM<-inci.matPEtetra(Xp,tetra,r=1.25)  #uses the default M="CM"
IM<-inci.matPEtetra(Xp,tetra,r=1.25,M)
IM
dom.num.greedy(IM)
Idom.num.up.bnd(IM,3) #try also dom.num.exact(IM)  #this might take a long time for large n

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5

Xp<-runif.tetra(n,tetra)$g  #try also Xp<-c(.5,.5,.5)

M<-"CM"  #try also M<-"CC"
r<-1.5

IM<-inci.matPEtetra(Xp,tetra,r=1.25)  #uses the default M="CM"
IM<-inci.matPEtetra(Xp,tetra,r=1.25,M)
IM
dom.num.greedy(IM)
Idom.num.up.bnd(IM,3) #try also dom.num.exact(IM)  #this might take a long time for large n

## End(Not run)

Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns the incidence matrix for the PE-PCD whose vertices are the given 2D numerical data set, Xp, in the triangle tri $=T(v=1,v=2,v=3)$ .

PE proximity regions are constructed with respect to triangle tri with expansion parameter $r \ge 1$ and vertex regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri. Loops are allowed, so the diagonal entries are all equal to 1.

See also (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)).

Usage

inci.matPEtri(Xp, tri, r, M = c(1, 1, 1))
inci.matPEtri(Xp, tri, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of PE-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.

Value

Incidence matrix for the PE-PCD with vertices being 2D data set, Xp, in the triangle tri with vertex regions based on center M

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)
IM<-inci.matPEtri(Xp,Tr,r=1.25,M)

IM
dom.num.greedy(IM) #try also dom.num.exact(IM)
Idom.num.up.bnd(IM,3)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)
IM<-inci.matPEtri(Xp,Tr,r=1.25,M)

IM
dom.num.greedy(IM) #try also dom.num.exact(IM)
Idom.num.up.bnd(IM,3)

## End(Not run)

Region index inside the Gamma-1 region

Description

Returns the region index of the point p for the 6 regions in standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ , starting with 1 on the first one-sixth of the triangle, and numbering follows the counter-clockwise direction (see the plot in the examples). These regions are in the inner hexagon which is the Gamma-1 region for CS-PCD with $t=1$ if p is not in any of the 6 regions the function returns NA.

Usage

index.six.Te(p)
index.six.Te(p)

Arguments

`p`	A 2D point whose index for the 6 regions in standard equilateral triangle $T_e$ is determined.

Value

rel An integer between 1-6 (inclusive) or NA

Author(s)

Elvan Ceyhan

Examples

## Not run: 
P<-c(.4,.2)
index.six.Te(P)

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

h1<-c(1/2,sqrt(3)/18); h2<-c(2/3, sqrt(3)/9); h3<-c(2/3, 2*sqrt(3)/9);
h4<-c(1/2, 5*sqrt(3)/18); h5<-c(1/3, 2*sqrt(3)/9); h6<-c(1/3, sqrt(3)/9);

r1<-(h1+h6+CM)/3;r2<-(h1+h2+CM)/3;r3<-(h2+h3+CM)/3;
r4<-(h3+h4+CM)/3;r5<-(h4+h5+CM)/3;r6<-(h5+h6+CM)/3;

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
polygon(rbind(h1,h2,h3,h4,h5,h6))

txt<-rbind(h1,h2,h3,h4,h5,h6)
xc<-txt[,1]+c(-.02,.02,.02,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,0,0,0)
txt.str<-c("h1","h2","h3","h4","h5","h6")
text(xc,yc,txt.str)

txt<-rbind(Te,CM,r1,r2,r3,r4,r5,r6)
xc<-txt[,1]+c(-.02,.02,.02,0,0,0,0,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,0,0,0,0,0,0,0)
txt.str<-c("A","B","C","CM","1","2","3","4","5","6")
text(xc,yc,txt.str)

n<-10  #try also n<-40
Xp<-runif.std.tri(n)$gen.points

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

rsix<-vector()
for (i in 1:n)
  rsix<-c(rsix,index.six.Te(Xp[i,]))
rsix

plot(A,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,pch=".")
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
polygon(rbind(h1,h2,h3,h4,h5,h6))
text(Xp,labels=factor(rsix))

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.02,.02,0)
yc<-txt[,2]+c(.02,.02,.02,-.05)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
P<-c(.4,.2)
index.six.Te(P)

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

h1<-c(1/2,sqrt(3)/18); h2<-c(2/3, sqrt(3)/9); h3<-c(2/3, 2*sqrt(3)/9);
h4<-c(1/2, 5*sqrt(3)/18); h5<-c(1/3, 2*sqrt(3)/9); h6<-c(1/3, sqrt(3)/9);

r1<-(h1+h6+CM)/3;r2<-(h1+h2+CM)/3;r3<-(h2+h3+CM)/3;
r4<-(h3+h4+CM)/3;r5<-(h4+h5+CM)/3;r6<-(h5+h6+CM)/3;

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
polygon(rbind(h1,h2,h3,h4,h5,h6))

txt<-rbind(h1,h2,h3,h4,h5,h6)
xc<-txt[,1]+c(-.02,.02,.02,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,0,0,0)
txt.str<-c("h1","h2","h3","h4","h5","h6")
text(xc,yc,txt.str)

txt<-rbind(Te,CM,r1,r2,r3,r4,r5,r6)
xc<-txt[,1]+c(-.02,.02,.02,0,0,0,0,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,0,0,0,0,0,0,0)
txt.str<-c("A","B","C","CM","1","2","3","4","5","6")
text(xc,yc,txt.str)

n<-10  #try also n<-40
Xp<-runif.std.tri(n)$gen.points

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

rsix<-vector()
for (i in 1:n)
  rsix<-c(rsix,index.six.Te(Xp[i,]))
rsix

plot(A,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,pch=".")
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
polygon(rbind(h1,h2,h3,h4,h5,h6))
text(Xp,labels=factor(rsix))

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.02,.02,0)
yc<-txt[,2]+c(.02,.02,.02,-.05)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

## End(Not run)

The points of intersection of a line and a circle

Description

Returns the intersection point(s) of a line and a circle. The line is determined by the two points p1 and p2 and the circle is centered at point cent and has radius rad. If the circle does not intersect the line, the function yields NULL; if the circle intersects at only one point, it yields only that point; otherwise it yields both intersection points as output. When there are two intersection points, they are listed in the order of the $x$ -coordinates of p1 and p2; and if the $x$ -coordinates of p1 and p2 are equal, intersection points are listed in the order of $y$ -coordinates of p1 and p2.

Usage

intersect.line.circle(p1, p2, cent, rad)
intersect.line.circle(p1, p2, cent, rad)

Arguments

`p1`, `p2`	2D points that determine the straight line (i.e., through which the straight line passes).
`cent`	A 2D point representing the center of the circle.
`rad`	A positive real number representing the radius of the circle.

Value

point(s) of intersection between the circle and the line (if they do not intersect, the function yields NULL as the output)

Author(s)

Elvan Ceyhan

Examples

## Not run: 
P1<-c(.3,.2)*100
P2<-c(.6,.3)*100
cent<-c(1.1,1.1)*100
rad<-2*100

intersect.line.circle(P1,P2,cent,rad)
intersect.line.circle(P2,P1,cent,rad)
intersect.line.circle(P1,P1+c(0,1),cent,rad)
intersect.line.circle(P1+c(0,1),P1,cent,rad)

dist.point2line(cent,P1,P2)
rad2<-dist.point2line(cent,P1,P2)$d
intersect.line.circle(P1,P2,cent,rad2)
intersect.line.circle(P1,P2,cent,rad=.8)
intersect.line.circle(P1,P2,cent,rad=.78)

#plot of the line and the circle
A<-c(.3,.2); B<-c(.6,.3); cent<-c(1,1); rad<-2 #check dist.point2line(cent,A,B)$dis, .3

IPs<-intersect.line.circle(A,B,cent,rad)

xr<-range(A[1],B[1],cent[1])
xf<-(xr[2]-xr[1])*.1 #how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-rad-xf,xr[2]+rad+xf,l=20)  #try also l=100
lnAB<-Line(A,B,x)
y<-lnAB$y

Xlim<-range(x,cent[1])
Ylim<-range(y,A[2],B[2],cent[2]-rad,cent[2]+rad)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(rbind(A,B,cent),pch=1,asp=1,xlab="x",ylab="y",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
lines(x,y,lty=1)
interp::circles(cent[1],cent[2],rad)
IP.txt<-c()
if (!is.null(IPs))
{
  for (i in 1:(length(IPs)/2))
    IP.txt<-c(IP.txt,paste("I",i, sep = ""))
}
txt<-rbind(A,B,cent,IPs)
text(txt+cbind(rep(xd*.03,nrow(txt)),rep(-yd*.03,nrow(txt))),c("A","B","M",IP.txt))

## End(Not run)

## Not run: 
P1<-c(.3,.2)*100
P2<-c(.6,.3)*100
cent<-c(1.1,1.1)*100
rad<-2*100

intersect.line.circle(P1,P2,cent,rad)
intersect.line.circle(P2,P1,cent,rad)
intersect.line.circle(P1,P1+c(0,1),cent,rad)
intersect.line.circle(P1+c(0,1),P1,cent,rad)

dist.point2line(cent,P1,P2)
rad2<-dist.point2line(cent,P1,P2)$d
intersect.line.circle(P1,P2,cent,rad2)
intersect.line.circle(P1,P2,cent,rad=.8)
intersect.line.circle(P1,P2,cent,rad=.78)

#plot of the line and the circle
A<-c(.3,.2); B<-c(.6,.3); cent<-c(1,1); rad<-2 #check dist.point2line(cent,A,B)$dis, .3

IPs<-intersect.line.circle(A,B,cent,rad)

xr<-range(A[1],B[1],cent[1])
xf<-(xr[2]-xr[1])*.1 #how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-rad-xf,xr[2]+rad+xf,l=20)  #try also l=100
lnAB<-Line(A,B,x)
y<-lnAB$y

Xlim<-range(x,cent[1])
Ylim<-range(y,A[2],B[2],cent[2]-rad,cent[2]+rad)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(rbind(A,B,cent),pch=1,asp=1,xlab="x",ylab="y",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
lines(x,y,lty=1)
interp::circles(cent[1],cent[2],rad)
IP.txt<-c()
if (!is.null(IPs))
{
  for (i in 1:(length(IPs)/2))
    IP.txt<-c(IP.txt,paste("I",i, sep = ""))
}
txt<-rbind(A,B,cent,IPs)
text(txt+cbind(rep(xd*.03,nrow(txt)),rep(-yd*.03,nrow(txt))),c("A","B","M",IP.txt))

## End(Not run)

The point of intersection of a line and a plane

Description

Returns the point of the intersection of the line determined by the 3D points $p_1$ and $p_2$ and the plane spanned by 3D points p3, p4, and p5.

Usage

intersect.line.plane(p1, p2, p3, p4, p5)
intersect.line.plane(p1, p2, p3, p4, p5)

Arguments

`p1`, `p2`	3D points that determine the straight line (i.e., through which the straight line passes).
`p3`, `p4`, `p5`	3D points that determine the plane (i.e., through which the plane passes).

Value

The coordinates of the point of intersection the line determined by the 3D points $p_1$ and $p_2$ and the plane determined by 3D points p3, p4, and p5.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
L1<-c(2,4,6); L2<-c(1,3,5);
A<-c(1,10,3); B<-c(1,1,3); C<-c(3,9,12)

Pint<-intersect.line.plane(L1,L2,A,B,C)
Pint
pts<-rbind(L1,L2,A,B,C,Pint)

tr<-max(Dist(L1,L2),Dist(L1,Pint),Dist(L2,Pint))
tf<-tr*1.1 #how far to go at the lower and upper ends
in the x-coordinate
tsq<-seq(-tf,tf,l=5)  #try also l=10, 20, or 100

lnAB3D<-Line3D(L1,L2,tsq)
xl<-lnAB3D$x
yl<-lnAB3D$y
zl<-lnAB3D$z

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
xp<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
yp<-seq(yr[1]-yf,yr[2]+yf,l=5)  #try also l=10, 20, or 100

plABC<-Plane(A,B,C,xp,yp)
z.grid<-plABC$z

res<-persp(xp,yp,z.grid, xlab="x",ylab="y",zlab="z",theta = -30,
phi = 30, expand = 0.5,
col = "lightblue", ltheta = 120, shade = 0.05, ticktype = "detailed")
lines (trans3d(xl, yl, zl, pmat = res), col = 3)

Xlim<-range(xl,pts[,1])
Ylim<-range(yl,pts[,2])
Zlim<-range(zl,pts[,3])

xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::persp3D(z = z.grid, x = xp, y = yp, theta =225, phi = 30,
ticktype = "detailed"
,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),zlim=Zlim+zd*c(-.1,.1),
expand = 0.7, facets = FALSE, scale = TRUE)
        #plane spanned by points A, B, C
#add the defining points
plot3D::points3D(pts[,1],pts[,2],pts[,3], pch = ".", col = "black",
bty = "f", cex = 5,add=TRUE)
plot3D::points3D(Pint[1],Pint[2],Pint[3], pch = "*", col = "red",
bty = "f", cex = 5,add=TRUE)
plot3D::lines3D(xl, yl, zl, bty = "g", cex = 2,
ticktype = "detailed",add=TRUE)

## End(Not run)

## Not run: 
L1<-c(2,4,6); L2<-c(1,3,5);
A<-c(1,10,3); B<-c(1,1,3); C<-c(3,9,12)

Pint<-intersect.line.plane(L1,L2,A,B,C)
Pint
pts<-rbind(L1,L2,A,B,C,Pint)

tr<-max(Dist(L1,L2),Dist(L1,Pint),Dist(L2,Pint))
tf<-tr*1.1 #how far to go at the lower and upper ends
in the x-coordinate
tsq<-seq(-tf,tf,l=5)  #try also l=10, 20, or 100

lnAB3D<-Line3D(L1,L2,tsq)
xl<-lnAB3D$x
yl<-lnAB3D$y
zl<-lnAB3D$z

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
xp<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
yp<-seq(yr[1]-yf,yr[2]+yf,l=5)  #try also l=10, 20, or 100

plABC<-Plane(A,B,C,xp,yp)
z.grid<-plABC$z

res<-persp(xp,yp,z.grid, xlab="x",ylab="y",zlab="z",theta = -30,
phi = 30, expand = 0.5,
col = "lightblue", ltheta = 120, shade = 0.05, ticktype = "detailed")
lines (trans3d(xl, yl, zl, pmat = res), col = 3)

Xlim<-range(xl,pts[,1])
Ylim<-range(yl,pts[,2])
Zlim<-range(zl,pts[,3])

xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::persp3D(z = z.grid, x = xp, y = yp, theta =225, phi = 30,
ticktype = "detailed"
,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),zlim=Zlim+zd*c(-.1,.1),
expand = 0.7, facets = FALSE, scale = TRUE)
        #plane spanned by points A, B, C
#add the defining points
plot3D::points3D(pts[,1],pts[,2],pts[,3], pch = ".", col = "black",
bty = "f", cex = 5,add=TRUE)
plot3D::points3D(Pint[1],Pint[2],Pint[3], pch = "*", col = "red",
bty = "f", cex = 5,add=TRUE)
plot3D::lines3D(xl, yl, zl, bty = "g", cex = 2,
ticktype = "detailed",add=TRUE)

## End(Not run)

The point of intersection of two lines defined by two pairs of points

Description

Returns the intersection of two lines, first line passing through points p1 and q1 and second line passing through points p2 and q2. The points are chosen so that lines are well defined.

Usage

intersect2lines(p1, q1, p2, q2)
intersect2lines(p1, q1, p2, q2)

Arguments

`p1`, `q1`	2D points that determine the first straight line (i.e., through which the first straight line passes).
`p2`, `q2`	2D points that determine the second straight line (i.e., through which the second straight line passes).

Value

The coordinates of the point of intersection of the two lines, first passing through points p1 and q1 and second passing through points p2 and q2.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(-1.22,-2.33); B<-c(2.55,3.75); C<-c(0,6); D<-c(3,-2)

ip<-intersect2lines(A,B,C,D)
ip
pts<-rbind(A,B,C,D,ip)
xr<-range(pts[,1])
xf<-abs(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
lnAB<-Line(A,B,x)
lnCD<-Line(C,D,x)

y1<-lnAB$y
y2<-lnCD$y
Xlim<-range(x,pts)
Ylim<-range(y1,y2,pts)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
pf<-c(xd,-yd)*.025

#plot of the line joining A and B
plot(rbind(A,B,C,D),pch=1,xlab="x",ylab="y",
main="Point of Intersection of Two Lines",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
lines(x,y1,lty=1,col=1)
lines(x,y2,lty=1,col=2)
text(rbind(A+pf,B+pf),c("A","B"))
text(rbind(C+pf,D+pf),c("C","D"))
text(rbind(ip+pf),c("intersection\n point"))

## End(Not run)

## Not run: 
A<-c(-1.22,-2.33); B<-c(2.55,3.75); C<-c(0,6); D<-c(3,-2)

ip<-intersect2lines(A,B,C,D)
ip
pts<-rbind(A,B,C,D,ip)
xr<-range(pts[,1])
xf<-abs(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
lnAB<-Line(A,B,x)
lnCD<-Line(C,D,x)

y1<-lnAB$y
y2<-lnCD$y
Xlim<-range(x,pts)
Ylim<-range(y1,y2,pts)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
pf<-c(xd,-yd)*.025

#plot of the line joining A and B
plot(rbind(A,B,C,D),pch=1,xlab="x",ylab="y",
main="Point of Intersection of Two Lines",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
lines(x,y1,lty=1,col=1)
lines(x,y2,lty=1,col=2)
text(rbind(A+pf,B+pf),c("A","B"))
text(rbind(C+pf,D+pf),c("C","D"))
text(rbind(ip+pf),c("intersection\n point"))

## End(Not run)

Indices of the intervals where the 1D point(s) reside

Description

Returns the indices of intervals for all the points in 1D data set, Xp, as a vector.

Intervals are based on Yp and left end interval is labeled as 1, the next interval as 2, and so on.

Usage

interval.indices.set(Xp, Yp)
interval.indices.set(Xp, Yp)

Arguments

`Xp`	A set of 1D points for which the indices of intervals are to be determined.
`Yp`	A set of 1D points from which intervals are constructed.

Value

The vector of indices of the intervals in which points in the 1D data set, Xp, reside

Author(s)

Elvan Ceyhan

Examples

## Not run: 
a<-0; b<-10; int<-c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1
Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)  #try also Yp<-runif(ny,a+1,b-1)

ind<-interval.indices.set(Xp,Yp)
ind

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(a,b,Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0), xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit),pch=".")
abline(h=0)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
text(Xp,yjit,labels=factor(ind))

## End(Not run)

## Not run: 
a<-0; b<-10; int<-c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1
Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)  #try also Yp<-runif(ny,a+1,b-1)

ind<-interval.indices.set(Xp,Yp)
ind

jit<-.1
yjit<-runif(nx,-jit,jit)

Xlim<-range(a,b,Xp,Yp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0), xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),ylim=3*c(-jit,jit),pch=".")
abline(h=0)
points(Xp, yjit,pch=".",cex=3)
abline(v=Yp,lty=2)
text(Xp,yjit,labels=factor(ind))

## End(Not run)

Check a point belong to a given data set

Description

returns TRUE if the point p of any dimension is inside the data set Xp of the same dimension as p; otherwise returns FALSE.

Usage

is.in.data(p, Xp)
is.in.data(p, Xp)

Arguments

`p`	A 2D point for which the function checks membership to the data set `Xp`.
`Xp`	A set of 2D points representing the set of data points.

Value

TRUE if p belongs to the data set Xp.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
n<-10
Xp<-cbind(runif(n),runif(n))

P<-Xp[7,]
is.in.data(P,Xp)
is.in.data(P,Xp[7,])

P<-Xp[7,]+10^(-7)
is.in.data(P,Xp)

P<-Xp[7,]+10^(-9)
is.in.data(P,Xp)

is.in.data(P,P)

is.in.data(c(2,2),c(2,2))

#for 1D data
n<-10
Xp<-runif(n)

P<-Xp[7]
is.in.data(P,Xp[7])  #this works because both entries are treated as 1D vectors but
#is.in.data(P,Xp) does not work since entries are treated as vectors of different dimensions

Xp<-as.matrix(Xp)
is.in.data(P,Xp)
#this works, because P is a 1D point, and Xp is treated as a set of 10 1D points

P<-Xp[7]+10^(-7)
is.in.data(P,Xp)

P<-Xp[7]+10^(-9)
is.in.data(P,Xp)

is.in.data(P,P)

#for 3D data
n<-10
Xp<-cbind(runif(n),runif(n),runif(n))

P<-Xp[7,]
is.in.data(P,Xp)
is.in.data(P,Xp[7,])

P<-Xp[7,]+10^(-7)
is.in.data(P,Xp)

P<-Xp[7,]+10^(-9)
is.in.data(P,Xp)

is.in.data(P,P)

n<-10
Xp<-cbind(runif(n),runif(n))
P<-Xp[7,]
is.in.data(P,Xp)

## End(Not run)

## Not run: 
n<-10
Xp<-cbind(runif(n),runif(n))

P<-Xp[7,]
is.in.data(P,Xp)
is.in.data(P,Xp[7,])

P<-Xp[7,]+10^(-7)
is.in.data(P,Xp)

P<-Xp[7,]+10^(-9)
is.in.data(P,Xp)

is.in.data(P,P)

is.in.data(c(2,2),c(2,2))

#for 1D data
n<-10
Xp<-runif(n)

P<-Xp[7]
is.in.data(P,Xp[7])  #this works because both entries are treated as 1D vectors but
#is.in.data(P,Xp) does not work since entries are treated as vectors of different dimensions

Xp<-as.matrix(Xp)
is.in.data(P,Xp)
#this works, because P is a 1D point, and Xp is treated as a set of 10 1D points

P<-Xp[7]+10^(-7)
is.in.data(P,Xp)

P<-Xp[7]+10^(-9)
is.in.data(P,Xp)

is.in.data(P,P)

#for 3D data
n<-10
Xp<-cbind(runif(n),runif(n),runif(n))

P<-Xp[7,]
is.in.data(P,Xp)
is.in.data(P,Xp[7,])

P<-Xp[7,]+10^(-7)
is.in.data(P,Xp)

P<-Xp[7,]+10^(-9)
is.in.data(P,Xp)

is.in.data(P,P)

n<-10
Xp<-cbind(runif(n),runif(n))
P<-Xp[7,]
is.in.data(P,Xp)

## End(Not run)

Check the argument is a point of a given dimension

Description

Returns TRUE if the argument p is a numeric point of dimension dim (default is dim=2); otherwise returns FALSE.

Usage

is.point(p, dim = 2)
is.point(p, dim = 2)

Arguments

`p`	A `vector` to be checked to see it is a point of dimension `dim` or not.
`dim`	A positive integer representing the dimension of the argument `p`.

Value

TRUE if p is a vector of dimension dim.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(-1.22,-2.33); B<-c(2.55,3.75,4)
is.point(A)
is.point(A,1)

is.point(B)
is.point(B,3)

## End(Not run)

## Not run: 
A<-c(-1.22,-2.33); B<-c(2.55,3.75,4)
is.point(A)
is.point(A,1)

is.point(B)
is.point(B,3)

## End(Not run)

Check whether a triangle is a standard equilateral triangle

Description

Checks whether the triangle, tri, is the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ or not.

Usage

is.std.eq.tri(tri)
is.std.eq.tri(tri)

Arguments

tri

A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

TRUE if tri is a standard equilateral triangle, else FALSE.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)  #try adding +10^(-16) to each vertex
is.std.eq.tri(Te)

is.std.eq.tri(rbind(B,C,A))

Tr<-rbind(A,B,-C)
is.std.eq.tri(Tr)

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
is.std.eq.tri(Tr)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)  #try adding +10^(-16) to each vertex
is.std.eq.tri(Te)

is.std.eq.tri(rbind(B,C,A))

Tr<-rbind(A,B,-C)
is.std.eq.tri(Tr)

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
is.std.eq.tri(Tr)

## End(Not run)

The `k` furthest points in a data set from vertices in each $CC$ -vertex region in a triangle

Description

An object of class "Extrema". Returns the k furthest data points among the data set, Xp, in each $CC$ -vertex region from the vertex in the triangle, tri $=T(A,B,C)$ , vertices are stacked row-wise. Vertex region labels/numbers correspond to the row number of the vertex in tri.

ch.all.intri is for checking whether all data points are inside tri (default is FALSE). If some of the data points are not inside tri and ch.all.intri=TRUE, then the function yields an error message. If some of the data points are not inside tri and ch.all.intri=FALSE, then the function yields the closest points to edges among the data points inside tri (yields NA if there are no data points inside tri).

In the extrema, $ext$ , in the output, the first k entries are the k furthest points from vertex 1, second k entries are k furthest points are from vertex 2, and last k entries are the k furthest points from vertex 3. If data size does not allow, NA's are inserted for some or all of the k furthest points for each vertex.

Usage

kfr2vertsCCvert.reg(Xp, tri, k, ch.all.intri = FALSE)
kfr2vertsCCvert.reg(Xp, tri, k, ch.all.intri = FALSE)

Arguments

`Xp`	A set of 2D points representing the set of data points.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`k`	A positive integer. `k` furthest data points in each $CC$ -vertex region are to be found if exists, else `NA` are provided for (some of) the `k` furthest points.
`ch.all.intri`	A logical argument (default=`FALSE`) to check whether all data points are inside the triangle `tri`. So, if it is `TRUE`, the function checks if all data points are inside the closure of the triangle (i.e., interior and boundary combined) else it does not.

Value

A list with the elements

`txt1`	Vertex labels are $A=1$ , $B=2$ , and $C=3$ (correspond to row number in Extremum Points).
`txt2`	A shorter description of the distances as `"Distances of k furthest points in the vertex regions to Vertices"`.
`type`	Type of the extrema points
`desc`	A short description of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, `k` furthest points from vertices in each $CC$ -vertex region in the triangle `tri`.
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, it is `tri` for this function.
`cent`	The center point used for construction of vertex regions
`ncent`	Name of the center, `cent`, it is circumcenter `"CC"` for this function.
`regions`	Vertex regions inside the triangle, `tri`, provided as a `list`
`region.names`	Names of the vertex regions as `"vr=1"`, `"vr=2"`, and `"vr=3"`
`region.centers`	Centers of mass of the vertex regions inside $T_b$ .
`dist2ref`	Distances from `k` furthest points in each vertex region to the corresponding vertex (each row representing a vertex in `tri`). Among the distances the first `k` entries are the distances from the `k` furthest points from vertex 1 to vertex 1, second `k` entries are distances from the `k` furthest points from vertex 2 to vertex 2, and the last `k` entries are the distances from the `k` furthest points from vertex 3 to vertex 3.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20
k<-3

set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-kfr2vertsCCvert.reg(Xp,Tr,k)
Ext
summary(Ext)
plot(Ext)

Xp2<-rbind(Xp,c(.2,.4))
kfr2vertsCCvert.reg(Xp2,Tr,k)
#try also kfr2vertsCCvert.reg(Xp2,Tr,k,ch.all.intri = TRUE)

kf2v<-Ext

CC<-circumcenter.tri(Tr)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main=paste(k," Furthest Points in CC-Vertex Regions \n from the Vertices",sep=""),
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(kf2v$ext,pch=4,col=2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.06,.08,.05,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.04,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20
k<-3

set.seed(1)
Xp<-runif.tri(n,Tr)$g

Ext<-kfr2vertsCCvert.reg(Xp,Tr,k)
Ext
summary(Ext)
plot(Ext)

Xp2<-rbind(Xp,c(.2,.4))
kfr2vertsCCvert.reg(Xp2,Tr,k)
#try also kfr2vertsCCvert.reg(Xp2,Tr,k,ch.all.intri = TRUE)

kf2v<-Ext

CC<-circumcenter.tri(Tr)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",
main=paste(k," Furthest Points in CC-Vertex Regions \n from the Vertices",sep=""),
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(kf2v$ext,pch=4,col=2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.06,.08,.05,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.04,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The line joining two distinct 2D points `a` and `b`

Description

An object of class "Lines". Returns the equation, slope, intercept, and $y$ -coordinates of the line crossing two distinct 2D points a and b with $x$ -coordinates provided in vector x.

This function is different from the line function in the standard stats package in R in the sense that Line(a,b,x) fits the line passing through points a and b and returns various quantities (see below) for this line and x is the $x$ -coordinates of the points we want to find on the Line(a,b,x) while line(a,b) fits the line robustly whose $x$ -coordinates are in a and $y$ -coordinates are in b.

Line(a,b,x) and line(x,Line(A,B,x)$y) would yield the same straight line (i.e., the line with the same coefficients.)

Usage

Line(a, b, x)
Line(a, b, x)

Arguments

`a`, `b`	2D points that determine the straight line (i.e., through which the straight line passes).
`x`	A scalar or a `vector` of scalars representing the $x$ -coordinates of the line.

Value

A list with the elements

`desc`	A description of the line
`mtitle`	The `"main"` title for the plot of the line
`points`	The input points `a` and `b` through which the straight line passes (stacked row-wise, i.e., row 1 is point `a` and row 2 is point `b`).
`x`	The input scalar or `vector` which constitutes the $x$ -coordinates of the point(s) of interest on the line.
`y`	The output scalar or `vector` which constitutes the $y$ -coordinates of the point(s) of interest on the line. If `x` is a scalar, then `y` will be a scalar and if `x` is a `vector` of scalars, then `y` will be a `vector` of scalars.
`slope`	Slope of the line, `Inf` is allowed, passing through points `a` and `b`
`intercept`	Intercept of the line passing through points `a` and `b`
`equation`	Equation of the line passing through points `a` and `b`

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(-1.22,-2.33); B<-c(2.55,3.75)

xr<-range(A,B);
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100

lnAB<-Line(A,B,x)
lnAB
summary(lnAB)
plot(lnAB)

line(A,B)
#this takes vector A as the x points and vector B as the y points and fits the line
#for example, try
x=runif(100); y=x+(runif(100,-.05,.05))
plot(x,y)
line(x,y)

x<-lnAB$x
y<-lnAB$y
Xlim<-range(x,A,B)
if (!is.na(y[1])) {Ylim<-range(y,A,B)} else {Ylim<-range(A,B)}
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
pf<-c(xd,-yd)*.025

#plot of the line joining A and B
plot(rbind(A,B),pch=1,xlab="x",ylab="y",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
if (!is.na(y[1])) {lines(x,y,lty=1)} else {abline(v=A[1])}
text(rbind(A+pf,B+pf),c("A","B"))
int<-round(lnAB$intercep,2)  #intercept
sl<-round(lnAB$slope,2)  #slope
text(rbind((A+B)/2+pf*3),ifelse(is.na(int),paste("x=",A[1]),
ifelse(sl==0,paste("y=",int),
ifelse(sl==1,ifelse(sign(int)<0,paste("y=x",int),paste("y=x+",int)),
ifelse(sign(int)<0,paste("y=",sl,"x",int),paste("y=",sl,"x+",int))))))

## End(Not run)

## Not run: 
A<-c(-1.22,-2.33); B<-c(2.55,3.75)

xr<-range(A,B);
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100

lnAB<-Line(A,B,x)
lnAB
summary(lnAB)
plot(lnAB)

line(A,B)
#this takes vector A as the x points and vector B as the y points and fits the line
#for example, try
x=runif(100); y=x+(runif(100,-.05,.05))
plot(x,y)
line(x,y)

x<-lnAB$x
y<-lnAB$y
Xlim<-range(x,A,B)
if (!is.na(y[1])) {Ylim<-range(y,A,B)} else {Ylim<-range(A,B)}
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
pf<-c(xd,-yd)*.025

#plot of the line joining A and B
plot(rbind(A,B),pch=1,xlab="x",ylab="y",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
if (!is.na(y[1])) {lines(x,y,lty=1)} else {abline(v=A[1])}
text(rbind(A+pf,B+pf),c("A","B"))
int<-round(lnAB$intercep,2)  #intercept
sl<-round(lnAB$slope,2)  #slope
text(rbind((A+B)/2+pf*3),ifelse(is.na(int),paste("x=",A[1]),
ifelse(sl==0,paste("y=",int),
ifelse(sl==1,ifelse(sign(int)<0,paste("y=x",int),paste("y=x+",int)),
ifelse(sign(int)<0,paste("y=",sl,"x",int),paste("y=",sl,"x+",int))))))

## End(Not run)

The line crossing 3D point `p` in the direction of `vector` `v` (or if `v` is a point, in direction of $v-r_0$ )

Description

An object of class "Lines3D". Returns the equation, $x$ -, $y$ -, and $z$ -coordinates of the line crossing 3D point $r_0$ in the direction of vector v (of if v is a point, in the direction of $v-r_0$ ) with the parameter t being provided in vector t.

Usage

Line3D(p, v, t, dir.vec = TRUE)
Line3D(p, v, t, dir.vec = TRUE)

Arguments

`p`	A 3D point through which the straight line passes.
`v`	A 3D `vector` which determines the direction of the straight line (i.e., the straight line would be parallel to this vector) if the `dir.vec=TRUE`, otherwise it is 3D point and $v-r_0$ determines the direction of the the straight line.
`t`	A scalar or a `vector` of scalars representing the parameter of the coordinates of the line (for the form: $x=p_0 + a t$ , $y=y_0 + b t$ , and $z=z_0 + c t$ where $r_0=(p_0,y_0,z_0)$ and $v=(a,b,c)$ if `dir.vec=TRUE`, else $v-r_0=(a,b,c)$ ).
`dir.vec`	A logical argument about `v`, if `TRUE` `v` is treated as a vector, else `v` is treated as a point and so the direction `vector` is taken to be $v-r_0$ .

Value

A list with the elements

`desc`	A description of the line
`mtitle`	The `"main"` title for the plot of the line
`pts`	The input points that determine a line and/or a plane, `NULL` for this function.
`pnames`	The names of the input points that determine a line and/or a plane, `NULL` for this function.
`vecs`	The point `p` and the `vector` `v` (if `dir.vec=TRUE`) or the point `v` (if `dir.vec=FALSE`). The first row is `p` and the second row is `v`.
`vec.names`	The names of the point `p` and the `vector` `v` (if `dir.vec=TRUE`) or the point `v` (if `dir.vec=FALSE`).
`x`, `y`, `z`	The $x$ -, $y$ -, and $z$ -coordinates of the point(s) of interest on the line.
`tsq`	The scalar or the `vector` of the parameter in defining each coordinate of the line for the form: $x=p_0 + a t$ , $y=y_0 + b t$ , and $z=z_0 + c t$ where $r_0=(p_0,y_0,z_0)$ and $v=(a,b,c)$ if `dir.vec=TRUE`, else $v-r_0=(a,b,c)$ .
`equation`	Equation of the line passing through point `p` in the direction of the `vector` `v` (if `dir.vec=TRUE`) else in the direction of $v-r_0$ . The line equation is in the form: $x=p_0 + a t$ , $y=y_0 + b t$ , and $z=z_0 + c t$ where $r_0=(p_0,y_0,z_0)$ and $v=(a,b,c)$ if `dir.vec=TRUE`, else $v-r_0=(a,b,c)$ .

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1,10,3); B<-c(1,1,3);

vecs<-rbind(A,B)

Line3D(A,B,.1)
Line3D(A,B,.1,dir.vec=FALSE)

tr<-range(vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=5)  #try also l=10, 20, or 100

lnAB3D<-Line3D(A,B,tsq)
#try also lnAB3D<-Line3D(A,B,tsq,dir.vec=FALSE)
lnAB3D
summary(lnAB3D)
plot(lnAB3D)

x<-lnAB3D$x
y<-lnAB3D$y
z<-lnAB3D$z

zr<-range(z)
zf<-(zr[2]-zr[1])*.2
Bv<-B*tf*5

Xlim<-range(x)
Ylim<-range(y)
Zlim<-range(z)

xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

Dr<-A+min(tsq)*B

plot3D::lines3D(x, y, z, phi = 0, bty = "g",
main="Line Crossing A \n in the Direction of OB",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.1,.1),
        pch = 20, cex = 2, ticktype = "detailed")
plot3D::arrows3D(Dr[1],Dr[2],Dr[3]+zf,Dr[1]+Bv[1],
Dr[2]+Bv[2],Dr[3]+zf+Bv[3], add=TRUE)
plot3D::points3D(A[1],A[2],A[3],add=TRUE)
plot3D::arrows3D(A[1],A[2],A[3]-2*zf,A[1],A[2],A[3],lty=2, add=TRUE)
plot3D::text3D(A[1],A[2],A[3]-2*zf,labels="initial point",add=TRUE)
plot3D::text3D(A[1],A[2],A[3]+zf/2,labels=expression(r[0]),add=TRUE)
plot3D::arrows3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+3*zf+Bv[3]/2,
Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+zf+Bv[3]/2,lty=2, add=TRUE)
plot3D::text3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+3*zf+Bv[3]/2,
labels="direction vector",add=TRUE)
plot3D::text3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,
Dr[3]+zf+Bv[3]/2,labels="v",add=TRUE)
plot3D::text3D(0,0,0,labels="O",add=TRUE)

## End(Not run)

## Not run: 
A<-c(1,10,3); B<-c(1,1,3);

vecs<-rbind(A,B)

Line3D(A,B,.1)
Line3D(A,B,.1,dir.vec=FALSE)

tr<-range(vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=5)  #try also l=10, 20, or 100

lnAB3D<-Line3D(A,B,tsq)
#try also lnAB3D<-Line3D(A,B,tsq,dir.vec=FALSE)
lnAB3D
summary(lnAB3D)
plot(lnAB3D)

x<-lnAB3D$x
y<-lnAB3D$y
z<-lnAB3D$z

zr<-range(z)
zf<-(zr[2]-zr[1])*.2
Bv<-B*tf*5

Xlim<-range(x)
Ylim<-range(y)
Zlim<-range(z)

xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

Dr<-A+min(tsq)*B

plot3D::lines3D(x, y, z, phi = 0, bty = "g",
main="Line Crossing A \n in the Direction of OB",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.1,.1),
        pch = 20, cex = 2, ticktype = "detailed")
plot3D::arrows3D(Dr[1],Dr[2],Dr[3]+zf,Dr[1]+Bv[1],
Dr[2]+Bv[2],Dr[3]+zf+Bv[3], add=TRUE)
plot3D::points3D(A[1],A[2],A[3],add=TRUE)
plot3D::arrows3D(A[1],A[2],A[3]-2*zf,A[1],A[2],A[3],lty=2, add=TRUE)
plot3D::text3D(A[1],A[2],A[3]-2*zf,labels="initial point",add=TRUE)
plot3D::text3D(A[1],A[2],A[3]+zf/2,labels=expression(r[0]),add=TRUE)
plot3D::arrows3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+3*zf+Bv[3]/2,
Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+zf+Bv[3]/2,lty=2, add=TRUE)
plot3D::text3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+3*zf+Bv[3]/2,
labels="direction vector",add=TRUE)
plot3D::text3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,
Dr[3]+zf+Bv[3]/2,labels="v",add=TRUE)
plot3D::text3D(0,0,0,labels="O",add=TRUE)

## End(Not run)

The vertices of the Arc Slice (AS) Proximity Region in the standard basic triangle

Description

Returns the end points of the line segments and arc-slices that constitute the boundary of AS proximity region for a point in the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Vertex regions are based on the center M="CC" for circumcenter of $T_b$ ; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_b$ ; default is M="CC" the circumcenter of $T_b$ . rv is the index of the vertex region p resides, with default=NULL.

If p is outside $T_b$ , it returns NULL for the proximity region. dec is the number of decimals (default is 4) to round the barycentric coordinates when checking whether the end points fall on the boundary of the triangle $T_b$ or not (so as not to miss the intersection points due to precision in the decimals).

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010)).

Usage

NASbasic.tri(p, c1, c2, M = "CC", rv = NULL, dec = 4)
NASbasic.tri(p, c1, c2, M = "CC", rv = NULL, dec = 4)

Arguments

`p`	A 2D point whose AS proximity region is to be computed.
`c1`, `c2`	Positive real numbers representing the top vertex in standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ , $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle $T_b$ or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$ ; default is `M="CC"` i.e., the circumcenter of $T_b$ .
`rv`	The index of the `M`-vertex region containing the point, either `1,2,3` or `NULL` (default is `NULL`).
`dec`	a positive integer the number of decimals (default is 4) to round the barycentric coordinates when checking whether the end points fall on the boundary of the triangle $T_b$ or not.

Value

A list with the elements

`L`, `R`	The end points of the line segments on the boundary of the AS proximity region. Each row in `L` and `R` constitute a line segment on the boundary.
`Arc.Slices`	The end points of the arc-slices on the circular parts of the AS proximity region. Here points in row 1 and row 2 constitute the end points of one arc-slice, points on row 3 and row 4 constitute the end points for the next arc-slice and so on.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6  #try also c1<-.2; c2<-.2;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)

set.seed(1)
M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g);  #try also P1<-c(.3,.2)
NASbasic.tri(P1,c1,c2)  #default with M="CC"
NASbasic.tri(P1,c1,c2,M)

#or try
Rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
NASbasic.tri(P1,c1,c2,M,Rv)

NASbasic.tri(c(3,5),c1,c2,M)

P2<-c(.5,.4)
NASbasic.tri(P2,c1,c2,M)

P3<-c(1.5,.4)
NASbasic.tri(P3,c1,c2,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

#plot of the NAS region
P1<-as.numeric(runif.basic.tri(1,c1,c2)$g);
CC<-circumcenter.basic.tri(c1,c2)

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges.basic.tri(c1,c2,M)
rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
}
RV<-Tb[rv,]
rad<-Dist(P1,RV)

Int.Pts<-NASbasic.tri(P1,c1,c2,M)

Xlim<-range(Tb[,1],P1[1]+rad,P1[1]-rad)
Ylim<-range(Tb[,2],P1[2]+rad,P1[2]-rad)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(rbind(Tb,P1,rbind(Int.Pts$L,Int.Pts$R)))
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
interp::circles(P1[1],P1[2],rad,lty=2)
L<-Int.Pts$L; R<-Int.Pts$R
segments(L[,1], L[,2], R[,1], R[,2], lty=1,col=2)
Arcs<-Int.Pts$a;
if (!is.null(Arcs))
{
  K<-nrow(Arcs)/2
  for (i in 1:K)
  {A1<-Arcs[2*i-1,]; A2<-Arcs[2*i,];
  angles<-angle.str2end(A1,P1,A2)$c

  plotrix::draw.arc(P1[1],P1[2],rad,angle1=angles[1],angle2=angles[2],col=2)
  }
}

#proximity region with the triangle (i.e., for labeling the vertices of the NAS)
IP.txt<-intpts<-c()
if (!is.null(Int.Pts$a))
{
 intpts<-unique(round(Int.Pts$a,7))
  #this part is for labeling the intersection points of the spherical
  for (i in 1:(length(intpts)/2))
    IP.txt<-c(IP.txt,paste("I",i+1, sep = ""))
}
txt<-rbind(Tb,P1,cent,intpts)
txt.str<-c("A","B","C","P1",cent.name,IP.txt)
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.03,nrow(txt))),txt.str)

c1<-.4; c2<-.6;
P1<-c(.3,.2)
NASbasic.tri(P1,c1,c2,M)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6  #try also c1<-.2; c2<-.2;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)

set.seed(1)
M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g);  #try also P1<-c(.3,.2)
NASbasic.tri(P1,c1,c2)  #default with M="CC"
NASbasic.tri(P1,c1,c2,M)

#or try
Rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
NASbasic.tri(P1,c1,c2,M,Rv)

NASbasic.tri(c(3,5),c1,c2,M)

P2<-c(.5,.4)
NASbasic.tri(P2,c1,c2,M)

P3<-c(1.5,.4)
NASbasic.tri(P3,c1,c2,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

#plot of the NAS region
P1<-as.numeric(runif.basic.tri(1,c1,c2)$g);
CC<-circumcenter.basic.tri(c1,c2)

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges.basic.tri(c1,c2,M)
rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
}
RV<-Tb[rv,]
rad<-Dist(P1,RV)

Int.Pts<-NASbasic.tri(P1,c1,c2,M)

Xlim<-range(Tb[,1],P1[1]+rad,P1[1]-rad)
Ylim<-range(Tb[,2],P1[2]+rad,P1[2]-rad)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(rbind(Tb,P1,rbind(Int.Pts$L,Int.Pts$R)))
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
interp::circles(P1[1],P1[2],rad,lty=2)
L<-Int.Pts$L; R<-Int.Pts$R
segments(L[,1], L[,2], R[,1], R[,2], lty=1,col=2)
Arcs<-Int.Pts$a;
if (!is.null(Arcs))
{
  K<-nrow(Arcs)/2
  for (i in 1:K)
  {A1<-Arcs[2*i-1,]; A2<-Arcs[2*i,];
  angles<-angle.str2end(A1,P1,A2)$c

  plotrix::draw.arc(P1[1],P1[2],rad,angle1=angles[1],angle2=angles[2],col=2)
  }
}

#proximity region with the triangle (i.e., for labeling the vertices of the NAS)
IP.txt<-intpts<-c()
if (!is.null(Int.Pts$a))
{
 intpts<-unique(round(Int.Pts$a,7))
  #this part is for labeling the intersection points of the spherical
  for (i in 1:(length(intpts)/2))
    IP.txt<-c(IP.txt,paste("I",i+1, sep = ""))
}
txt<-rbind(Tb,P1,cent,intpts)
txt.str<-c("A","B","C","P1",cent.name,IP.txt)
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.03,nrow(txt))),txt.str)

c1<-.4; c2<-.6;
P1<-c(.3,.2)
NASbasic.tri(P1,c1,c2,M)

## End(Not run)

The vertices of the Arc Slice (AS) Proximity Region in a general triangle

Description

Returns the end points of the line segments and arc-slices that constitute the boundary of AS proximity region for a point in the triangle tri $=T(A,B,C)=$ (rv=1,rv=2,rv=3).

If p is outside of tri, it returns NULL for the proximity region. dec is the number of decimals (default is 4) to round the barycentric coordinates when checking the points fall on the boundary of the triangle tri or not (so as not to miss the intersection points due to precision in the decimals).

See also (Ceyhan (2005, 2010)).

Usage

NAStri(p, tri, M = "CC", rv = NULL, dec = 4)
NAStri(p, tri, M = "CC", rv = NULL, dec = 4)

Arguments

`p`	A 2D point whose AS proximity region is to be computed.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is `M="CC"` i.e., the circumcenter of `tri`.
`rv`	Index of the `M`-vertex region containing the point `p`, either `1,2,3` or `NULL` (default is `NULL`).
`dec`	a positive integer the number of decimals (default is 4) to round the barycentric coordinates when checking whether the end points fall on the boundary of the triangle `tri` or not.

Value

A list with the elements

`L`, `R`	End points of the line segments on the boundary of the AS proximity region. Each row in `L` and `R` constitute a pair of points that determine a line segment on the boundary.
`arc.slices`	The end points of the arc-slices on the circular parts of the AS proximity region. Here points in rows 1 and 2 constitute the end points of the first arc-slice, points on rows 3 and 4 constitute the end points for the next arc-slice and so on.
`Angles`	The angles (in radians) between the vectors joining arc slice end points to the point `p` with the horizontal line crossing the point `p`

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(1.3,1.2)
NAStri(P1,Tr,M)

#or try
Rv<-rel.vert.triCC(P1,Tr)$rv
NAStri(P1,Tr,M,Rv)

NAStri(c(3,5),Tr,M)

P2<-c(1.5,1.4)
NAStri(P2,Tr,M)

P3<-c(1.5,.4)
NAStri(P3,Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

CC<-circumcenter.tri(Tr)  #the circumcenter

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
rv<-rel.vert.triCC(P1,Tr)$rv
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
rv<-rel.vert.tri(P1,Tr,M)$rv
}
RV<-Tr[rv,]
rad<-Dist(P1,RV)

Int.Pts<-NAStri(P1,Tr,M)

#plot of the NAS region
Xlim<-range(Tr[,1],P1[1]+rad,P1[1]-rad)
Ylim<-range(Tr[,2],P1[2]+rad,P1[2]-rad)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
#asp=1 must be the case to have the arc properly placed in the figure
polygon(Tr)
points(rbind(Tr,P1,rbind(Int.Pts$L,Int.Pts$R)))
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
interp::circles(P1[1],P1[2],rad,lty=2)
L<-Int.Pts$L; R<-Int.Pts$R
segments(L[,1], L[,2], R[,1], R[,2], lty=1,col=2)
Arcs<-Int.Pts$a;
if (!is.null(Arcs))
{
  K<-nrow(Arcs)/2
  for (i in 1:K)
  {A1<-Int.Pts$arc[2*i-1,]; A2<-Int.Pts$arc[2*i,];
  angles<-angle.str2end(A1,P1,A2)$c

  test.ang1<-angles[1]+(.01)*(angles[2]-angles[1])
  test.Pnt<-P1+rad*c(cos(test.ang1),sin(test.ang1))
  if (!in.triangle(test.Pnt,Tr,boundary = TRUE)$i) {angles<-c(min(angles),max(angles)-2*pi)}
  plotrix::draw.arc(P1[1],P1[2],rad,angle1=angles[1],angle2=angles[2],col=2)
  }
}

#proximity region with the triangle (i.e., for labeling the vertices of the NAS)
IP.txt<-intpts<-c()
if (!is.null(Int.Pts$a))
{
 intpts<-unique(round(Int.Pts$a,7))
  #this part is for labeling the intersection points of the spherical
  for (i in 1:(length(intpts)/2))
    IP.txt<-c(IP.txt,paste("I",i+1, sep = ""))
}
txt<-rbind(Tr,P1,cent,intpts)
txt.str<-c("A","B","C","P1",cent.name,IP.txt)
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.03,nrow(txt))),txt.str)

P1<-c(.3,.2)
NAStri(P1,Tr,M)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(1.3,1.2)
NAStri(P1,Tr,M)

#or try
Rv<-rel.vert.triCC(P1,Tr)$rv
NAStri(P1,Tr,M,Rv)

NAStri(c(3,5),Tr,M)

P2<-c(1.5,1.4)
NAStri(P2,Tr,M)

P3<-c(1.5,.4)
NAStri(P3,Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

CC<-circumcenter.tri(Tr)  #the circumcenter

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
rv<-rel.vert.triCC(P1,Tr)$rv
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
rv<-rel.vert.tri(P1,Tr,M)$rv
}
RV<-Tr[rv,]
rad<-Dist(P1,RV)

Int.Pts<-NAStri(P1,Tr,M)

#plot of the NAS region
Xlim<-range(Tr[,1],P1[1]+rad,P1[1]-rad)
Ylim<-range(Tr[,2],P1[2]+rad,P1[2]-rad)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
#asp=1 must be the case to have the arc properly placed in the figure
polygon(Tr)
points(rbind(Tr,P1,rbind(Int.Pts$L,Int.Pts$R)))
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
interp::circles(P1[1],P1[2],rad,lty=2)
L<-Int.Pts$L; R<-Int.Pts$R
segments(L[,1], L[,2], R[,1], R[,2], lty=1,col=2)
Arcs<-Int.Pts$a;
if (!is.null(Arcs))
{
  K<-nrow(Arcs)/2
  for (i in 1:K)
  {A1<-Int.Pts$arc[2*i-1,]; A2<-Int.Pts$arc[2*i,];
  angles<-angle.str2end(A1,P1,A2)$c

  test.ang1<-angles[1]+(.01)*(angles[2]-angles[1])
  test.Pnt<-P1+rad*c(cos(test.ang1),sin(test.ang1))
  if (!in.triangle(test.Pnt,Tr,boundary = TRUE)$i) {angles<-c(min(angles),max(angles)-2*pi)}
  plotrix::draw.arc(P1[1],P1[2],rad,angle1=angles[1],angle2=angles[2],col=2)
  }
}

#proximity region with the triangle (i.e., for labeling the vertices of the NAS)
IP.txt<-intpts<-c()
if (!is.null(Int.Pts$a))
{
 intpts<-unique(round(Int.Pts$a,7))
  #this part is for labeling the intersection points of the spherical
  for (i in 1:(length(intpts)/2))
    IP.txt<-c(IP.txt,paste("I",i+1, sep = ""))
}
txt<-rbind(Tr,P1,cent,intpts)
txt.str<-c("A","B","C","P1",cent.name,IP.txt)
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.03,nrow(txt))),txt.str)

P1<-c(.3,.2)
NAStri(P1,Tr,M)

## End(Not run)

The end points of the Central Similarity (CS) Proximity Region for a point - one interval case

Description

Returns the end points of the interval which constitutes the CS proximity region for a point in the interval int $=(a,b)=$ (rv=1,rv=2).

CS proximity region is constructed with respect to the interval int with expansion parameter $t>0$ and centrality parameter $c \in (0,1)$ .

Vertex regions are based on the (parameterized) center, $M_c$ , which is $M_c=a+c(b-a)$ for the interval, int $=(a,b)$ . The CS proximity region is constructed whether x is inside or outside the interval int.

See also (Ceyhan (2016)).

Usage

NCSint(x, int, t, c = 0.5)
NCSint(x, int, t, c = 0.5)

Arguments

`x`	A 1D point for which CS proximity region is constructed.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

The interval which constitutes the CS proximity region for the point x

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

NCSint(7,int,t,c)
NCSint(17,int,t,c)
NCSint(1,int,t,c)
NCSint(-1,int,t,c)

NCSint(3,int,t,c)
NCSint(4,int,t,c)
NCSint(a,int,t,c)

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

NCSint(7,int,t,c)
NCSint(17,int,t,c)
NCSint(1,int,t,c)
NCSint(-1,int,t,c)

NCSint(3,int,t,c)
NCSint(4,int,t,c)
NCSint(a,int,t,c)

The vertices of the Central Similarity (CS) Proximity Region in a general triangle

Description

Returns the vertices of the CS proximity region (which is itself a triangle) for a point in the triangle tri $=T(A,B,C)=$ (rv=1,rv=2,rv=3).

CS proximity region is defined with respect to the triangle tri with expansion parameter $t>0$ and edge regions based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is $M=(1,1,1)$ i.e., the center of mass of tri.

Edge regions are labeled as 1,2,3 rowwise for the corresponding vertices of the triangle tri. re is the index of the edge region p resides, with default=NULL. If p is outside of tri, it returns NULL for the proximity region.

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

NCStri(p, tri, t, M = c(1, 1, 1), re = NULL)
NCStri(p, tri, t, M = c(1, 1, 1), re = NULL)

Arguments

`p`	A 2D point whose CS proximity region is to be computed.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`.
`re`	Index of the `M`-edge region containing the point `p`, either `1,2,3` or `NULL` (default is `NULL`).

Value

Vertices of the triangular region which constitutes the CS proximity region with expansion parameter $t>0$ and center M for a point p

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
tau<-1.5

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

n<-3
set.seed(1)
Xp<-runif.tri(n,Tr)$g

NCStri(Xp[1,],Tr,tau,M)

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(.4,.2)
NCStri(P1,Tr,tau,M)

#or try
re<-rel.edges.tri(P1,Tr,M)$re
NCStri(P1,Tr,tau,M,re)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
tau<-1.5

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

n<-3
set.seed(1)
Xp<-runif.tri(n,Tr)$g

NCStri(Xp[1,],Tr,tau,M)

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(.4,.2)
NCStri(P1,Tr,tau,M)

#or try
re<-rel.edges.tri(P1,Tr,M)$re
NCStri(P1,Tr,tau,M,re)

## End(Not run)

The vertices of the Proportional Edge (PE) Proximity Region in a standard basic triangle

Description

Returns the vertices of the PE proximity region (which is itself a triangle) for a point in the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))=$ (rv=1,rv=2,rv=3).

PE proximity region is defined with respect to the standard basic triangle $T_b$ with expansion parameter $r \ge 1$ and vertex regions based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the basic triangle $T_b$ or based on the circumcenter of $T_b$ ; default is $M=(1,1,1)$ , i.e., the center of mass of $T_b$ .

Vertex regions are labeled as $1,2,3$ rowwise for the vertices of the triangle $T_b$ . rv is the index of the vertex region p resides, with default=NULL. If p is outside of tri, it returns NULL for the proximity region.

See also (Ceyhan (2005, 2010)).

Usage

NPEbasic.tri(p, r, c1, c2, M = c(1, 1, 1), rv = NULL)
NPEbasic.tri(p, r, c1, c2, M = c(1, 1, 1), rv = NULL)

Arguments

`p`	A 2D point whose PE proximity region is to be computed.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c1`, `c2`	Positive real numbers representing the top vertex in standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ , $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle $T_b$ or the circumcenter of $T_b$ which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of $T_b$ .
`rv`	Index of the `M`-vertex region containing the point `p`, either `1,2,3` or `NULL` (default is `NULL`).

Value

Vertices of the triangular region which constitutes the PE proximity region with expansion parameter r and center M for a point p

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

r<-2

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also P1<-c(.4,.2)
NPEbasic.tri(P1,r,c1,c2,M)

#or try
Rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
NPEbasic.tri(P1,r,c1,c2,M,Rv)

P1<-c(1.4,1.2)
P2<-c(1.5,1.26)
NPEbasic.tri(P1,r,c1,c2,M) #gives an error if M=c(1.3,1.3)
#since center is not the circumcenter or not in the interior of the triangle

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

r<-2

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also P1<-c(.4,.2)
NPEbasic.tri(P1,r,c1,c2,M)

#or try
Rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
NPEbasic.tri(P1,r,c1,c2,M,Rv)

P1<-c(1.4,1.2)
P2<-c(1.5,1.26)
NPEbasic.tri(P1,r,c1,c2,M) #gives an error if M=c(1.3,1.3)
#since center is not the circumcenter or not in the interior of the triangle

## End(Not run)

The end points of the Proportional Edge (PE) Proximity Region for a point - one interval case

Description

Returns the end points of the interval which constitutes the PE proximity region for a point in the interval int $=(a,b)=$ (rv=1,rv=2). PE proximity region is constructed with respect to the interval int with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ .

Vertex regions are based on the (parameterized) center, $M_c$ , which is $M_c=a+c(b-a)$ for the interval, int $=(a,b)$ . The PE proximity region is constructed whether x is inside or outside the interval int.

See also (Ceyhan (2012)).

Usage

NPEint(x, int, r, c = 0.5)
NPEint(x, int, r, c = 0.5)

Arguments

`x`	A 1D point for which PE proximity region is constructed.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

The interval which constitutes the PE proximity region for the point x

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

NPEint(7,int,r,c)
NPEint(17,int,r,c)
NPEint(1,int,r,c)
NPEint(-1,int,r,c)

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

NPEint(7,int,r,c)
NPEint(17,int,r,c)
NPEint(1,int,r,c)
NPEint(-1,int,r,c)

The vertices of the Proportional Edge (PE) Proximity Region in the standard regular tetrahedron

Description

Returns the vertices of the PE proximity region (which is itself a tetrahedron) for a point in the standard regular tetrahedron $T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))=$ (rv=1,rv=2,rv=3,rv=4).

PE proximity region is defined with respect to the tetrahedron $T_h$ with expansion parameter $r \ge 1$ and vertex regions based on the circumcenter of $T_h$ (which is equivalent to the center of mass in the standard regular tetrahedron).

Vertex regions are labeled as 1,2,3,4 rowwise for the vertices of the tetrahedron $T_h$ . rv is the index of the vertex region p resides, with default=NULL. If p is outside of $T_h$ , it returns NULL for the proximity region.

See also (Ceyhan (2005, 2010)).

Usage

NPEstd.tetra(p, r, rv = NULL)
NPEstd.tetra(p, r, rv = NULL)

Arguments

`p`	A 3D point whose PE proximity region is to be computed.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`rv`	Index of the vertex region containing the point, either `1,2,3,4` or `NULL` (default is `NULL`).

Value

Vertices of the tetrahedron which constitutes the PE proximity region with expansion parameter r and circumcenter (or center of mass) for a point p in the standard regular tetrahedron

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-3
Xp<-runif.std.tetra(n)$g
r<-1.5
NPEstd.tetra(Xp[1,],r)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
NPEstd.tetra(Xp[1,],r,rv=RV)

NPEstd.tetra(c(-1,-1,-1),r,rv=NULL)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-3
Xp<-runif.std.tetra(n)$g
r<-1.5
NPEstd.tetra(Xp[1,],r)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
NPEstd.tetra(Xp[1,],r,rv=RV)

NPEstd.tetra(c(-1,-1,-1),r,rv=NULL)

## End(Not run)

The vertices of the Proportional Edge (PE) Proximity Region in a tetrahedron

Description

Returns the vertices of the PE proximity region (which is itself a tetrahedron) for a point in the tetrahedron th.

PE proximity region is defined with respect to the tetrahedron th with expansion parameter $r \ge 1$ and vertex regions based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM".

Vertex regions are labeled as 1,2,3,4 rowwise for the vertices of the tetrahedron th. rv is the index of the vertex region p resides, with default=NULL. If p is outside of th, it returns NULL for the proximity region.

See also (Ceyhan (2005, 2010)).

Usage

NPEtetra(p, th, r, M = "CM", rv = NULL)
NPEtetra(p, th, r, M = "CM", rv = NULL)

Arguments

`p`	A 3D point whose PE proximity region is to be computed.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.
`rv`	Index of the vertex region containing the point, either `1,2,3,4` (default is `NULL`).

Value

Vertices of the tetrahedron which constitutes the PE proximity region with expansion parameter r and circumcenter (or center of mass) for a point p in the tetrahedron

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
set.seed(1)
tetra<-rbind(A,B,C,D)+matrix(runif(12,-.25,.25),ncol=3)
n<-3  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.5

NPEtetra(Xp[1,],tetra,r)  #uses the default M="CM"
NPEtetra(Xp[1,],tetra,r,M="CC")

#or try
RV<-rel.vert.tetraCM(Xp[1,],tetra)$rv
NPEtetra(Xp[1,],tetra,r,M,rv=RV)

P1<-c(.1,.1,.1)
NPEtetra(P1,tetra,r,M)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
set.seed(1)
tetra<-rbind(A,B,C,D)+matrix(runif(12,-.25,.25),ncol=3)
n<-3  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.5

NPEtetra(Xp[1,],tetra,r)  #uses the default M="CM"
NPEtetra(Xp[1,],tetra,r,M="CC")

#or try
RV<-rel.vert.tetraCM(Xp[1,],tetra)$rv
NPEtetra(Xp[1,],tetra,r,M,rv=RV)

P1<-c(.1,.1,.1)
NPEtetra(P1,tetra,r,M)

## End(Not run)

The vertices of the Proportional Edge (PE) Proximity Region in a general triangle

Description

Returns the vertices of the PE proximity region (which is itself a triangle) for a point in the triangle tri $=T(A,B,C)=$ (rv=1,rv=2,rv=3).

PE proximity region is defined with respect to the triangle tri with expansion parameter $r \ge 1$ and vertex regions based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri or based on the circumcenter of tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri.

Vertex regions are labeled as $1,2,3$ rowwise for the vertices of the triangle tri. rv is the index of the vertex region p resides, with default=NULL. If p is outside of tri, it returns NULL for the proximity region.

See also (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)).

Usage

NPEtri(p, tri, r, M = c(1, 1, 1), rv = NULL)
NPEtri(p, tri, r, M = c(1, 1, 1), rv = NULL)

Arguments

`p`	A 2D point whose PE proximity region is to be computed.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.
`rv`	Index of the `M`-vertex region containing the point `p`, either `1,2,3` or `NULL` (default is `NULL`).

Value

Vertices of the triangular region which constitutes the PE proximity region with expansion parameter r and center M for a point p

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5

n<-3
set.seed(1)
Xp<-runif.tri(n,Tr)$g

NPEtri(Xp[3,],Tr,r,M)

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(.4,.2)
NPEtri(P1,Tr,r,M)

M<-c(1.3,1.3)
r<-2

P1<-c(1.4,1.2)
P2<-c(1.5,1.26)
NPEtri(P1,Tr,r,M)
NPEtri(P2,Tr,r,M)

#or try
Rv<-rel.vert.tri(P1,Tr,M)$rv
NPEtri(P1,Tr,r,M,Rv)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5

n<-3
set.seed(1)
Xp<-runif.tri(n,Tr)$g

NPEtri(Xp[3,],Tr,r,M)

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(.4,.2)
NPEtri(P1,Tr,r,M)

M<-c(1.3,1.3)
r<-2

P1<-c(1.4,1.2)
P2<-c(1.5,1.26)
NPEtri(P1,Tr,r,M)
NPEtri(P2,Tr,r,M)

#or try
Rv<-rel.vert.tri(P1,Tr,M)$rv
NPEtri(P1,Tr,r,M,Rv)

## End(Not run)

Number of arcs of Arc Slice Proximity Catch Digraphs (AS-PCDs) and related quantities of the induced subdigraphs for points in the Delaunay triangles - multiple triangle case

Description

An object of class "NumArcs". Returns the number of arcs and various other quantities related to the Delaunay triangles for Arc Slice Proximity Catch Digraph (AS-PCD) whose vertices are the data points in Xp in the multiple triangle case.

AS proximity regions are defined with respect to the Delaunay triangles based on Yp points and vertex regions in each triangle are based on the center M="CC" for circumcenter of each Delaunay triangle or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle; default is M="CC" i.e., circumcenter of each triangle.

See (Ceyhan (2005, 2010)) for more on AS-PCDs. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

num.arcsAS(Xp, Yp, M = "CC")
num.arcsAS(Xp, Yp, M = "CC")

Arguments

`Xp`	A set of 2D points which constitute the vertices of the AS-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`M`	The center of the triangle. `"CC"` stands for circumcenter of each Delaunay triangle or 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle; default is `M="CC"` i.e., the circumcenter of each triangle.

Value

A list with the elements

`desc`	A short description of the output: number of arcs and related quantities for the induced subdigraphs in the Delaunay triangles
`num.arcs`	Total number of arcs in all triangles, i.e., the number of arcs for the entire AS-PCD
`num.in.conhull`	Number of `Xp` points in the convex hull of `Yp` points
`num.in.tris`	The vector of number of `Xp` points in the Delaunay triangles based on `Yp` points
`weight.vec`	The `vector` of the areas of Delaunay triangles based on `Yp` points
`tri.num.arcs`	The `vector` of the number of arcs of the component of the AS-PCD in the Delaunay triangles based on `Yp` points
`del.tri.ind`	A matrix of indices of Delaunay triangles based on `Yp` points, each column corresponds to the vector of indices of the vertices of one of the Delaunay triangle.
`data.tri.ind`	A `vector` of indices of vertices of the Delaunay triangles in which data points reside, i.e., column number of `del.tri.ind` for each `Xp` point.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the Delaunay triangulation based on `Yp` points.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-"CC"  #try also M<-c(1,1,1)
Narcs = num.arcsAS(Xp,Yp,M)
Narcs
summary(Narcs)
plot(Narcs)


## End(Not run)

## Not run: 
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-"CC"  #try also M<-c(1,1,1)
Narcs = num.arcsAS(Xp,Yp,M)
Narcs
summary(Narcs)
plot(Narcs)


## End(Not run)

Number of arcs of Arc Slice Proximity Catch Digraphs (AS-PCDs) and quantities related to the triangle - one triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Arc Slice Proximity Catch Digraphs (AS-PCDs) whose vertices are the 2D data set, Xp. It also provides number of vertices (i.e., number of data points inside the triangle) and indices of the data points that reside in the triangle.

The data points could be inside or outside a general triangle tri $=T(A,B,C)=$ (rv=1,rv=2,rv=3), with vertices of tri stacked row-wise.

See also (Ceyhan (2005, 2010)).

Usage

num.arcsAStri(Xp, tri, M = "CC")
num.arcsAStri(Xp, tri, M = "CC")

Arguments

`Xp`	A set of 2D points which constitute the vertices of the digraph (i.e., AS-PCD).
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of `tri`; default is `M="CC"` i.e., the circumcenter of `tri`.

Value

A list with the elements

`desc`	A short description of the output: number of arcs and quantities related to the triangle
`num.arcs`	Number of arcs of the AS-PCD
`num.in.tri`	Number of `Xp` points in the triangle, `tri`
`ind.in.tri`	The vector of indices of the `Xp` points that reside in the triangle
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Narcs = num.arcsAStri(Xp,Tr,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Narcs = num.arcsAStri(Xp,Tr,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and related quantities of the induced subdigraphs for points in the Delaunay triangles - multiple triangle case

Description

An object of class "NumArcs". Returns the number of arcs and various other quantities related to the Delaunay triangles for Central Similarity Proximity Catch Digraph (CS-PCD) whose vertices are the data points in Xp in the multiple triangle case.

CS proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter $t>0$ and edge regions in each triangle is based on the center $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle or based on circumcenter of each Delaunay triangle (default for $M=(1,1,1)$ which is the center of mass of the triangle). Each Delaunay triangle is first converted to an (nonscaled) basic triangle so that M will be the same type of center for each Delaunay triangle (this conversion is not necessary when M is $CM$ ).

Usage

num.arcsCS(Xp, Yp, t, M = c(1, 1, 1))
num.arcsCS(Xp, Yp, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle, default for $M=(1,1,1)$ which is the center of mass of each triangle.

Value

A list with the elements

`desc`	A short description of the output: number of arcs and related quantities for the induced subdigraphs in the Delaunay triangles
`num.arcs`	Total number of arcs in all triangles, i.e., the number of arcs for the entire PE-PCD
`num.in.conhull`	Number of `Xp` points in the convex hull of `Yp` points
`num.in.tris`	The vector of number of `Xp` points in the Delaunay triangles based on `Yp` points
`weight.vec`	The `vector` of the areas of Delaunay triangles based on `Yp` points
`tri.num.arcs`	The `vector` of the number of arcs of the component of the PE-PCD in the Delaunay triangles based on `Yp` points
`del.tri.ind`	A matrix of indices of vertices of the Delaunay triangles based on `Yp` points, each column corresponds to the vector of indices of the vertices of one triangle.
`data.tri.ind`	A `vector` of indices of vertices of the Delaunay triangles in which data points reside, i.e., column number of `del.tri.ind` for each `Xp` point.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the Delaunay triangulation based on `Yp` points.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2014). “Comparison of Relative Density of Two Random Geometric Digraph Families in Testing Spatial Clustering.” TEST, 23(1), 100-134.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

Narcs = num.arcsCS(Xp,Yp,t=1,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

Narcs = num.arcsCS(Xp,Yp,t=1,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and related quantities of the induced subdigraphs for points in the partition intervals - multiple interval case

Description

An object of class "NumArcs". Returns the number of arcs and various other quantities related to the partition intervals for Central Similarity Proximity Catch Digraph (CS-PCD) whose vertices are the data points in Xp in the multiple interval case.

For this function, CS proximity regions are constructed data points inside or outside the intervals based on Yp points with expansion parameter $t \ge 0$ and centrality parameter $c \in (0,1)$ . That is, for this function, arcs may exist for points in the middle or end intervals.

Range (or convex hull) of Yp (i.e., the interval $(\min(Yp),\max(Yp))$ ) is partitioned by the spacings based on Yp points (i.e., multiple intervals are these partition intervals based on the order statistics of Yp points whose union constitutes the range of Yp points). For the number of arcs, loops are not counted.

Usage

num.arcsCS1D(Xp, Yp, t, c = 0.5)
num.arcsCS1D(Xp, Yp, t, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the CS-PCD.
`Yp`	A set or `vector` of 1D points which constitute the end points of the partition intervals.
`t`	A positive real number which serves as the expansion parameter in CS proximity region; must be $> 0$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside the middle (partition) intervals with the default `c=.5`. For an interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`desc`	A short description of the output: number of arcs and related quantities for the induced subdigraphs in the partition intervals
`num.arcs`	Total number of arcs in all intervals (including the end intervals), i.e., the number of arcs for the entire CS-PCD
`num.in.range`	Number of `Xp` points in the range or convex hull of `Yp` points
`num.in.ints`	The vector of number of `Xp` points in the partition intervals (including the end intervals) based on `Yp` points
`weight.vec`	The `vector` of the lengths of the middle partition intervals (i.e., end intervals excluded) based on `Yp` points
`int.num.arcs`	The `vector` of the number of arcs of the component of the CS-PCD in the partition intervals (including the end intervals) based on `Yp` points
`part.int`	A list of partition intervals based on `Yp` points
`data.int.ind`	A `vector` of indices of partition intervals in which data points reside, i.e., column number of `part.int` is provided for each `Xp` point. Partition intervals are numbered from left to right with 1 being the left end interval.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the partition intervals based on `Yp` points.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

There are no references for Rd macro ⁠\insertAllCites⁠ on this help page.

Examples

tau<-1.5
c<-.4
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Narcs = num.arcsCS1D(Xp,Yp,tau,c)
Narcs
summary(Narcs)
plot(Narcs)

tau<-1.5
c<-.4
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Narcs = num.arcsCS1D(Xp,Yp,tau,c)
Narcs
summary(Narcs)
plot(Narcs)

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) - end interval case

Description

Returns the number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) whose vertices are a 1D numerical data set, Xp, outside the interval int $=(a,b)$ .

CS proximity region is constructed only with expansion parameter $t>0$ for points outside the interval $(a,b)$ .

End vertex regions are based on the end points of the interval, i.e., the corresponding end vertex region is an interval as $(-\infty,a)$ or $(b,\infty)$ for the interval $(a,b)$ . For the number of arcs, loops are not allowed, so arcs are only possible for points outside the interval, int, for this function.

See also (Ceyhan (2016)).

Usage

num.arcsCSend.int(Xp, int, t)
num.arcsCSend.int(Xp, int, t)

Arguments

`Xp`	A `vector` of 1D points which constitute the vertices of the digraph.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.

Value

Number of arcs for the CS-PCD with vertices being 1D data set, Xp, expansion parameter, t, for the end intervals.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

a<-0; b<-10; int<-c(a,b)

n<-5
XpL<-runif(n,a-5,a)
XpR<-runif(n,b,b+5)
Xp<-c(XpL,XpR)

num.arcsCSend.int(Xp,int,t=2)

num.arcsCSend.int(Xp,int,t=1.2)

num.arcsCSend.int(Xp,int,t=4)

num.arcsCSend.int(Xp,int,t=2+5)
#num.arcsCSend.int(Xp,int,t=c(-5,15))

n<-10  #try also n<-20
Xp2<-runif(n,a-5,b+5)
num.arcsCSend.int(Xp2,int,t=2)

t<-.5
num.arcsCSend.int(Xp,int,t)

a<-0; b<-10; int<-c(a,b)

n<-5
XpL<-runif(n,a-5,a)
XpR<-runif(n,b,b+5)
Xp<-c(XpL,XpR)

num.arcsCSend.int(Xp,int,t=2)

num.arcsCSend.int(Xp,int,t=1.2)

num.arcsCSend.int(Xp,int,t=4)

num.arcsCSend.int(Xp,int,t=2+5)
#num.arcsCSend.int(Xp,int,t=c(-5,15))

n<-10  #try also n<-20
Xp2<-runif(n,a-5,b+5)
num.arcsCSend.int(Xp2,int,t=2)

t<-.5
num.arcsCSend.int(Xp,int,t)

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the interval - one interval case

Description

An object of class "NumArcs". Returns the number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) whose vertices are the data points in Xp in the one middle interval case. It also provides number of vertices (i.e., number of data points inside the intervals) and indices of the data points that reside in the intervals.

The data points could be inside or outside the interval is int $=(a,b)$ .

CS proximity region is constructed with an expansion parameter $t>0$ and a centrality parameter $c \in (0,1)$ . CS proximity region is constructed for both points inside and outside the interval, hence the arcs may exist for all points inside or outside the interval.

See also (Ceyhan (2016)).

Usage

num.arcsCSint(Xp, int, t, c = 0.5)
num.arcsCSint(Xp, int, t, c = 0.5)

Arguments

`Xp`	A set of 1D points which constitute the vertices of CS-PCD.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`desc`	A short description of the output: number of arcs and quantities related to the interval
`num.arcs`	Total number of arcs in all intervals (including the end intervals), i.e., the number of arcs for the entire CS-PCD
`num.in.range`	Number of `Xp` points in the interval `int`
`num.in.ints`	The vector of number of `Xp` points in the partition intervals (including the end intervals)
`int.num.arcs`	The `vector` of the number of arcs of the component of the CS-PCD in the partition intervals (including the end intervals)
`data.int.ind`	A `vector` of indices of partition intervals in which data points reside. Partition intervals are numbered from left to right with 1 being the left end interval.
`ind.left.end`, `ind.mid`, `ind.right.end`	Indices of data points in the left end interval, middle interval, and right end interval (respectively)
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support interval.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

n<-10
set.seed(1)
Xp<-runif(n,a,b)
Narcs = num.arcsCSint(Xp,int,t,c)
Narcs
summary(Narcs)
plot(Narcs)

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

n<-10
set.seed(1)
Xp<-runif(n,a,b)
Narcs = num.arcsCSint(Xp,int,t,c)
Narcs
summary(Narcs)
plot(Narcs)

Number of Arcs of of Central Similarity Proximity Catch Digraphs (CS-PCDs) - middle interval case

Description

Returns the number of arcs of of Central Similarity Proximity Catch Digraphs (CS-PCDs) whose vertices are the given 1D numerical data set, Xp.

CS proximity region $N_{CS}(x,t,c)$ is defined with respect to the interval int $=(a,b)$ for this function. CS proximity region is constructed with expansion parameter $t>0$ and centrality parameter $c \in (0,1)$ .

Vertex regions are based on the center associated with the centrality parameter $c \in (0,1)$ . For the interval, int $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ and for the number of arcs, loops are not allowed so arcs are only possible for points inside the middle interval int for this function.

See also (Ceyhan (2016)).

Usage

num.arcsCSmid.int(Xp, int, t, c = 0.5)
num.arcsCSmid.int(Xp, int, t, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of CS-PCD.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

Number of arcs for the PE-PCD whose vertices are the 1D data set, Xp, with expansion parameter, $r \ge 1$ , and centrality parameter, $c \in (0,1)$ . PE proximity regions are defined only for Xp points inside the interval int, i.e., arcs are possible for such points only.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

n<-10
Xp<-runif(n,a,b)
num.arcsCSmid.int(Xp,int,t,c)

num.arcsCSmid.int(Xp,int,t,c=.3)

num.arcsCSmid.int(Xp,int,t=1.5,c)

#num.arcsCSmid.int(Xp,int,t,c+5) #gives error
#num.arcsCSmid.int(Xp,int,t,c+10)

n<-10  #try also n<-20
Xp<-runif(n,a-5,b+5)
num.arcsCSint(Xp,int,t,c)

Xp<-runif(n,a+10,b+10)
num.arcsCSmid.int(Xp,int,t,c)

n<-10
Xp<-runif(n,a,b)
num.arcsCSmid.int(Xp,int,t,c)

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

n<-10
Xp<-runif(n,a,b)
num.arcsCSmid.int(Xp,int,t,c)

num.arcsCSmid.int(Xp,int,t,c=.3)

num.arcsCSmid.int(Xp,int,t=1.5,c)

#num.arcsCSmid.int(Xp,int,t,c+5) #gives error
#num.arcsCSmid.int(Xp,int,t,c+10)

n<-10  #try also n<-20
Xp<-runif(n,a-5,b+5)
num.arcsCSint(Xp,int,t,c)

Xp<-runif(n,a+10,b+10)
num.arcsCSmid.int(Xp,int,t,c)

n<-10
Xp<-runif(n,a,b)
num.arcsCSmid.int(Xp,int,t,c)

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the triangle - standard equilateral triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) whose vertices are the given 2D numerical data set, Xp. It also provides number of vertices (i.e., number of data points inside the standard equilateral triangle $T_e$ ) and indices of the data points that reside in $T_e$ .

CS proximity region $N_{CS}(x,t)$ is defined with respect to the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with expansion parameter $t>0$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$ . For the number of arcs, loops are not allowed so arcs are only possible for points inside $T_e$ for this function.

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

num.arcsCSstd.tri(Xp, t, M = c(1, 1, 1))
num.arcsCSstd.tri(Xp, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the digraph.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates. which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .

Value

A list with the elements

`desc`	A short description of the output: number of arcs and quantities related to the standard equilateral triangle
`num.arcs`	Number of arcs of the CS-PCD
`num.in.tri`	Number of `Xp` points in the standard equilateral triangle, $T_e$
`ind.in.tri`	The vector of indices of the `Xp` points that reside in $T_e$
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle $T_e$ .
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

Narcs = num.arcsCSstd.tri(Xp,t=.5,M)
Narcs
summary(Narcs)
par(pty="s")
plot(Narcs,asp=1)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

Narcs = num.arcsCSstd.tri(Xp,t=.5,M)
Narcs
summary(Narcs)
par(pty="s")
plot(Narcs,asp=1)

## End(Not run)

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the triangle - one triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) whose vertices are the given 2D numerical data set, Xp. It also provides number of vertices (i.e., number of data points inside the triangle) and indices of the data points that reside in the triangle.

CS proximity region $N_{CS}(x,t)$ is defined with respect to the triangle, tri with expansion parameter $t>0$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of tri; default is $M=(1,1,1)$ i.e., the center of mass of tri. For the number of arcs, loops are not allowed so arcs are only possible for points inside tri for this function.

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

num.arcsCStri(Xp, tri, t, M = c(1, 1, 1))
num.arcsCStri(Xp, tri, t, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of CS-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is $M=(1,1,1)$ i.e. the center of mass of `tri`.

Value

A list with the elements

`desc`	A short description of the output: number of arcs and quantities related to the triangle
`num.arcs`	Number of arcs of the CS-PCD
`num.in.tri`	Number of `Xp` points in the triangle, `tri`
`ind.in.tri`	The vector of indices of the `Xp` points that reside in the triangle
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Narcs = num.arcsCStri(Xp,Tr,t=.5,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Narcs = num.arcsCStri(Xp,Tr,t=.5,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and related quantities of the induced subdigraphs for points in the Delaunay triangles - multiple triangle case

Description

An object of class "NumArcs". Returns the number of arcs and various other quantities related to the Delaunay triangles for Proportional Edge Proximity Catch Digraph (PE-PCD) whose vertices are the data points in Xp in the multiple triangle case.

PE proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter $r \ge 1$ and vertex regions in each triangle is based on the center $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle or based on circumcenter of each Delaunay triangle (default for $M=(1,1,1)$ which is the center of mass of the triangle). Each Delaunay triangle is first converted to an (nonscaled) basic triangle so that M will be the same type of center for each Delaunay triangle (this conversion is not necessary when M is $CM$ ).

See (Ceyhan (2005); Ceyhan et al. (2006)) for more on PE-PCDs. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

num.arcsPE(Xp, Yp, r, M = c(1, 1, 1))
num.arcsPE(Xp, Yp, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as `M="CC"`), default for $M=(1,1,1)$ which is the center of mass of each triangle.

Value

A list with the elements

`desc`	A short description of the output: number of arcs and related quantities for the induced subdigraphs in the Delaunay triangles
`num.arcs`	Total number of arcs in all triangles, i.e., the number of arcs for the entire PE-PCD
`num.in.conhull`	Number of `Xp` points in the convex hull of `Yp` points
`num.in.tris`	The vector of number of `Xp` points in the Delaunay triangles based on `Yp` points
`weight.vec`	The `vector` of the areas of Delaunay triangles based on `Yp` points
`tri.num.arcs`	The `vector` of the number of arcs of the component of the PE-PCD in the Delaunay triangles based on `Yp` points
`del.tri.ind`	A matrix of indices of vertices of the Delaunay triangles based on `Yp` points, each column corresponds to the vector of indices of the vertices of one triangle.
`data.tri.ind`	A `vector` of indices of vertices of the Delaunay triangles in which data points reside, i.e., column number of `del.tri.ind` for each `Xp` point.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the Delaunay triangulation based on `Yp` points.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

Narcs = num.arcsPE(Xp,Yp,r=1.25,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

Narcs = num.arcsPE(Xp,Yp,r=1.25,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and related quantities of the induced subdigraphs for points in the partition intervals - multiple interval case

Description

An object of class "NumArcs". Returns the number of arcs and various other quantities related to the partition intervals for Proportional Edge Proximity Catch Digraph (PE-PCD) whose vertices are the data points in Xp in the multiple interval case.

See also (Ceyhan (2012)).

Usage

num.arcsPE1D(Xp, Yp, r, c = 0.5)
num.arcsPE1D(Xp, Yp, r, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of the PE-PCD.
`Yp`	A set or `vector` of 1D points which constitute the end points of the partition intervals.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside the middle (partition) intervals with the default `c=.5`. For an interval, $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`desc`	A short description of the output: number of arcs and related quantities for the induced subdigraphs in the partition intervals
`num.arcs`	Total number of arcs in all intervals (including the end intervals), i.e., the number of arcs for the entire PE-PCD
`num.in.range`	Number of `Xp` points in the range or convex hull of `Yp` points
`num.in.ints`	The vector of number of `Xp` points in the partition intervals (including the end intervals) based on `Yp` points
`weight.vec`	The `vector` of the lengths of the middle partition intervals (i.e., end intervals excluded) based on `Yp` points
`int.num.arcs`	The `vector` of the number of arcs of the components of the PE-PCD in the partition intervals (including the end intervals) based on `Yp` points
`part.int`	A matrix with columns corresponding to the partition intervals based on `Yp` points.
`data.int.ind`	A `vector` of indices of partition intervals in which data points reside, i.e., column number of `part.int` is provided for each `Xp` point. Partition intervals are numbered from left to right with 1 being the left end interval.
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the partition intervals based on `Yp` points.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Narcs = num.arcsPE1D(Xp,Yp,r,c)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Narcs = num.arcsPE1D(Xp,Yp,r,c)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - end interval case

Description

Returns the number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertices are a 1D numerical data set, Xp, outside the interval int $=(a,b)$ .

PE proximity region is constructed only with expansion parameter $r \ge 1$ for points outside the interval $(a,b)$ . End vertex regions are based on the end points of the interval, i.e., the corresponding vertex region is an interval as $(-\infty,a)$ or $(b,\infty)$ for the interval $(a,b)$ . For the number of arcs, loops are not allowed, so arcs are only possible for points outside the interval, int, for this function.

See also (Ceyhan (2012)).

Usage

num.arcsPEend.int(Xp, int, r)
num.arcsPEend.int(Xp, int, r)

Arguments

`Xp`	A `vector` of 1D points which constitute the vertices of the digraph.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .

Value

Number of arcs for the PE-PCD with vertices being 1D data set, Xp, expansion parameter, $r \ge 1$ , for the end intervals.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
a<-0; b<-10; int<-c(a,b)

n<-5
XpL<-runif(n,a-5,a)
XpR<-runif(n,b,b+5)
Xp<-c(XpL,XpR)

r<-1.2
num.arcsPEend.int(Xp,int,r)
num.arcsPEend.int(Xp,int,r=2)

## End(Not run)

## Not run: 
a<-0; b<-10; int<-c(a,b)

n<-5
XpL<-runif(n,a-5,a)
XpR<-runif(n,b,b+5)
Xp<-c(XpL,XpR)

r<-1.2
num.arcsPEend.int(Xp,int,r)
num.arcsPEend.int(Xp,int,r=2)

## End(Not run)

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the interval - one interval case

Description

An object of class "NumArcs". Returns the number of arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) whose vertices are the data points in Xp in the one middle interval case. It also provides number of vertices (i.e., number of data points inside the intervals) and indices of the data points that reside in the intervals.

The data points could be inside or outside the interval is int $=(a,b)$ . PE proximity region is constructed with an expansion parameter $r \ge 1$ and a centrality parameter $c \in (0,1)$ . int determines the end points of the interval.

The PE proximity region is constructed for both points inside and outside the interval, hence the arcs may exist for all points inside or outside the interval.

See also (Ceyhan (2012)).

Usage

num.arcsPEint(Xp, int, r, c = 0.5)
num.arcsPEint(Xp, int, r, c = 0.5)

Arguments

`Xp`	A set of 1D points which constitute the vertices of PE-PCD.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with the elements

`desc`	A short description of the output: number of arcs and quantities related to the interval
`num.arcs`	Total number of arcs in all intervals (including the end intervals), i.e., the number of arcs for the entire PE-PCD
`num.in.range`	Number of `Xp` points in the interval `int`
`num.in.ints`	The vector of number of `Xp` points in the partition intervals (including the end intervals)
`int.num.arcs`	The `vector` of the number of arcs of the components of the PE-PCD in the partition intervals (including the end intervals)
`data.int.ind`	A `vector` of indices of partition intervals in which data points reside. Partition intervals are numbered from left to right with 1 being the left end interval.
`ind.left.end`, `ind.mid`, `ind.right.end`	Indices of data points in the left end interval, middle interval, and right end interval (respectively)
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support interval.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

xf<-(int[2]-int[1])*.1

set.seed(123)

n<-10
Xp<-runif(n,a-xf,b+xf)
Narcs = num.arcsPEint(Xp,int,r,c)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

xf<-(int[2]-int[1])*.1

set.seed(123)

n<-10
Xp<-runif(n,a-xf,b+xf)
Narcs = num.arcsPEint(Xp,int,r,c)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

Number of Arcs for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case

Description

Returns the number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertices are the given 1D numerical data set, Xp. PE proximity region $N_{PE}(x,r,c)$ is defined with respect to the interval int $=(a,b)$ for this function.

PE proximity region is constructed with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ .

See also (Ceyhan (2012)).

Usage

num.arcsPEmid.int(Xp, int, r, c = 0.5)
num.arcsPEmid.int(Xp, int, r, c = 0.5)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of PE-PCD.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

n<-10
Xp<-runif(n,a,b)
num.arcsPEmid.int(Xp,int,r,c)
num.arcsPEmid.int(Xp,int,r=1.5,c)

## End(Not run)

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

n<-10
Xp<-runif(n,a,b)
num.arcsPEmid.int(Xp,int,r,c)
num.arcsPEmid.int(Xp,int,r=1.5,c)

## End(Not run)

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the triangle - standard equilateral triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertices are the given 2D numerical data set, Xp in the standard equilateral triangle. It also provides number of vertices (i.e., number of data points inside the standard equilateral triangle $T_e$ ) and indices of the data points that reside in $T_e$ .

PE proximity region $N_{PE}(x,r)$ is defined with respect to the standard equilateral triangle $T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with expansion parameter $r \ge 1$ and vertex regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ ; default is $M=(1,1,1)$ , i.e., the center of mass of $T_e$ . For the number of arcs, loops are not allowed so arcs are only possible for points inside $T_e$ for this function.

Usage

num.arcsPEstd.tri(Xp, r, M = c(1, 1, 1))
num.arcsPEstd.tri(Xp, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`r`	A positive real number which serves as the expansion parameter for PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ ; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$ .

Value

A list with the elements

`desc`	A short description of the output: number of arcs and quantities related to the standard equilateral triangle
`num.arcs`	Number of arcs of the PE-PCD
`num.in.tri`	Number of `Xp` points in the standard equilateral triangle, $T_e$
`ind.in.tri`	The vector of indices of the `Xp` points that reside in $T_e$
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle $T_e$ .
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-c(.6,.2)  #try also M<-c(1,1,1)

Narcs = num.arcsPEstd.tri(Xp,r=1.25,M)
Narcs
summary(Narcs)
par(pty="s")
plot(Narcs,asp=1)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-c(.6,.2)  #try also M<-c(1,1,1)

Narcs = num.arcsPEstd.tri(Xp,r=1.25,M)
Narcs
summary(Narcs)
par(pty="s")
plot(Narcs,asp=1)

## End(Not run)

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the tetrahedron - one tetrahedron case

Description

An object of class "NumArcs". Returns the number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertices are the given 3D numerical data set, Xp. It also provides number of vertices (i.e., number of data points inside the tetrahedron) and indices of the data points that reside in the tetrahedron.

PE proximity region is constructed with respect to the tetrahedron th and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM". For the number of arcs, loops are not allowed so arcs are only possible for points inside the tetrahedron th for this function.

See also (Ceyhan (2005, 2010)).

Usage

num.arcsPEtetra(Xp, th, r, M = "CM")
num.arcsPEtetra(Xp, th, r, M = "CM")

Arguments

`Xp`	A set of 3D points which constitute the vertices of PE-PCD.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.

Value

A list with the elements

`desc`	A short description of the output: number of arcs and quantities related to the tetrahedron
`num.arcs`	Number of arcs of the PE-PCD
`num.in.tetra`	Number of `Xp` points in the tetrahedron, `th`
`ind.in.tetra`	The vector of indices of the `Xp` points that reside in the tetrahedron
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support tetrahedron.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.25

Narcs = num.arcsPEtetra(Xp,tetra,r,M)
Narcs
summary(Narcs)
#plot(Narcs)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.25

Narcs = num.arcsPEtetra(Xp,tetra,r,M)
Narcs
summary(Narcs)
#plot(Narcs)

## End(Not run)

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the triangle - one triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertices are the given 2D numerical data set, Xp. It also provides number of vertices (i.e., number of data points inside the triangle) and indices of the data points that reside in the triangle.

PE proximity region $N_{PE}(x,r)$ is defined with respect to the triangle, tri with expansion parameter $r \ge 1$ and vertex regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri or based on circumcenter of tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri. For the number of arcs, loops are not allowed so arcs are only possible for points inside the triangle tri for this function.

Usage

num.arcsPEtri(Xp, tri, r, M = c(1, 1, 1))
num.arcsPEtri(Xp, tri, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of PE-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.

Value

A list with the elements

`desc`	A short description of the output: number of arcs and quantities related to the triangle
`num.arcs`	Number of arcs of the PE-PCD
`num.in.tri`	Number of `Xp` points in the triangle, `tri`
`ind.in.tri`	The vector of indices of the `Xp` points that reside in the triangle
`tess.points`	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle.
`vertices`	Vertices of the digraph, `Xp`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Narcs = num.arcsPEtri(Xp,Tr,r=1.25,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Narcs = num.arcsPEtri(Xp,Tr,r=1.25,M)
Narcs
summary(Narcs)
plot(Narcs)

## End(Not run)

Number of Delaunay triangles based on a 2D data set

Description

Returns the number of Delaunay triangles based on the 2D set of points Yp. See (Okabe et al. (2000); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

num.delaunay.tri(Yp)
num.delaunay.tri(Yp)

Arguments

`Yp`	A set of 2D points which constitute the vertices of Delaunay triangles.

Value

Number of Delaunay triangles based on Yp points.

Author(s)

Elvan Ceyhan

References

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

ny<-10

set.seed(1)
Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

num.delaunay.tri(Yp)

ny<-10

set.seed(1)
Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

num.delaunay.tri(Yp)

The line at a point `p` parallel to the line segment joining two distinct 2D points `a` and `b`

Description

An object of class "Lines". Returns the equation, slope, intercept, and $y$ -coordinates of the line crossing the point p and parallel to the line passing through the points a and b with $x$ -coordinates are provided in vector x.

Usage

paraline(p, a, b, x)
paraline(p, a, b, x)

Arguments

`p`	A 2D point at which the parallel line to line segment joining `a` and `b` crosses.
`a`, `b`	2D points that determine the line segment (the line will be parallel to this line segment).
`x`	A scalar or a `vector` of scalars representing the $x$ -coordinates of the line parallel to `ab` and crossing `p`.

Value

A list with the elements

`desc`	Description of the line passing through point `p` and parallel to line segment joining `a` and `b`
`mtitle`	The `"main"` title for the plot of the line passing through point `p` and parallel to line segment joining `a` and `b`
`points`	The input points `p`, `a`, and `b` (stacked row-wise, i.e., point `p` is in row 1, point `a` is in row 2 and point `b` is in row 3). Line parallel to `ab` crosses p.
`x`	The input vector. It can be a scalar or a `vector` of scalars, which constitute the $x$ -coordinates of the point(s) of interest on the line passing through point `p` and parallel to line segment joining `a` and `b`.
`y`	The output scalar or `vector` which constitutes the $y$ -coordinates of the point(s) of interest on the line passing through point `p` and parallel to line segment joining `a` and `b`. If `x` is a scalar, then `y` will be a scalar and if `x` is a `vector` of scalars, then `y` will be a `vector` of scalars.
`slope`	Slope of the line, `Inf` is allowed, passing through point `p` and parallel to line segment joining `a` and `b`
`intercept`	Intercept of the line passing through point `p` and parallel to line segment joining `a` and `b`
`equation`	Equation of the line passing through point `p` and parallel to line segment joining `a` and `b`

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1.1,1.2); B<-c(2.3,3.4); p<-c(.51,2.5)

paraline(p,A,B,.45)

pts<-rbind(A,B,p)
xr<-range(pts[,1])
xf<-(xr[2]-xr[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100

plnAB<-paraline(p,A,B,x)
plnAB
summary(plnAB)
plot(plnAB)

y<-plnAB$y
Xlim<-range(x,pts[,1])
if (!is.na(y[1])) {Ylim<-range(y,pts[,2])} else {Ylim<-range(pts[,2])}
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
pf<-c(xd,-yd)*.025

plot(A,pch=".",xlab="",ylab="",main="Line Crossing P and Parallel to AB",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(pts)
txt.str<-c("A","B","p")
text(pts+rbind(pf,pf,pf),txt.str)

segments(A[1],A[2],B[1],B[2],lty=2)
if (!is.na(y[1])) {lines(x,y,type="l",lty=1,xlim=Xlim,ylim=Ylim)} else {abline(v=p[1])}
tx<-(A[1]+B[1])/2;
if (!is.na(y[1])) {ty<-paraline(p,A,B,tx)$y} else {ty=p[2]}
text(tx,ty,"line parallel to AB\n and crossing p")

## End(Not run)

## Not run: 
A<-c(1.1,1.2); B<-c(2.3,3.4); p<-c(.51,2.5)

paraline(p,A,B,.45)

pts<-rbind(A,B,p)
xr<-range(pts[,1])
xf<-(xr[2]-xr[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100

plnAB<-paraline(p,A,B,x)
plnAB
summary(plnAB)
plot(plnAB)

y<-plnAB$y
Xlim<-range(x,pts[,1])
if (!is.na(y[1])) {Ylim<-range(y,pts[,2])} else {Ylim<-range(pts[,2])}
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
pf<-c(xd,-yd)*.025

plot(A,pch=".",xlab="",ylab="",main="Line Crossing P and Parallel to AB",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(pts)
txt.str<-c("A","B","p")
text(pts+rbind(pf,pf,pf),txt.str)

segments(A[1],A[2],B[1],B[2],lty=2)
if (!is.na(y[1])) {lines(x,y,type="l",lty=1,xlim=Xlim,ylim=Ylim)} else {abline(v=p[1])}
tx<-(A[1]+B[1])/2;
if (!is.na(y[1])) {ty<-paraline(p,A,B,tx)$y} else {ty=p[2]}
text(tx,ty,"line parallel to AB\n and crossing p")

## End(Not run)

The line crossing the 3D point `p` and parallel to line joining 3D points `a` and `b`

Description

An object of class "Lines3D". Returns the equation, $x$ -, $y$ -, and $z$ -coordinates of the line crossing 3D point p and parallel to the line joining 3D points a and b (i.e., the line is in the direction of vector b-a) with the parameter t being provided in vector t.

Usage

paraline3D(p, a, b, t)
paraline3D(p, a, b, t)

Arguments

`p`	A 3D point through which the straight line passes.
`a`, `b`	3D points which determine the straight line to which the line passing through point `p` would be parallel (i.e., $b-a$ determines the direction of the straight line passing through `p`).
`t`	A scalar or a `vector` of scalars representing the parameter of the coordinates of the line (for the form: $x=p_0 + A t$ , $y=y_0 + B t$ , and $z=z_0 + C t$ where $p=(p_0,y_0,z_0)$ and $b-a=(A,B,C)$ ).

Value

A list with the elements

`desc`	A description of the line
`mtitle`	The `"main"` title for the plot of the line
`points`	The input points that determine the line to which the line crossing point `p` would be parallel.
`pnames`	The names of the input points that determine the line to which the line crossing point `p` would be parallel.
`vecs`	The points `p`, `a`, and `b` stacked row-wise in this order.
`vec.names`	The names of the points `p`, `a`, and `b`.
`x`, `y`, `z`	The $x$ -, $y$ -, and $z$ -coordinates of the point(s) of interest on the line parallel to the line determined by points `a` and `b`.
`tsq`	The scalar or the `vector` of the parameter in defining each coordinate of the line for the form: $x=p_0 + A t$ , $y=y_0 + B t$ , and $z=z_0 + C t$ where $p=(p_0,y_0,z_0)$ and $b-a=(A,B,C)$ .
`equation`	Equation of the line passing through point `p` and parallel to the line joining points `a` and `b` (i.e., in the direction of the `vector` `b`-`a`). The line equation is in the form: $x=p_0 + A t$ , $y=y_0 + B t$ , and $z=z_0 + C t$ where $p=(p_0,y_0,z_0)$ and $b-a=(A,B,C)$ .

Author(s)

Elvan Ceyhan

Examples

## Not run: 
P<-c(1,10,4); Q<-c(1,1,3); R<-c(3,9,12)

vecs<-rbind(P,R-Q)
pts<-rbind(P,Q,R)
paraline3D(P,Q,R,.1)

tr<-range(pts,vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=5)  #try also l=10, 20, or 100

pln3D<-paraline3D(P,Q,R,tsq)
pln3D
summary(pln3D)
plot(pln3D)

x<-pln3D$x
y<-pln3D$y
z<-pln3D$z

zr<-range(z)
zf<-(zr[2]-zr[1])*.2
Qv<-(R-Q)*tf*5

Xlim<-range(x,pts[,1])
Ylim<-range(y,pts[,2])
Zlim<-range(z,pts[,3])

xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

Dr<-P+min(tsq)*(R-Q)

plot3D::lines3D(x, y, z, phi = 0, bty = "g",
main="Line Crossing P \n in the direction of R-Q",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.1,.1)+c(-zf,zf),
        pch = 20, cex = 2, ticktype = "detailed")
plot3D::arrows3D(Dr[1],Dr[2],Dr[3]+zf,Dr[1]+Qv[1],
Dr[2]+Qv[2],Dr[3]+zf+Qv[3], add=TRUE)
plot3D::points3D(pts[,1],pts[,2],pts[,3],add=TRUE)
plot3D::text3D(pts[,1],pts[,2],pts[,3],labels=c("P","Q","R"),add=TRUE)
plot3D::arrows3D(P[1],P[2],P[3]-2*zf,P[1],P[2],P[3],lty=2, add=TRUE)
plot3D::text3D(P[1],P[2],P[3]-2*zf,labels="initial point",add=TRUE)
plot3D::arrows3D(Dr[1]+Qv[1]/2,Dr[2]+Qv[2]/2,
Dr[3]+3*zf+Qv[3]/2,Dr[1]+Qv[1]/2,
Dr[2]+Qv[2]/2,Dr[3]+zf+Qv[3]/2,lty=2, add=TRUE)
plot3D::text3D(Dr[1]+Qv[1]/2,Dr[2]+Qv[2]/2,Dr[3]+3*zf+Qv[3]/2,
labels="direction vector",add=TRUE)
plot3D::text3D(Dr[1]+Qv[1]/2,Dr[2]+Qv[2]/2,
Dr[3]+zf+Qv[3]/2,labels="R-Q",add=TRUE)

## End(Not run)

## Not run: 
P<-c(1,10,4); Q<-c(1,1,3); R<-c(3,9,12)

vecs<-rbind(P,R-Q)
pts<-rbind(P,Q,R)
paraline3D(P,Q,R,.1)

tr<-range(pts,vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=5)  #try also l=10, 20, or 100

pln3D<-paraline3D(P,Q,R,tsq)
pln3D
summary(pln3D)
plot(pln3D)

x<-pln3D$x
y<-pln3D$y
z<-pln3D$z

zr<-range(z)
zf<-(zr[2]-zr[1])*.2
Qv<-(R-Q)*tf*5

Xlim<-range(x,pts[,1])
Ylim<-range(y,pts[,2])
Zlim<-range(z,pts[,3])

xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

Dr<-P+min(tsq)*(R-Q)

plot3D::lines3D(x, y, z, phi = 0, bty = "g",
main="Line Crossing P \n in the direction of R-Q",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.1,.1)+c(-zf,zf),
        pch = 20, cex = 2, ticktype = "detailed")
plot3D::arrows3D(Dr[1],Dr[2],Dr[3]+zf,Dr[1]+Qv[1],
Dr[2]+Qv[2],Dr[3]+zf+Qv[3], add=TRUE)
plot3D::points3D(pts[,1],pts[,2],pts[,3],add=TRUE)
plot3D::text3D(pts[,1],pts[,2],pts[,3],labels=c("P","Q","R"),add=TRUE)
plot3D::arrows3D(P[1],P[2],P[3]-2*zf,P[1],P[2],P[3],lty=2, add=TRUE)
plot3D::text3D(P[1],P[2],P[3]-2*zf,labels="initial point",add=TRUE)
plot3D::arrows3D(Dr[1]+Qv[1]/2,Dr[2]+Qv[2]/2,
Dr[3]+3*zf+Qv[3]/2,Dr[1]+Qv[1]/2,
Dr[2]+Qv[2]/2,Dr[3]+zf+Qv[3]/2,lty=2, add=TRUE)
plot3D::text3D(Dr[1]+Qv[1]/2,Dr[2]+Qv[2]/2,Dr[3]+3*zf+Qv[3]/2,
labels="direction vector",add=TRUE)
plot3D::text3D(Dr[1]+Qv[1]/2,Dr[2]+Qv[2]/2,
Dr[3]+zf+Qv[3]/2,labels="R-Q",add=TRUE)

## End(Not run)

The plane at a point and parallel to the plane spanned by three distinct 3D points `a`, `b`, and `c`

Description

An object of class "Planes". Returns the equation and $z$ -coordinates of the plane passing through point p and parallel to the plane spanned by three distinct 3D points a, b, and c with $x$ - and $y$ -coordinates are provided in vectors x and y, respectively.

Usage

paraplane(p, a, b, c, x, y)
paraplane(p, a, b, c, x, y)

Arguments

`p`	A 3D point which the plane parallel to the plane spanned by three distinct 3D points `a`, `b`, and `c` crosses.
`a`, `b`, `c`	3D points that determine the plane to which the plane crossing point `p` is parallel to.
`x`, `y`	Scalars or `vectors` of scalars representing the $x$ - and $y$ -coordinates of the plane parallel to the plane spanned by points `a`, `b`, and `c` and passing through point `p`.

Value

A list with the elements

`desc`	Description of the plane passing through point `p` and parallel to plane spanned by points `a`, `b` and `c`
`points`	The input points `a`, `b`, `c`, and `p`. Plane is parallel to the plane spanned by `a`, `b`, and `c` and passes through point `p` (stacked row-wise, i.e., row 1 is point `a`, row 2 is point `b`, row 3 is point `c`, and row 4 is point `p`).
`x`, `y`	The input vectors which constitutes the $x$ - and $y$ -coordinates of the point(s) of interest on the plane. `x` and `y` can be scalars or vectors of scalars.
`z`	The output `vector` which constitutes the $z$ -coordinates of the point(s) of interest on the plane. If `x` and `y` are scalars, `z` will be a scalar and if `x` and `y` are vectors of scalars, then `z` needs to be a `matrix` of scalars, containing the $z$ -coordinate for each pair of `x` and `y` values.
`coeff`	Coefficients of the plane (in the $z = A x+B y+C$ form).
`equation`	Equation of the plane in long form
`equation2`	Equation of the plane in short form, to be inserted on the plot

Author(s)

Elvan Ceyhan

Examples

## Not run: 
Q<-c(1,10,3); R<-c(1,1,3); S<-c(3,9,12); P<-c(1,1,0)

pts<-rbind(Q,R,S,P)
paraplane(P,Q,R,S,.1,.2)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.25
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5)  #try also l=10, 20, or 100

plP2QRS<-paraplane(P,Q,R,S,x,y)
plP2QRS
summary(plP2QRS)
plot(plP2QRS,theta = 225, phi = 30, expand = 0.7, facets = FALSE, scale = TRUE)

paraplane(P,Q,R,Q+R,.1,.2)

z.grid<-plP2QRS$z

plQRS<-Plane(Q,R,S,x,y)
plQRS
pl.grid<-plQRS$z

zr<-max(z.grid)-min(z.grid)
Pts<-rbind(Q,R,S,P)+rbind(c(0,0,zr*.1),c(0,0,zr*.1),
c(0,0,zr*.1),c(0,0,zr*.1))
Mn.pts<-apply(Pts[1:3,],2,mean)

plot3D::persp3D(z = pl.grid, x = x, y = y, theta =225, phi = 30,
ticktype = "detailed",
main="Plane Crossing Points Q, R, S\n and Plane Passing P Parallel to it")
#plane spanned by points Q, R, S
plot3D::persp3D(z = z.grid, x = x, y = y,add=TRUE)
#plane parallel to the original plane and passing thru point \code{P}

plot3D::persp3D(z = z.grid, x = x, y = y, theta =225, phi = 30,
ticktype = "detailed",
main="Plane Crossing Point P \n and Parallel to the Plane Crossing Q, R, S")
#plane spanned by points Q, R, S
#add the defining points
plot3D::points3D(Pts[,1],Pts[,2],Pts[,3], add=TRUE)
plot3D::text3D(Pts[,1],Pts[,2],Pts[,3], c("Q","R","S","P"),add=TRUE)
plot3D::text3D(Mn.pts[1],Mn.pts[2],Mn.pts[3],plP2QRS$equation,add=TRUE)
plot3D::polygon3D(Pts[1:3,1],Pts[1:3,2],Pts[1:3,3], add=TRUE)

## End(Not run)

## Not run: 
Q<-c(1,10,3); R<-c(1,1,3); S<-c(3,9,12); P<-c(1,1,0)

pts<-rbind(Q,R,S,P)
paraplane(P,Q,R,S,.1,.2)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.25
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5)  #try also l=10, 20, or 100

plP2QRS<-paraplane(P,Q,R,S,x,y)
plP2QRS
summary(plP2QRS)
plot(plP2QRS,theta = 225, phi = 30, expand = 0.7, facets = FALSE, scale = TRUE)

paraplane(P,Q,R,Q+R,.1,.2)

z.grid<-plP2QRS$z

plQRS<-Plane(Q,R,S,x,y)
plQRS
pl.grid<-plQRS$z

zr<-max(z.grid)-min(z.grid)
Pts<-rbind(Q,R,S,P)+rbind(c(0,0,zr*.1),c(0,0,zr*.1),
c(0,0,zr*.1),c(0,0,zr*.1))
Mn.pts<-apply(Pts[1:3,],2,mean)

plot3D::persp3D(z = pl.grid, x = x, y = y, theta =225, phi = 30,
ticktype = "detailed",
main="Plane Crossing Points Q, R, S\n and Plane Passing P Parallel to it")
#plane spanned by points Q, R, S
plot3D::persp3D(z = z.grid, x = x, y = y,add=TRUE)
#plane parallel to the original plane and passing thru point \code{P}

plot3D::persp3D(z = z.grid, x = x, y = y, theta =225, phi = 30,
ticktype = "detailed",
main="Plane Crossing Point P \n and Parallel to the Plane Crossing Q, R, S")
#plane spanned by points Q, R, S
#add the defining points
plot3D::points3D(Pts[,1],Pts[,2],Pts[,3], add=TRUE)
plot3D::text3D(Pts[,1],Pts[,2],Pts[,3], c("Q","R","S","P"),add=TRUE)
plot3D::text3D(Mn.pts[1],Mn.pts[2],Mn.pts[3],plP2QRS$equation,add=TRUE)
plot3D::polygon3D(Pts[1:3,1],Pts[1:3,2],Pts[1:3,3], add=TRUE)

## End(Not run)

pcds: A package for Proximity Catch Digraphs and Their Applications

Description

pcds is a package for construction and visualization of proximity catch digraphs (PCDs) and computation of two graph invariants of the PCDs and testing spatial patterns using these invariants. The PCD families considered are Arc-Slice (AS) PCDs, Proportional-Edge (PE) PCDs and Central Similarity (CS) PCDs.

Details

The graph invariants used in testing spatial point data are the domination number (Ceyhan (2011)) and arc density (Ceyhan et al. (2006); Ceyhan et al. (2007)) of for two-dimensional data.

The pcds package also contains the functions for generating patterns of segregation, association, CSR (complete spatial randomness) and Uniform data in one, two and three dimensional cases, for testing these patterns based on two invariants of various families of the proximity catch digraphs (PCDs), (see (Ceyhan (2005)).

Moreover, the package has visualization tools for these digraphs for 1D-3D vertices. The AS-PCD related tools are provided for 1D and 2D data; PE-PCD related tools are provided for 1D-3D data, and CS-PCD tools are provided for 1D and 2D data.

The `pcds` functions

The pcds functions can be grouped as Auxiliary Functions, AS-PCD Functions, PE-PCD Functions, and CS-PCD Functions.

Auxiliary Functions

Contains the auxiliary (or utility) functions for constructing and visualizing Delaunay tessellations in 1D and 2D settings, computing the domination number, constructing the geometrical tools, such as equation of lines for two points, distances between lines and points, checking points inside the triangle etc., finding the (local) extrema (restricted to Delaunay cells or vertex or edge regions in them).

Arc-Slice PCD Functions

Contains the functions used in AS-PCD construction, estimation of domination number, arc density, etc in the 2D setting.

Proportional-Edge PCD Functions

Contains the functions used in PE-PCD construction, estimation of domination number, arc density, etc in the 1D-3D settings.

Central-Similarity PCD Functions

Contains the functions used in CS-PCD construction, estimation of domination number, arc density, etc in the 1D and 2D setting.

Point Generation Functions

Contains functions for generation of points from uniform (or CSR), segregation and association patterns.

In all these functions points are vectors, and data sets are either matrices or data frames.

References

The asymptotic probability of domination number $= 2$ for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case

Description

Returns the asymptotic $P($ domination number $\le 1)$ for PE-PCD whose vertices are a uniform data set in a finite interval $(a,b)$ .

The PE proximity region $N_{PE}(x,r,c)$ is defined with respect to $(a,b)$ with centrality parameter c in $(0,1)$ and expansion parameter $r=1/\max(c,1-c)$ .

Usage

Pdom.num2PE1Dasy(c)
Pdom.num2PE1Dasy(c)

Arguments

`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ . For the interval, $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

The asymptotic $P($ domination number $\le 1)$ for PE-PCD whose vertices are a uniform data set in a finite interval $(a,b)$

Author(s)

Elvan Ceyhan

Examples

c<-.5

Pdom.num2PE1Dasy(c)

Pdom.num2PE1Dasy(c=1/1.5)
Pdom.num2PE1D(r=1.5,c=1/1.5,n=10)
Pdom.num2PE1D(r=1.5,c=1/1.5,n=100)

c<-.5

Pdom.num2PE1Dasy(c)

Pdom.num2PE1Dasy(c=1/1.5)
Pdom.num2PE1D(r=1.5,c=1/1.5,n=10)
Pdom.num2PE1D(r=1.5,c=1/1.5,n=100)

Asymptotic probability that domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs) equals 2 where vertices of the digraph are uniform points in a triangle

Description

Returns $P($ domination number $=2)$ for PE-PCD for uniform data in a triangle, when the sample size $n$ goes to infinity (i.e., asymptotic probability of domination number $= 2$ ).

PE proximity regions are constructed with respect to the triangle with the expansion parameter $r \ge 1$ and $M$ -vertex regions where $M$ is the vertex that renders the asymptotic distribution of the domination number non-degenerate for the given value of r in $(1,1.5]$ .

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011)).

Usage

Pdom.num2PEtri(r)
Pdom.num2PEtri(r)

Arguments

`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,1.5]$ to attain non-degenerate asymptotic distribution for the domination number.

Value

$P($ domination number $=2)$ for PE-PCD for uniform data on an triangle as the sample size $n$ goes to infinity

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
Pdom.num2PEtri(r=1.5)
Pdom.num2PEtri(r=1.4999999999)

Pdom.num2PEtri(r=1.5) / Pdom.num2PEtri(r=1.4999999999)

rseq<-seq(1.01,1.49999999999,l=20)  #try also l=100
lrseq<-length(rseq)

pg2<-vector()
for (i in 1:lrseq)
{
  pg2<-c(pg2,Pdom.num2PEtri(rseq[i]))
}

plot(rseq, pg2,type="l",xlab="r",
ylab=expression(paste("P(", gamma, "=2)")),
     lty=1,xlim=range(rseq)+c(0,.01),ylim=c(0,1))
points(rbind(c(1.50,Pdom.num2PEtri(1.50))),pch=".",cex=3)

## End(Not run)

## Not run: 
Pdom.num2PEtri(r=1.5)
Pdom.num2PEtri(r=1.4999999999)

Pdom.num2PEtri(r=1.5) / Pdom.num2PEtri(r=1.4999999999)

rseq<-seq(1.01,1.49999999999,l=20)  #try also l=100
lrseq<-length(rseq)

pg2<-vector()
for (i in 1:lrseq)
{
  pg2<-c(pg2,Pdom.num2PEtri(rseq[i]))
}

plot(rseq, pg2,type="l",xlab="r",
ylab=expression(paste("P(", gamma, "=2)")),
     lty=1,xlim=range(rseq)+c(0,.01),ylim=c(0,1))
points(rbind(c(1.50,Pdom.num2PEtri(1.50))),pch=".",cex=3)

## End(Not run)

A test of segregation/association based on arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data

Description

Under the null hypothesis of uniformity of Xp points in the convex hull of Yp points, arc density of PE-PCD whose vertices are Xp points equals to its expected value under the uniform distribution and alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association) or right-sided (i.e., data is accumulated around the centers of the triangles, or segregation).

PE proximity region is constructed with the expansion parameter $r \ge 1$ and $CM$ -vertex regions (i.e., the test is not available for a general center $M$ at this version of the function).

**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are assumed to be fixed (i.e., the test is conditional on Yp points). Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points, say at least 5 times more. This test is more appropriate when supports of Xp and Yp have a substantial overlap. Currently, the Xp points outside the convex hull of Yp points are handled with a convex hull correction factor (see the description below and the function code.) However, in the special case of no Xp points in the convex hull of Yp points, arc density is taken to be 1, as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is an ongoing topic of research of the author of the package.

ch.cor is for convex hull correction (default is "no convex hull correction", i.e., ch.cor=FALSE) which is recommended when both Xp and Yp have the same rectangular support.

See also (Ceyhan (2005); Ceyhan et al. (2006)) for more on the test based on the arc density of PE-PCDs.

Usage

PEarc.dens.test(
  Xp,
  Yp,
  r,
  ch.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
PEarc.dens.test(
  Xp,
  Yp,
  r,
  ch.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`ch.cor`	A logical argument for convex hull correction, default `ch.cor=FALSE`, recommended when both `Xp` and `Yp` have the same rectangular support.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the arc density of PE-PCD based on the 2D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`
`conf.int`	Confidence interval for the arc density at the given confidence level `conf.level` and depends on the type of `alternative`.
`estimate`	Estimate of the parameter, i.e., arc density
`null.value`	Hypothesized value for the parameter, i.e., the null arc density, which is usually the mean arc density under uniform distribution.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab="")

PEarc.dens.test(Xp,Yp,r=1.25)
PEarc.dens.test(Xp,Yp,r=1.25,ch=TRUE)
#since Y points are not uniform, convex hull correction is invalid here

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab="")

PEarc.dens.test(Xp,Yp,r=1.25)
PEarc.dens.test(Xp,Yp,r=1.25,ch=TRUE)
#since Y points are not uniform, convex hull correction is invalid here

## End(Not run)

A test of uniformity of 1D data in a given interval based on Proportional Edge Proximity Catch Digraph (PE-PCD)

Description

An object of class "htest". This is an "htest" (i.e., hypothesis test) function which performs a hypothesis test of uniformity of 1D data in one interval based on the normal approximation of the arc density of the PE-PCD with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ .

The null hypothesis is that data is uniform in a finite interval (i.e., arc density of PE-PCD equals to its expected value under uniform distribution) and alternative could be two-sided, or left-sided (i.e., data is accumulated around the end points) or right-sided (i.e., data is accumulated around the mid point or center $M_c$ ).

See also (Ceyhan (2012, 2016)).

Usage

PEarc.dens.test.int(
  Xp,
  int,
  r,
  c = 0.5,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
PEarc.dens.test.int(
  Xp,
  int,
  r,
  c = 0.5,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set or `vector` of 1D points which constitute the vertices of PE-PCD.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the arc density of PE-PCD based on the 1D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`
`conf.int`	Confidence interval for the arc density at the given confidence level `conf.level` and depends on the type of `alternative`.
`estimate`	Estimate of the parameter, i.e., arc density
`null.value`	Hypothesized value for the parameter, i.e., the null arc density, which is usually the mean arc density under uniform distribution.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

n<-100  #try also n<-20, 1000
Xp<-runif(n,a,b)

PEarc.dens.test.int(Xp,int,r,c)
PEarc.dens.test.int(Xp,int,r,c,alt="g")
PEarc.dens.test.int(Xp,int,r,c,alt="l")

## End(Not run)

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

n<-100  #try also n<-20, 1000
Xp<-runif(n,a,b)

PEarc.dens.test.int(Xp,int,r,c)
PEarc.dens.test.int(Xp,int,r,c,alt="g")
PEarc.dens.test.int(Xp,int,r,c,alt="l")

## End(Not run)

A test of segregation/association based on arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data

Description

Under the null hypothesis of uniformity of Xp points in the range of Yp points, arc density of PE-PCD whose vertices are Xp points equals to its expected value under the uniform distribution and alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association) or right-sided (i.e., data is accumulated around the centers of the intervals, or segregation).

**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are assumed to be fixed (i.e., the test is conditional on Yp points). Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points, say at least 5 times more. This test is more appropriate when supports of Xp and Yp have a substantial overlap. Currently, the Xp points outside the range of Yp points are handled with a range correction (or end interval correction) factor (see the description below and the function code.) However, in the special case of no Xp points in the range of Yp points, arc density is taken to be 1, as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is an ongoing line of research of the author of the package.

end.int.cor is for end interval correction, (default is "no end interval correction", i.e., end.int.cor=FALSE), recommended when both Xp and Yp have the same interval support.

See also (Ceyhan (2012)) for more on the uniformity test based on the arc density of PE-PCDs.

Usage

PEarc.dens.test1D(
  Xp,
  Yp,
  r,
  c = 0.5,
  support.int = NULL,
  end.int.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
PEarc.dens.test1D(
  Xp,
  Yp,
  r,
  c = 0.5,
  support.int = NULL,
  end.int.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set of 1D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 1D points which constitute the end points of the partition intervals.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number which serves as the centrality parameter in PE proximity region; must be in $(0,1)$ (default `c=.5`).
`support.int`	Support interval $(a,b)$ with $a<b$ . Uniformity of `Xp` points in this interval is tested. Default is `NULL`.
`end.int.cor`	A logical argument for end interval correction, default is `FALSE`, recommended when both `Xp` and `Yp` have the same interval support.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the arc density PE-PCD whose vertices are the 1D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`.
`conf.int`	Confidence interval for the arc density at the given confidence level `conf.level` and depends on the type of `alternative`.
`estimate`	Estimate of the parameter, i.e., arc density
`null.value`	Hypothesized value for the parameter, i.e., the null arc density, which is usually the mean arc density under uniform distribution.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int=c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

PEarc.dens.test1D(Xp,Yp,r,c,int)
#try also PEarc.dens.test1D(Xp,Yp,r,c,int,alt="l") and PEarc.dens.test1D(Xp,Yp,r,c,int,alt="g")

PEarc.dens.test1D(Xp,Yp,r,c,int,end.int.cor = TRUE)

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int=c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

PEarc.dens.test1D(Xp,Yp,r,c,int)
#try also PEarc.dens.test1D(Xp,Yp,r,c,int,alt="l") and PEarc.dens.test1D(Xp,Yp,r,c,int,alt="g")

PEarc.dens.test1D(Xp,Yp,r,c,int,end.int.cor = TRUE)

## End(Not run)

Arc density of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns the arc density of PE-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the tetrahedron th.

PE proximity region is constructed with respect to the tetrahedron th and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM". For the number of arcs, loops are not allowed so arcs are only possible for points inside the tetrahedron th for this function.

th.cor is a logical argument for tetrahedron correction (default is TRUE), if TRUE, only the points inside the tetrahedron are considered (i.e., digraph induced by these vertices are considered) in computing the arc density, otherwise all points are considered (for the number of vertices in the denominator of arc density).

See also (Ceyhan (2005, 2010)).

Usage

PEarc.dens.tetra(Xp, th, r, M = "CM", th.cor = FALSE)
PEarc.dens.tetra(Xp, th, r, M = "CM", th.cor = FALSE)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.
`th.cor`	A logical argument for computing the arc density for only the points inside the tetrahedron, `th`. (default is `th.cor=FALSE`), i.e., if `th.cor=TRUE` only the induced digraph with the vertices inside `th` are considered in the computation of arc density.

Value

Arc density of PE-PCD whose vertices are the 2D numerical data set, Xp; PE proximity regions are defined with respect to the tetrahedron th and M-vertex regions

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.5

num.arcsPEtetra(Xp,tetra,r,M)
PEarc.dens.tetra(Xp,tetra,r,M)
PEarc.dens.tetra(Xp,tetra,r,M,th.cor = FALSE)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.5

num.arcsPEtetra(Xp,tetra,r,M)
PEarc.dens.tetra(Xp,tetra,r,M)
PEarc.dens.tetra(Xp,tetra,r,M,th.cor = FALSE)

## End(Not run)

Arc density of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns the arc density of PE-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

PE proximity regions is defined with respect to tri with expansion parameter $r \ge 1$ and vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri or based on circumcenter of tri; default is $M=(1,1,1)$ , i.e., the center of mass of tri. The function also provides arc density standardized by the mean and asymptotic variance of the arc density of PE-PCD for uniform data in the triangle tri only when M is the center of mass. For the number of arcs, loops are not allowed.

See also (Ceyhan (2005); Ceyhan et al. (2006)).

Usage

PEarc.dens.tri(Xp, tri, r, M = c(1, 1, 1), tri.cor = FALSE)
PEarc.dens.tri(Xp, tri, r, M = c(1, 1, 1), tri.cor = FALSE)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.
`tri.cor`	A logical argument for computing the arc density for only the points inside the triangle, `tri`. (default is `tri.cor=FALSE`), i.e., if `tri.cor=TRUE` only the induced digraph with the vertices inside `tri` are considered in the computation of arc density.

Value

A list with the elements

`arc.dens`	Arc density of PE-PCD whose vertices are the 2D numerical data set, `Xp`; PE proximity regions are defined with respect to the triangle `tri` and `M`-vertex regions
`std.arc.dens`	Arc density standardized by the mean and asymptotic variance of the arc density of PE-PCD for uniform data in the triangle `tri`. This will only be returned, if `M` is the center of mass.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

num.arcsPEtri(Xp,Tr,r=1.5,M)
PEarc.dens.tri(Xp,Tr,r=1.5,M)
PEarc.dens.tri(Xp,Tr,r=1.5,M,tri.cor = TRUE)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

num.arcsPEtri(Xp,Tr,r=1.5,M)
PEarc.dens.tri(Xp,Tr,r=1.5,M)
PEarc.dens.tri(Xp,Tr,r=1.5,M,tri.cor = TRUE)

## End(Not run)

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - multiple triangle case

Description

Returns the domination number, indices of a minimum dominating set of PE-PCD whose vertices are the data points in Xp in the multiple triangle case and domination numbers for the Delaunay triangles based on Yp points.

See (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)) for more on the domination number of PE-PCDs. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

PEdom.num(Xp, Yp, r, M = c(1, 1, 1))
PEdom.num(Xp, Yp, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as `M="CC"`), default for $M=(1,1,1)$ which is the center of mass of each triangle.

Value

A list with three elements

dom.num

Domination number of the PE-PCD whose vertices are Xp points. PE proximity regions are constructed with respect to the Delaunay triangles based on the Yp points with expansion parameter $r \ge 1$ .

`mds`	A minimum dominating set of the PE-PCD whose vertices are `Xp` points
`ind.mds`	The vector of data indices of the minimum dominating set of the PE-PCD whose vertices are `Xp` points.
`tri.dom.nums`	The vector of domination numbers of the PE-PCD components for the Delaunay triangles.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Ceyhan E, Priebe CE (2007). “On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs.” Model Assisted Statistics and Applications, 1(4), 231-255.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)
r<-1.5  #try also r<-2
PEdom.num(Xp,Yp,r,M)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)
r<-1.5  #try also r<-2
PEdom.num(Xp,Yp,r,M)

## End(Not run)

A test of segregation/association based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - Binomial Approximation

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of Xp points in the convex hull of Yp points against the alternatives of segregation (where Xp points cluster away from Yp points i.e., cluster around the centers of the Delaunay triangles) and association (where Xp points cluster around Yp points) based on the (asymptotic) binomial distribution of the domination number of PE-PCD for uniform 2D data in the convex hull of Yp points.

The function yields the test statistic, $p$ -value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is $Pr($ domination number $\le 2)$ ), and method and name of the data set used.

Under the null hypothesis of uniformity of Xp points in the convex hull of Yp points, probability of success (i.e., $Pr($ domination number $\le 2)$ ) equals to its expected value under the uniform distribution) and alternative could be two-sided, or right-sided (i.e., data is accumulated around the Yp points, or association) or left-sided (i.e., data is accumulated around the centers of the triangles, or segregation).

PE proximity region is constructed with the expansion parameter $r \ge 1$ and $M$ -vertex regions where $M$ is a center that yields non-degenerate asymptotic distribution of the domination number.

The test statistic is based on the binomial distribution, when success is defined as domination number being less than or equal to 2 in the one triangle case (i.e., number of failures is equal to number of times restricted domination number = 3 in the triangles). That is, the test statistic is based on the domination number for Xp points inside convex hull of Yp points for the PE-PCD and default convex hull correction, ch.cor, is FALSE where M is the center that yields nondegenerate asymptotic distribution for the domination number. For this approximation to work, number of Xp points must be at least 7 times more than number of Yp points.

PE proximity region is constructed with the expansion parameter $r \ge 1$ and $CM$ -vertex regions (i.e., the test is not available for a general center $M$ at this version of the function).

**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are assumed to be fixed (i.e., the test is conditional on Yp points). Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points, say at least 7 times more. This test is more appropriate when supports of Xp and Yp have a substantial overlap. Currently, the Xp points outside the convex hull of Yp points are handled with a convex hull correction factor (see the description below and the function code.) Removing the conditioning and extending it to the case of non-concurring supports is an ongoing topic of research of the author of the package.

Usage

PEdom.num.binom.test(
  Xp,
  Yp,
  r,
  ch.cor = FALSE,
  ndt = NULL,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
PEdom.num.binom.test(
  Xp,
  Yp,
  r,
  ch.cor = FALSE,
  ndt = NULL,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,1.5]$ .
`ch.cor`	A logical argument for convex hull correction, default `ch.cor=FALSE`, recommended when both `Xp` and `Yp` have the same rectangular support.
`ndt`	Number of Delaunay triangles based on `Yp` points, default is `NULL`.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the probability of success (i.e., $Pr($ domination number $=3)$ for PE-PCD whose vertices are the 2D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`
`conf.int`	Confidence interval for $Pr($ Domination Number $\le 2)$ at the given level `conf.level` and depends on the type of `alternative`.
`estimate`	A `vector` with two entries: first is is the estimate of the parameter, i.e., $Pr($ Domination Number $=3)$ and second is the domination number
`null.value`	Hypothesized value for the parameter, i.e., the null value for $Pr($ Domination Number $\le 2)$
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Examples

## Not run: 
nx<-100; ny<-5 #try also nx<-1000; ny<-10
r<-1.4  #try also r<-1.5

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab="")
PEdom.num.binom.test(Xp,Yp,r) #try also #PEdom.num.binom.test(Xp,Yp,r,alt="l") and
# PEdom.num.binom.test(Xp,Yp,r,alt="g")
PEdom.num.binom.test(Xp,Yp,r,ch=TRUE)

#or try
ndt<-num.delaunay.tri(Yp)
PEdom.num.binom.test(Xp,Yp,r,ndt=ndt)
#values might differ due to the random of choice of the three centers M1,M2,M3
#for the non-degenerate asymptotic distribution of the domination number

## End(Not run)

## Not run: 
nx<-100; ny<-5 #try also nx<-1000; ny<-10
r<-1.4  #try also r<-1.5

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab="")
PEdom.num.binom.test(Xp,Yp,r) #try also #PEdom.num.binom.test(Xp,Yp,r,alt="l") and
# PEdom.num.binom.test(Xp,Yp,r,alt="g")
PEdom.num.binom.test(Xp,Yp,r,ch=TRUE)

#or try
ndt<-num.delaunay.tri(Yp)
PEdom.num.binom.test(Xp,Yp,r,ndt=ndt)
#values might differ due to the random of choice of the three centers M1,M2,M3
#for the non-degenerate asymptotic distribution of the domination number

## End(Not run)

A test of segregation/association based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data - Binomial Approximation

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of Xp points within the partition intervals based on Yp points (both residing in the support interval $(a,b)$ ). The test is for testing the spatial interaction between Xp and Yp points.

The null hypothesis is uniformity of Xp points on $(y_{\min},y_{\max})$ (by default) where $y_{\min}$ and $y_{\max}$ are minimum and maximum of Yp points, respectively. Yp determines the end points of the intervals (i.e., partition the real line via its spacings called intervalization) where end points are the order statistics of Yp points.

The alternatives are segregation (where Xp points cluster away from Yp points i.e., cluster around the centers of the partition intervals) and association (where Xp points cluster around Yp points). The test is based on the (asymptotic) binomial distribution of the domination number of PE-PCD for uniform 1D data in the partition intervals based on Yp points.

The test by default is restricted to the range of Yp points, and so ignores Xp points outside this range. However, a correction for the Xp points outside the range of Yp points is available by setting end.int.cor=TRUE, which is recommended when both Xp and Yp have the same interval support.

The function yields the test statistic, $p$ -value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is $Pr($ domination number $\le 1)$ ), and method and name of the data set used.

Under the null hypothesis of uniformity of Xp points in the intervals based on Yp points, probability of success (i.e., $Pr($ domination number $\le 1)$ ) equals to its expected value) and alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association) or right-sided (i.e., data is accumulated around the centers of the partition intervals, or segregation).

PE proximity region is constructed with the expansion parameter $r \ge 1$ and centrality parameter c which yields $M$ -vertex regions. More precisely, for a middle interval $(y_{(i)},y_{(i+1)})$ , the center is $M=y_{(i)}+c(y_{(i+1)}-y_{(i)})$ for the centrality parameter c. For a given $c \in (0,1)$ , the expansion parameter $r$ is taken to be $1/\max(c,1-c)$ which yields non-degenerate asymptotic distribution of the domination number.

The test statistic is based on the binomial distribution, when success is defined as domination number being less than or equal to 1 in the one interval case (i.e., number of successes is equal to domination number $\le 1$ in the partition intervals). That is, the test statistic is based on the domination number for Xp points inside range of Yp points (the domination numbers are summed over the $|Yp|-1$ middle intervals) for the PE-PCD and default end interval correction, end.int.cor, is FALSE and the center $Mc$ is chosen so that asymptotic distribution for the domination number is nondegenerate. For this test to work, Xp must be at least 5 times more than Yp points (or Xp must be at least 5 or more per partition interval). Probability of success is the exact probability of success for the binomial distribution.

**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are assumed to be fixed (i.e., the test is conditional on Yp points). Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points, say at least 7 times more. This test is more appropriate when supports of Xp and Yp have a substantial overlap. Currently, the Xp points outside the range of Yp points are handled with an end interval correction factor (see the description below and the function code.) Removing the conditioning and extending it to the case of non-concurring supports is an ongoing line of research of the author of the package.

See also (Ceyhan (2020)) for more on the uniformity test based on the arc density of PE-PCDs.

Usage

PEdom.num.binom.test1D(
  Xp,
  Yp,
  c = 0.5,
  support.int = NULL,
  end.int.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
PEdom.num.binom.test1D(
  Xp,
  Yp,
  c = 0.5,
  support.int = NULL,
  end.int.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set of 1D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 1D points which constitute the end points of the partition intervals.
`c`	A positive real number which serves as the centrality parameter in PE proximity region; must be in $(0,1)$ (default `c=.5`).
`support.int`	Support interval $(a,b)$ with $a<b$ . Uniformity of `Xp` points in this interval is tested. Default is `NULL`.
`end.int.cor`	A logical argument for end interval correction, default is `FALSE`, recommended when both `Xp` and `Yp` have the same interval support.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the probability of success (i.e., $Pr($ domination number $\le 1)$ for PE-PCD whose vertices are the 1D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`.
`conf.int`	Confidence interval for $Pr($ domination number $\le 1)$ at the given level `conf.level` and depends on the type of `alternative`.
`estimate`	A `vector` with two entries: first is is the estimate of the parameter, i.e., $Pr($ domination number $\le 1)$ and second is the domination number
`null.value`	Hypothesized value for the parameter, i.e., the null value for $Pr($ domination number $\le 1)$
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

Ceyhan E (2020). “Domination Number of an Interval Catch Digraph Family and Its Use for Testing Uniformity.” Statistics, 54(2), 310-339.

Examples

## Not run: 
a<-0; b<-10; supp<-c(a,b)
c<-.4

r<-1/max(c,1-c)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)
PEdom.num.binom.test1D(Xp,Yp,c,supp)
PEdom.num.binom.test1D(Xp,Yp,c,supp,alt="l")
PEdom.num.binom.test1D(Xp,Yp,c,supp,alt="g")
PEdom.num.binom.test1D(Xp,Yp,c,supp,end=TRUE)

## End(Not run)

## Not run: 
a<-0; b<-10; supp<-c(a,b)
c<-.4

r<-1/max(c,1-c)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)
PEdom.num.binom.test1D(Xp,Yp,c,supp)
PEdom.num.binom.test1D(Xp,Yp,c,supp,alt="l")
PEdom.num.binom.test1D(Xp,Yp,c,supp,alt="g")
PEdom.num.binom.test1D(Xp,Yp,c,supp,end=TRUE)

## End(Not run)

A test of uniformity for 1D data based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - Binomial Approximation

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of uniformity of Xp points in the support interval $(a,b)$ ).

The support interval $(a,b)$ is partitioned as (b-a)*(0:nint)/nint where nint=round(sqrt(nx),0) and nx is number of Xp points, and the test is for testing the uniformity of Xp points in the interval $(a,b)$ .

The null hypothesis is uniformity of Xp points on $(a,b)$ . The alternative is deviation of distribution of Xp points from uniformity. The test is based on the (asymptotic) binomial distribution of the domination number of PE-PCD for uniform 1D data in the partition intervals based on partition of $(a,b)$ .

Under the null hypothesis of uniformity of Xp points in the support interval, probability of success (i.e., $Pr($ domination number $\le 1)$ ) equals to its expected value) and alternative could be two-sided, or left-sided (i.e., data is accumulated around the end points of the partition intervals of the support) or right-sided (i.e., data is accumulated around the centers of the partition intervals).

PE proximity region is constructed with the expansion parameter $r \ge 1$ and centrality parameter c which yields $M$ -vertex regions. More precisely $M_c=a+c(b-a)$ for the centrality parameter c and for a given $c \in (0,1)$ , the expansion parameter $r$ is taken to be $1/\max(c,1-c)$ which yields non-degenerate asymptotic distribution of the domination number.

The test statistic is based on the binomial distribution, when success is defined as domination number being less than or equal to 1 in the one interval case (i.e., number of failures is equal to number of times restricted domination number = 1 in the intervals). That is, the test statistic is based on the domination number for Xp points inside the partition intervals for the PE-PCD. For this approach to work, Xp must be large for each partition interval, but 5 or more per partition interval seems to work in practice.

Probability of success is chosen in the following way for various parameter choices. asy.bin is a logical argument for the use of asymptotic probability of success for the binomial distribution, default is asy.bin=FALSE. When asy.bin=TRUE, asymptotic probability of success for the binomial distribution is used. When asy.bin=FALSE, the finite sample probability of success for the binomial distribution is used with number of trials equals to expected number of Xp points per partition interval.

Usage

PEdom.num.binom.test1Dint(
  Xp,
  support.int,
  c = 0.5,
  asy.bin = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
PEdom.num.binom.test1Dint(
  Xp,
  support.int,
  c = 0.5,
  asy.bin = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set of 1D points which constitute the vertices of the PE-PCD.
`support.int`	Support interval $(a,b)$ with $a<b$ . Uniformity of `Xp` points in this interval is tested.
`c`	A positive real number which serves as the centrality parameter in PE proximity region; must be in $(0,1)$ (default `c=.5`).
`asy.bin`	A logical argument for the use of asymptotic probability of success for the binomial distribution, default `asy.bin=FALSE`. When `asy.bin=TRUE`, asymptotic probability of success for the binomial distribution is used. When `asy.bin=FALSE`, the finite sample asymptotic probability of success for the binomial distribution is used with number of trials equals to expected number of `Xp` points per partition interval.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the probability of success (i.e., $Pr($ domination number $\le 1)$ for PE-PCD whose vertices are the 1D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`
`conf.int`	Confidence interval for $Pr($ domination number $\le 1)$ at the given level `conf.level` and depends on the type of `alternative`.
`estimate`	A `vector` with two entries: first is is the estimate of the parameter, i.e., $Pr($ domination number $\le 1)$ and second is the domination number
`null.value`	Hypothesized value for the parameter, i.e., the null value for $Pr($ domination number $\le 1)$
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

There are no references for Rd macro ⁠\insertAllCites⁠ on this help page.

Examples

## Not run: 
a<-0; b<-10; supp<-c(a,b)
c<-.4

r<-1/max(c,1-c)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)

PEdom.num.binom.test1Dint(Xp,supp,c,alt="t")
PEdom.num.binom.test1Dint(Xp,support.int = supp,c=c,alt="t")
PEdom.num.binom.test1Dint(Xp,supp,c,alt="l")
PEdom.num.binom.test1Dint(Xp,supp,c,alt="g")
PEdom.num.binom.test1Dint(Xp,supp,c,alt="t",asy.bin = TRUE)

## End(Not run)

## Not run: 
a<-0; b<-10; supp<-c(a,b)
c<-.4

r<-1/max(c,1-c)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)

PEdom.num.binom.test1Dint(Xp,supp,c,alt="t")
PEdom.num.binom.test1Dint(Xp,support.int = supp,c=c,alt="t")
PEdom.num.binom.test1Dint(Xp,supp,c,alt="l")
PEdom.num.binom.test1Dint(Xp,supp,c,alt="g")
PEdom.num.binom.test1Dint(Xp,supp,c,alt="t",asy.bin = TRUE)

## End(Not run)

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) with non-degeneracy centers - multiple triangle case

Description

PE proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter $r \ge 1$ and vertex regions in each triangle are based on the center $M$ which is one of the 3 centers that renders the asymptotic distribution of domination number to be non-degenerate for a given value of r in $(1,1.5)$ and M is center of mass for $r=1.5$ .

See (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)) more on the domination number of PE-PCDs. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

PEdom.num.nondeg(Xp, Yp, r)
PEdom.num.nondeg(Xp, Yp, r)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,1.5]$ here.

Value

A list with three elements

dom.num

`mds`	A minimum dominating set of the PE-PCD whose vertices are `Xp` points.
`ind.mds`	The data indices of the minimum dominating set of the PE-PCD whose vertices are `Xp` points.
`tri.dom.nums`	Domination numbers of the PE-PCD components for the Delaunay triangles.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Ceyhan E, Priebe CE (2007). “On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs.” Model Assisted Statistics and Applications, 1(4), 231-255.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

r<-1.5  #try also r<-2

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

PEdom.num.nondeg(Xp,Yp,r)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

r<-1.5  #try also r<-2

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

PEdom.num.nondeg(Xp,Yp,r)

## End(Not run)

A test of segregation/association based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - Normal Approximation

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of Xp points in the convex hull of Yp points against the alternatives of segregation (where Xp points cluster away from Yp points i.e., cluster around the centers of the Delaunay triangles) and association (where Xp points cluster around Yp points) based on the normal approximation to the binomial distribution of the domination number of PE-PCD for uniform 2D data in the convex hull of Yp points

The function yields the test statistic, $p$ -value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is $Pr($ domination number $\le 2)$ ), and method and name of the data set used.

PE proximity region is constructed with the expansion parameter $r \ge 1$ and $M$ -vertex regions where M is a center that yields non-degenerate asymptotic distribution of the domination number.

The test statistic is based on the normal approximation to the binomial distribution, when success is defined as domination number being less than or equal to 2 in the one triangle case (i.e., number of failures is equal to number of times restricted domination number = 3 in the triangles). That is, the test statistic is based on the domination number for Xp points inside convex hull of Yp points for the PE-PCD and default convex hull correction, ch.cor, is FALSE where M is the center that yields nondegenerate asymptotic distribution for the domination number.

For this approximation to work, number of Yp points must be at least 5 (i.e., about 7 or more Delaunay triangles) and number of Xp points must be at least 7 times more than the number of Yp points.

Usage

PEdom.num.norm.test(
  Xp,
  Yp,
  r,
  ch.cor = FALSE,
  ndt = NULL,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)
PEdom.num.norm.test(
  Xp,
  Yp,
  r,
  ch.cor = FALSE,
  ndt = NULL,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,1.5]$ .
`ch.cor`	A logical argument for convex hull correction, default `ch.cor=FALSE`, recommended when both `Xp` and `Yp` have the same rectangular support.
`ndt`	Number of Delaunay triangles based on `Yp` points, default is `NULL`.
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`.
`conf.level`	Level of the confidence interval, default is `0.95`, for the domination number of PE-PCD whose vertices are the 2D data set `Xp`.

Value

A list with the elements

`statistic`	Test statistic
`p.value`	The $p$ -value for the hypothesis test for the corresponding `alternative`
`conf.int`	Confidence interval for the domination number at the given level `conf.level` and depends on the type of `alternative`.
`estimate`	A `vector` with two entries: first is the domination number, and second is the estimate of the parameter, i.e., $Pr($ Domination Number $=3)$
`null.value`	Hypothesized value for the parameter, i.e., the null value for expected domination number
`alternative`	Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`
`method`	Description of the hypothesis test
`data.name`	Name of the data set

Author(s)

Elvan Ceyhan

References

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Examples

## Not run: 
nx<-100; ny<-5 #try also nx<-1000; ny<-10
r<-1.5  #try also r<-2 or r<-1.25

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab="")
PEdom.num.norm.test(Xp,Yp,r) #try also PEdom.num.norm.test(Xp,Yp,r, alt="l")

PEdom.num.norm.test(Xp,Yp,1.25,ch=TRUE)

#or try
ndt<-num.delaunay.tri(Yp)
PEdom.num.norm.test(Xp,Yp,r,ndt=ndt)
#values might differ due to the random of choice of the three centers M1,M2,M3
#for the non-degenerate asymptotic distribution of the domination number

## End(Not run)

## Not run: 
nx<-100; ny<-5 #try also nx<-1000; ny<-10
r<-1.5  #try also r<-2 or r<-1.25

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab="")
PEdom.num.norm.test(Xp,Yp,r) #try also PEdom.num.norm.test(Xp,Yp,r, alt="l")

PEdom.num.norm.test(Xp,Yp,1.25,ch=TRUE)

#or try
ndt<-num.delaunay.tri(Yp)
PEdom.num.norm.test(Xp,Yp,r,ndt=ndt)
#values might differ due to the random of choice of the three centers M1,M2,M3
#for the non-degenerate asymptotic distribution of the domination number

## End(Not run)

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - one tetrahedron case

Description

Returns the domination number of PE-PCD whose vertices are the data points in Xp.

PE proximity region is defined with respect to the tetrahedron th with expansion parameter $r \ge 1$ and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM".

See also (Ceyhan (2005, 2010)).

Usage

PEdom.num.tetra(Xp, th, r, M = "CM")
PEdom.num.tetra(Xp, th, r, M = "CM")

Arguments

`Xp`	A set of 3D points which constitute the vertices of the digraph.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.

Value

A list with two elements

`dom.num`	Domination number of PE-PCD with vertex set = `Xp` and expansion parameter $r \ge 1$ and center `M`
`mds`	A minimum dominating set of PE-PCD with vertex set = `Xp` and expansion parameter $r \ge 1$ and center `M`
`ind.mds`	Indices of the minimum dominating set `mds`

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-10  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.25

PEdom.num.tetra(Xp,tetra,r,M)

P1<-c(.5,.5,.5)
PEdom.num.tetra(P1,tetra,r,M)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-10  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.25

PEdom.num.tetra(Xp,tetra,r,M)

P1<-c(.5,.5,.5)
PEdom.num.tetra(P1,tetra,r,M)

## End(Not run)

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - one triangle case

Description

Returns the domination number of PE-PCD whose vertices are the data points in Xp.

PE proximity region is defined with respect to the triangle tri with expansion parameter $r \ge 1$ and vertex regions are constructed with center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri or the circumcenter of tri.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

PEdom.num.tri(Xp, tri, r, M = c(1, 1, 1))
PEdom.num.tri(Xp, tri, r, M = c(1, 1, 1))

Arguments

`Xp`	A set of 2D points which constitute the vertices of the digraph.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $(1,1,1)$ , i.e., the center of mass.

Value

A list with two elements

`dom.num`	Domination number of PE-PCD with vertex set = `Xp` and expansion parameter $r \ge 1$ and center `M`
`mds`	A minimum dominating set of PE-PCD with vertex set = `Xp` and expansion parameter $r \ge 1$ and center `M`
`ind.mds`	Indices of the minimum dominating set `mds`

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2)
Tr<-rbind(A,B,C)
n<-10  #try also n<-20
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1,1,1)

r<-1.4

PEdom.num.tri(Xp,Tr,r,M)
IM<-inci.matPEtri(Xp,Tr,r,M)
dom.num.greedy #try also dom.num.exact(IM)

gr.gam<-dom.num.greedy(IM)
gr.gam
Xp[gr.gam$i,]

PEdom.num.tri(Xp,Tr,r,M=c(.4,.4))

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2)
Tr<-rbind(A,B,C)
n<-10  #try also n<-20
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1,1,1)

r<-1.4

PEdom.num.tri(Xp,Tr,r,M)
IM<-inci.matPEtri(Xp,Tr,r,M)
dom.num.greedy #try also dom.num.exact(IM)

gr.gam<-dom.num.greedy(IM)
gr.gam
Xp[gr.gam$i,]

PEdom.num.tri(Xp,Tr,r,M=c(.4,.4))

## End(Not run)

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data

Description

Returns the domination number, a minimum dominating set of PE-PCD whose vertices are the 1D data set Xp, and the domination numbers for partition intervals based on Yp.

Yp determines the end points of the intervals (i.e., partition the real line via intervalization). It also includes the domination numbers in the end intervals, with interval label 1 for the left end interval and $|Yp|+1$ for the right end interval.

PE proximity region is constructed with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ .

Usage

PEdom.num1D(Xp, Yp, r, c = 0.5)
PEdom.num1D(Xp, Yp, r, c = 0.5)

Arguments

`Xp`	A set of 1D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 1D points which constitute the end points of the intervals which partition the real line.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` (default `c=.5`).

Value

A list with three elements

`dom.num`	Domination number of PE-PCD with vertex set `Xp` and expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ .
`mds`	A minimum dominating set of the PE-PCD.
`ind.mds`	The data indices of the minimum dominating set of the PE-PCD whose vertices are `Xp` points.
`int.dom.nums`	Domination numbers of the PE-PCD components for the partition intervals.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
a<-0; b<-10
c<-.4
r<-2

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

PEdom.num1D(Xp,Yp,r,c)

PEdom.num1D(Xp,Yp,r,c=.25)
PEdom.num1D(Xp,Yp,r=1.25,c)

## End(Not run)

## Not run: 
a<-0; b<-10
c<-.4
r<-2

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

PEdom.num1D(Xp,Yp,r,c)

PEdom.num1D(Xp,Yp,r,c=.25)
PEdom.num1D(Xp,Yp,r=1.25,c)

## End(Not run)

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) with non-degeneracy centers - multiple interval case

Description

Returns the domination number, a minimum dominating set of PE-PCD whose vertices are the 1D data set Xp, and the domination numbers for partition intervals based on Yp when PE-PCD is constructed with vertex regions based on non-degeneracy centers.

Yp determines the end points of the intervals (i.e., partition the real line via intervalization).

PE proximity regions are defined with respect to the intervals based on Yp points with expansion parameter $r \ge 1$ and vertex regions in each interval are based on the centrality parameter c which is one of the 2 values of c (i.e., $c \in \{(r-1)/r,1/r\}$ ) that renders the asymptotic distribution of domination number to be non-degenerate for a given value of r in $(1,2)$ and c is center of mass for $r=2$ . These values are called non-degeneracy centrality parameters and the corresponding centers are called nondegeneracy centers.

Usage

PEdom.num1Dnondeg(Xp, Yp, r)
PEdom.num1Dnondeg(Xp, Yp, r)

Arguments

`Xp`	A set of 1D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 1D points which constitute the end points of the intervals which partition the real line.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,2]$ here.

Value

A list with three elements

`dom.num`	Domination number of PE-PCD with vertex set `Xp` and expansion parameter $r in (1,2]$ and centrality parameter $c \in \{(r-1)/r,1/r\}$ .
`mds`	A minimum dominating set of the PE-PCD.
`ind.mds`	The data indices of the minimum dominating set of the PE-PCD whose vertices are `Xp` points.
`int.dom.nums`	Domination numbers of the PE-PCD components for the partition intervals.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
a<-0; b<-10
r<-1.5

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

PEdom.num1Dnondeg(Xp,Yp,r)
PEdom.num1Dnondeg(Xp,Yp,r=1.25)

## End(Not run)

## Not run: 
a<-0; b<-10
r<-1.5

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

PEdom.num1Dnondeg(Xp,Yp,r)
PEdom.num1Dnondeg(Xp,Yp,r=1.25)

## End(Not run)

The line passing through a point and perpendicular to the line segment joining two points

Description

An object of class "Lines". Returns the equation, slope, intercept, and $y$ -coordinates of the line crossing the point p and perpendicular to the line passing through the points a and b with $x$ -coordinates are provided in vector x.

Usage

perpline(p, a, b, x)
perpline(p, a, b, x)

Arguments

`p`	A 2D point at which the perpendicular line to line segment joining `a` and `b` crosses.
`a`, `b`	2D points that determine the line segment (the line will be perpendicular to this line segment).
`x`	A scalar or a `vector` of scalars representing the $x$ -coordinates of the line perpendicular to line joining `a` and `b` and crossing `p`.

Value

A list with the elements

`desc`	Description of the line passing through point `p` and perpendicular to line joining `a` and `b`
`mtitle`	The `"main"` title for the plot of the line passing through point `p` and perpendicular to line joining `a` and `b`
`points`	The input points `a` and `b` (stacked row-wise, i.e., row 1 is point `a` and row 2 is point `b`). Line passing through point `p` is perpendicular to line joining `a` and `b`
`x`	The input vector, can be a scalar or a `vector` of scalars, which constitute the $x$ -coordinates of the point(s) of interest on the line passing through point `p` and perpendicular to line joining `a` and `b`
`y`	The output `vector` which constitutes the $y$ -coordinates of the point(s) of interest on the line passing through point `p` and perpendicular to line joining `a` and `b`. If `x` is a scalar, then `y` will be a scalar and if `x` is a `vector` of scalars, then `y` will be a `vector` of scalars.
`slope`	Slope of the line passing through point `p` and perpendicular to line joining `a` and `b`
`intercept`	Intercept of the line passing through point `p` and perpendicular to line joining `a` and `b`
`equation`	Equation of the line passing through point `p` and perpendicular to line joining `a` and `b`

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1.1,1.2); B<-c(2.3,3.4); p<-c(.51,2.5)

perpline(p,A,B,.45)

pts<-rbind(A,B,p)
xr<-range(pts[,1])
xf<-(xr[2]-xr[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100

plnAB<-perpline(p,A,B,x)
plnAB
summary(plnAB)
plot(plnAB,asp=1)

y<-plnAB$y
Xlim<-range(x,pts[,1])
if (!is.na(y[1])) {Ylim<-range(y,pts[,2])} else {Ylim<-range(pts[,2])}
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
pf<-c(xd,-yd)*.025

plot(A,asp=1,pch=".",xlab="",ylab="",
main="Line Crossing p and Perpendicular to AB",
xlim=Xlim+xd*c(-.5,.5),ylim=Ylim+yd*c(-.05,.05))
points(pts)
txt.str<-c("A","B","p")
text(pts+rbind(pf,pf,pf),txt.str)

segments(A[1],A[2],B[1],B[2],lty=2)
if (!is.na(y[1])) {lines(x,y,type="l",lty=1,
xlim=Xlim,ylim=Ylim)} else {abline(v=p[1])}
tx<-p[1]+abs(xf-p[1])/2;
if (!is.na(y[1])) {ty<-perpline(p,A,B,tx)$y} else {ty=p[2]}
text(tx,ty,"line perpendicular to AB\n and crossing p")

## End(Not run)

## Not run: 
A<-c(1.1,1.2); B<-c(2.3,3.4); p<-c(.51,2.5)

perpline(p,A,B,.45)

pts<-rbind(A,B,p)
xr<-range(pts[,1])
xf<-(xr[2]-xr[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100

plnAB<-perpline(p,A,B,x)
plnAB
summary(plnAB)
plot(plnAB,asp=1)

y<-plnAB$y
Xlim<-range(x,pts[,1])
if (!is.na(y[1])) {Ylim<-range(y,pts[,2])} else {Ylim<-range(pts[,2])}
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
pf<-c(xd,-yd)*.025

plot(A,asp=1,pch=".",xlab="",ylab="",
main="Line Crossing p and Perpendicular to AB",
xlim=Xlim+xd*c(-.5,.5),ylim=Ylim+yd*c(-.05,.05))
points(pts)
txt.str<-c("A","B","p")
text(pts+rbind(pf,pf,pf),txt.str)

segments(A[1],A[2],B[1],B[2],lty=2)
if (!is.na(y[1])) {lines(x,y,type="l",lty=1,
xlim=Xlim,ylim=Ylim)} else {abline(v=p[1])}
tx<-p[1]+abs(xf-p[1])/2;
if (!is.na(y[1])) {ty<-perpline(p,A,B,tx)$y} else {ty=p[2]}
text(tx,ty,"line perpendicular to AB\n and crossing p")

## End(Not run)

The line crossing the 3D point `p` and perpendicular to the plane spanned by 3D points `a`, `b`, and `c`

Description

An object of class "Lines3D". Returns the equation, $x$ -, $y$ -, and $z$ -coordinates of the line crossing 3D point p and perpendicular to the plane spanned by 3D points a, b, and c (i.e., the line is in the direction of normal vector of this plane) with the parameter t being provided in vector t.

Usage

perpline2plane(p, a, b, c, t)
perpline2plane(p, a, b, c, t)

Arguments

`p`	A 3D point through which the straight line passes.
`a`, `b`, `c`	3D points which determine the plane to which the line passing through point `p` would be perpendicular (i.e., the normal `vector` of this plane determines the direction of the straight line passing through `p`).
`t`	A scalar or a `vector` of scalars representing the parameter of the coordinates of the line (for the form: $x=p_0 + A t$ , $y=y_0 + B t$ , and $z=z_0 + C t$ where $p=(p_0,y_0,z_0)$ and normal vector $=(A,B,C)$ ).

Value

A list with the elements

`desc`	A description of the line
`mtitle`	The `"main"` title for the plot of the line
`points`	The input points that determine the line and plane, line crosses point `p` and plane is determined by 3D points `a`, `b`, and `c`.
`pnames`	The names of the input points that determine the line and plane; line would be perpendicular to the plane.
`vecs`	The point `p` and normal vector.
`vec.names`	The names of the point `p` and the second entry is "normal vector".
`x`, `y`, `z`	The $x$ -, $y$ -, and $z$ -coordinates of the point(s) of interest on the line perpendicular to the plane determined by points `a`, `b`, and `c`.
`tsq`	The scalar or the `vector` of the parameter in defining each coordinate of the line for the form: $x=p_0 + A t$ , $y=y_0 + B t$ , and $z=z_0 + C t$ where $p=(p_0,y_0,z_0)$ and normal vector $=(A,B,C)$ .
`equation`	Equation of the line passing through point `p` and perpendicular to the plane determined by points `a`, `b`, and `c` (i.e., line is in the direction of the normal `vector` N of the plane). The line equation is in the form: $x=p_0 + A t$ , $y=y_0 + B t$ , and $z=z_0 + C t$ where $p=(p_0,y_0,z_0)$ and normal vector $=(A,B,C)$ .

Author(s)

Elvan Ceyhan

Examples

## Not run: 
P<-c(1,1,1); Q<-c(1,10,4); R<-c(1,1,3); S<-c(3,9,12)

cf<-as.numeric(Plane(Q,R,S,1,1)$coeff)
a<-cf[1]; b<-cf[2]; c<- -1;

vecs<-rbind(Q,c(a,b,c))
pts<-rbind(P,Q,R,S)
perpline2plane(P,Q,R,S,.1)

tr<-range(pts,vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=5)  #try also l=10, 20, or 100

pln2pl<-perpline2plane(P,Q,R,S,tsq)
pln2pl
summary(pln2pl)
plot(pln2pl,theta = 225, phi = 30, expand = 0.7,
facets = FALSE, scale = TRUE)

xc<-pln2pl$x
yc<-pln2pl$y
zc<-pln2pl$z

zr<-range(zc)
zf<-(zr[2]-zr[1])*.2
Rv<- -c(a,b,c)*zf*5

Dr<-(Q+R+S)/3

pts2<-rbind(Q,R,S)
xr<-range(pts2[,1],xc); yr<-range(pts2[,2],yc)
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
xs<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
ys<-seq(yr[1]-yf,yr[2]+yf,l=5)  #try also l=10, 20, or 100

plQRS<-Plane(Q,R,S,xs,ys)
z.grid<-plQRS$z

Xlim<-range(xc,xs,pts[,1])
Ylim<-range(yc,ys,pts[,2])
Zlim<-range(zc,z.grid,pts[,3])

xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::persp3D(z = z.grid, x = xs, y = ys, theta =225, phi = 30,
main="Line Crossing P and \n Perpendicular to the Plane Defined by Q, R, S",
col="lightblue", ticktype = "detailed",
        xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
        zlim=Zlim+zd*c(-.05,.05))
        #plane spanned by points Q, R, S
plot3D::lines3D(xc, yc, zc, bty = "g",pch = 20, cex = 2,col="red",
ticktype = "detailed",add=TRUE)
plot3D::arrows3D(Dr[1],Dr[2],Dr[3],Dr[1]+Rv[1],Dr[2]+Rv[2],
Dr[3]+Rv[3], add=TRUE)
plot3D::points3D(pts[,1],pts[,2],pts[,3],add=TRUE)
plot3D::text3D(pts[,1],pts[,2],pts[,3],labels=c("P","Q","R","S"),add=TRUE)
plot3D::arrows3D(P[1],P[2],P[3]-zf,P[1],P[2],P[3],lty=2, add=TRUE)
plot3D::text3D(P[1],P[2],P[3]-zf,labels="initial point",add=TRUE)
plot3D::text3D(P[1],P[2],P[3]+zf/2,labels="P",add=TRUE)
plot3D::arrows3D(Dr[1],Dr[2],Dr[3],Dr[1]+Rv[1]/2,Dr[2]+Rv[2]/2,
Dr[3]+Rv[3]/2,lty=2, add=TRUE)
plot3D::text3D(Dr[1]+Rv[1]/2,Dr[2]+Rv[2]/2,Dr[3]+Rv[3]/2,
labels="normal vector",add=TRUE)

## End(Not run)
## Not run: 
P<-c(1,1,1); Q<-c(1,10,4); R<-c(1,1,3); S<-c(3,9,12)

cf<-as.numeric(Plane(Q,R,S,1,1)$coeff)
a<-cf[1]; b<-cf[2]; c<- -1;

vecs<-rbind(Q,c(a,b,c))
pts<-rbind(P,Q,R,S)
perpline2plane(P,Q,R,S,.1)

tr<-range(pts,vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=5)  #try also l=10, 20, or 100

pln2pl<-perpline2plane(P,Q,R,S,tsq)
pln2pl
summary(pln2pl)
plot(pln2pl,theta = 225, phi = 30, expand = 0.7,
facets = FALSE, scale = TRUE)

xc<-pln2pl$x
yc<-pln2pl$y
zc<-pln2pl$z

zr<-range(zc)
zf<-(zr[2]-zr[1])*.2
Rv<- -c(a,b,c)*zf*5

Dr<-(Q+R+S)/3

pts2<-rbind(Q,R,S)
xr<-range(pts2[,1],xc); yr<-range(pts2[,2],yc)
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
xs<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
ys<-seq(yr[1]-yf,yr[2]+yf,l=5)  #try also l=10, 20, or 100

plQRS<-Plane(Q,R,S,xs,ys)
z.grid<-plQRS$z

Xlim<-range(xc,xs,pts[,1])
Ylim<-range(yc,ys,pts[,2])
Zlim<-range(zc,z.grid,pts[,3])

xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::persp3D(z = z.grid, x = xs, y = ys, theta =225, phi = 30,
main="Line Crossing P and \n Perpendicular to the Plane Defined by Q, R, S",
col="lightblue", ticktype = "detailed",
        xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
        zlim=Zlim+zd*c(-.05,.05))
        #plane spanned by points Q, R, S
plot3D::lines3D(xc, yc, zc, bty = "g",pch = 20, cex = 2,col="red",
ticktype = "detailed",add=TRUE)
plot3D::arrows3D(Dr[1],Dr[2],Dr[3],Dr[1]+Rv[1],Dr[2]+Rv[2],
Dr[3]+Rv[3], add=TRUE)
plot3D::points3D(pts[,1],pts[,2],pts[,3],add=TRUE)
plot3D::text3D(pts[,1],pts[,2],pts[,3],labels=c("P","Q","R","S"),add=TRUE)
plot3D::arrows3D(P[1],P[2],P[3]-zf,P[1],P[2],P[3],lty=2, add=TRUE)
plot3D::text3D(P[1],P[2],P[3]-zf,labels="initial point",add=TRUE)
plot3D::text3D(P[1],P[2],P[3]+zf/2,labels="P",add=TRUE)
plot3D::arrows3D(Dr[1],Dr[2],Dr[3],Dr[1]+Rv[1]/2,Dr[2]+Rv[2]/2,
Dr[3]+Rv[3]/2,lty=2, add=TRUE)
plot3D::text3D(Dr[1]+Rv[1]/2,Dr[2]+Rv[2]/2,Dr[3]+Rv[3]/2,
labels="normal vector",add=TRUE)

## End(Not run)

The plane passing through three distinct 3D points `a`, `b`, and `c`

Description

An object of class "Planes". Returns the equation and $z$ -coordinates of the plane passing through three distinct 3D points a, b, and c with $x$ - and $y$ -coordinates are provided in vectors x and y, respectively.

Usage

Plane(a, b, c, x, y)
Plane(a, b, c, x, y)

Arguments

`a`, `b`, `c`	3D points that determine the plane (i.e., through which the plane is passing).
`x`, `y`	Scalars or vectors of scalars representing the $x$ - and $y$ -coordinates of the plane.

Value

A list with the elements

`desc`	A description of the plane
`points`	The input points `a`, `b`, and `c` through which the plane is passing (stacked row-wise, i.e., row 1 is point `a`, row 2 is point `b` and row 3 is point `c`).
`x`, `y`	The input vectors which constitutes the $x$ - and $y$ -coordinates of the point(s) of interest on the plane. `x` and `y` can be scalars or vectors of scalars.
`z`	The output `vector` which constitutes the $z$ -coordinates of the point(s) of interest on the plane. If `x` and `y` are scalars, `z` will be a scalar and if `x` and `y` are vectors of scalars, then `z` needs to be a `matrix` of scalars, containing the $z$ -coordinate for each pair of `x` and `y` values.
`coeff`	Coefficients of the plane (in the $z = A x+B y+C$ form).
`equation`	Equation of the plane in long form
`equation2`	Equation of the plane in short form, to be inserted on the plot

Author(s)

Elvan Ceyhan

Examples

## Not run: 
P1<-c(1,10,3); P2<-c(1,1,3); P3<-c(3,9,12) #also try P2=c(2,2,3)

pts<-rbind(P1,P2,P3)
Plane(P1,P2,P3,.1,.2)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5)  #try also l=10, 20, or 100

plP123<-Plane(P1,P2,P3,x,y)
plP123
summary(plP123)
plot(plP123,theta = 225, phi = 30, expand = 0.7, facets = FALSE, scale = TRUE)

z.grid<-plP123$z

persp(x,y,z.grid, xlab="x",ylab="y",zlab="z",
theta = -30, phi = 30, expand = 0.5, col = "lightblue",
      ltheta = 120, shade = 0.05, ticktype = "detailed")

zr<-max(z.grid)-min(z.grid)
Pts<-rbind(P1,P2,P3)+rbind(c(0,0,zr*.1),c(0,0,zr*.1),c(0,0,zr*.1))
Mn.pts<-apply(Pts,2,mean)

plot3D::persp3D(z = z.grid, x = x, y = y,theta = 225, phi = 30, expand = 0.3,
main = "Plane Crossing Points P1, P2, and P3", facets = FALSE, scale = TRUE)
#plane spanned by points P1, P2, P3
#add the defining points
plot3D::points3D(Pts[,1],Pts[,2],Pts[,3], add=TRUE)
plot3D::text3D(Pts[,1],Pts[,2],Pts[,3], c("P1","P2","P3"),add=TRUE)
plot3D::text3D(Mn.pts[1],Mn.pts[2],Mn.pts[3],plP123$equation,add=TRUE)
#plot3D::polygon3D(Pts[,1],Pts[,2],Pts[,3], add=TRUE)

## End(Not run)

## Not run: 
P1<-c(1,10,3); P2<-c(1,1,3); P3<-c(3,9,12) #also try P2=c(2,2,3)

pts<-rbind(P1,P2,P3)
Plane(P1,P2,P3,.1,.2)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5)  #try also l=10, 20, or 100

plP123<-Plane(P1,P2,P3,x,y)
plP123
summary(plP123)
plot(plP123,theta = 225, phi = 30, expand = 0.7, facets = FALSE, scale = TRUE)

z.grid<-plP123$z

persp(x,y,z.grid, xlab="x",ylab="y",zlab="z",
theta = -30, phi = 30, expand = 0.5, col = "lightblue",
      ltheta = 120, shade = 0.05, ticktype = "detailed")

zr<-max(z.grid)-min(z.grid)
Pts<-rbind(P1,P2,P3)+rbind(c(0,0,zr*.1),c(0,0,zr*.1),c(0,0,zr*.1))
Mn.pts<-apply(Pts,2,mean)

plot3D::persp3D(z = z.grid, x = x, y = y,theta = 225, phi = 30, expand = 0.3,
main = "Plane Crossing Points P1, P2, and P3", facets = FALSE, scale = TRUE)
#plane spanned by points P1, P2, P3
#add the defining points
plot3D::points3D(Pts[,1],Pts[,2],Pts[,3], add=TRUE)
plot3D::text3D(Pts[,1],Pts[,2],Pts[,3], c("P1","P2","P3"),add=TRUE)
plot3D::text3D(Mn.pts[1],Mn.pts[2],Mn.pts[3],plP123$equation,add=TRUE)
#plot3D::polygon3D(Pts[,1],Pts[,2],Pts[,3], add=TRUE)

## End(Not run)

Plot an `Extrema` `object`

Description

Plots the data points and extrema among these points together with the reference object (e.g., boundary of the support region)

Usage

## S3 method for class 'Extrema'
plot(x, asp = NA, xlab = "", ylab = "", zlab = "", ...)
## S3 method for class 'Extrema'
plot(x, asp = NA, xlab = "", ylab = "", zlab = "", ...)

Arguments

`x`	Object of class `Extrema`.
`asp`	A `numeric` value, giving the aspect ratio for $y$ -axis to $x$ -axis $y/x$ for the 2D case, it is redundant in the 3D case (default is `NA`), see the official help for `asp` by typing "`? asp`".
`xlab`, `ylab`, `zlab`	Titles for the $x$ and $y$ axes in the 2D case, and $x$ , $y$ , and $z$ axes in the 3D case, respectively (default is `""` for all).
`...`	Additional parameters for `plot`.

Value

None

Examples

## Not run: 
n<-10
Xp<-runif.std.tri(n)$gen.points
Ext<-cl2edges.std.tri(Xp)
Ext
plot(Ext,asp=1)

## End(Not run)

## Not run: 
n<-10
Xp<-runif.std.tri(n)$gen.points
Ext<-cl2edges.std.tri(Xp)
Ext
plot(Ext,asp=1)

## End(Not run)

Plot a `Lines` `object`

Description

Plots the line together with the defining points.

Usage

## S3 method for class 'Lines'
plot(x, asp = NA, xlab = "x", ylab = "y", ...)
## S3 method for class 'Lines'
plot(x, asp = NA, xlab = "x", ylab = "y", ...)

Arguments

`x`	Object of class `Lines`.
`asp`	A `numeric` value, giving the aspect ratio for $y$ -axis to $x$ -axis $y/x$ (default is `NA`), see the official help for `asp` by typing "`? asp`".
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default is `xlab="x"` and `ylab="y"`).
`...`	Additional parameters for `plot`.

Value

None

Examples

## Not run: 
A<-c(-1.22,-2.33); B<-c(2.55,3.75)
xr<-range(A,B);
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=3) #try also l=10, 20 or 100

lnAB<-Line(A,B,x)
lnAB
plot(lnAB)

## End(Not run)

## Not run: 
A<-c(-1.22,-2.33); B<-c(2.55,3.75)
xr<-range(A,B);
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=3) #try also l=10, 20 or 100

lnAB<-Line(A,B,x)
lnAB
plot(lnAB)

## End(Not run)

Plot a `Lines3D` `object`

Description

Plots the line together with the defining vectors (i.e., the initial and direction vectors).

Usage

## S3 method for class 'Lines3D'
plot(x, xlab = "x", ylab = "y", zlab = "z", phi = 40, theta = 40, ...)
## S3 method for class 'Lines3D'
plot(x, xlab = "x", ylab = "y", zlab = "z", phi = 40, theta = 40, ...)

Arguments

`x`	Object of class `Lines3D`.
`xlab`, `ylab`, `zlab`	Titles for the $x$ , $y$ , and $z$ axes, respectively (default is `xlab="x"`, `ylab="y"` and `zlab="z"`).
`theta`, `phi`	The angles defining the viewing direction. `theta` gives the azimuthal direction and `phi` the colatitude. See `persp3D` for more details.
`...`	Additional parameters for `plot`.

Value

None

Examples

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3);
vecs<-rbind(P,Q)
Line3D(P,Q,.1)
Line3D(P,Q,.1,dir.vec=FALSE)

tr<-range(vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=3) #try also l=10, 20 or 100

lnPQ3D<-Line3D(P,Q,tsq)
lnPQ3D
plot(lnPQ3D)

## End(Not run)

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3);
vecs<-rbind(P,Q)
Line3D(P,Q,.1)
Line3D(P,Q,.1,dir.vec=FALSE)

tr<-range(vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=3) #try also l=10, 20 or 100

lnPQ3D<-Line3D(P,Q,tsq)
lnPQ3D
plot(lnPQ3D)

## End(Not run)

Plot a `NumArcs` `object`

Description

Plots the scatter plot of the data points (i.e. vertices of the PCDs) and the Delaunay tessellation of the nontarget points marked with number of arcs in the centroid of the Delaunay cells.

Usage

## S3 method for class 'NumArcs'
plot(x, Jit = 0.1, ...)
## S3 method for class 'NumArcs'
plot(x, Jit = 0.1, ...)

Arguments

`x`	Object of class `NumArcs`.
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default is 0.1, for the 1D case, the vertices of the PCD are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of vertices and the interval end points; it is redundant in the 2D case.
`...`	Additional parameters for `plot`.

Value

None

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
plot(Arcs)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
plot(Arcs)

## End(Not run)

Plot a `Patterns` `object`

Description

Plots the points generated from the pattern (color coded for each class) together with the study window

Usage

## S3 method for class 'Patterns'
plot(x, asp = NA, xlab = "x", ylab = "y", ...)
## S3 method for class 'Patterns'
plot(x, asp = NA, xlab = "x", ylab = "y", ...)

Arguments

`x`	Object of class `Patterns`.
`asp`	A `numeric` value, giving the aspect ratio for $y$ -axis to $x$ -axis $y/x$ (default is `NA`), see the official help for `asp` by typing "`? asp`".
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default is `xlab="x"` and `ylab="y"`).
`...`	Additional parameters for `plot`.

Value

None

Examples

## Not run: 
nx<-10; #try also 100 and 1000
ny<-5; #try also 1
e<-.15;
Y<-cbind(runif(ny),runif(ny))
#with default bounding box (i.e., unit square)

Xdt<-rseg.circular(nx,Y,e)
Xdt
plot(Xdt,asp=1)

## End(Not run)

## Not run: 
nx<-10; #try also 100 and 1000
ny<-5; #try also 1
e<-.15;
Y<-cbind(runif(ny),runif(ny))
#with default bounding box (i.e., unit square)

Xdt<-rseg.circular(nx,Y,e)
Xdt
plot(Xdt,asp=1)

## End(Not run)

Plot a `PCDs` `object`

Description

Plots the vertices and the arcs of the PCD together with the vertices and boundaries of the partition cells (i.e., intervals in the 1D case and triangles in the 2D case)

Usage

## S3 method for class 'PCDs'
plot(x, Jit = 0.1, ...)
## S3 method for class 'PCDs'
plot(x, Jit = 0.1, ...)

Arguments

`x`	Object of class `PCDs`.
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default is 0.1, for the 1D case, the vertices of the PCD are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of vertices and the interval end points; it is redundant in the 2D case.
`...`	Additional parameters for `plot`.

Value

None

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
plot(Arcs)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
plot(Arcs)

## End(Not run)

Plot a `Planes` `object`

Description

Plots the plane together with the defining 3D points.

Usage

## S3 method for class 'Planes'
plot(
  x,
  x.grid.size = 10,
  y.grid.size = 10,
  xlab = "x",
  ylab = "y",
  zlab = "z",
  phi = 40,
  theta = 40,
  ...
)
## S3 method for class 'Planes'
plot(
  x,
  x.grid.size = 10,
  y.grid.size = 10,
  xlab = "x",
  ylab = "y",
  zlab = "z",
  phi = 40,
  theta = 40,
  ...
)

Arguments

`x`	Object of class `Planes`.
`x.grid.size`, `y.grid.size`	the size of the grids for the $x$ and $y$ axes, default is 10 for both
`xlab`, `ylab`, `zlab`	Titles for the $x$ , $y$ , and $z$ axes, respectively (default is `xlab="x"`, `ylab="y"`, and `zlab="z"`).
`theta`, `phi`	The angles defining the viewing direction, default is 40 for both. `theta` gives the azimuthal direction and `phi` the colatitude. see `persp`.
`...`	Additional parameters for `plot`.

Value

None

Examples

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3); C<-c(3,9,12)
pts<-rbind(P,Q,C)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5) #try also l=10, 20 or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5) #try also l=10, 20 or 100

plPQC<-Plane(P,Q,C,x,y)
plPQC
plot(plPQC,theta = 225, phi = 30, expand = 0.7,
facets = FALSE, scale = TRUE)

## End(Not run)

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3); C<-c(3,9,12)
pts<-rbind(P,Q,C)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5) #try also l=10, 20 or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5) #try also l=10, 20 or 100

plPQC<-Plane(P,Q,C,x,y)
plPQC
plot(plPQC,theta = 225, phi = 30, expand = 0.7,
facets = FALSE, scale = TRUE)

## End(Not run)

Plot a `TriLines` `object`

Description

Plots the line together with the defining triangle.

Usage

## S3 method for class 'TriLines'
plot(x, xlab = "x", ylab = "y", ...)
## S3 method for class 'TriLines'
plot(x, xlab = "x", ylab = "y", ...)

Arguments

`x`	Object of class `TriLines`.
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default is `xlab="x"` and `ylab="y"`).
`...`	Additional parameters for `plot`.

Value

None

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,l=3)

lnACM<-lineA2CMinTe(x)
lnACM
plot(lnACM)

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,l=3)

lnACM<-lineA2CMinTe(x)
lnACM
plot(lnACM)

Plot a `Uniform` `object`

Description

Plots the points generated from the uniform distribution together with the support region

Usage

## S3 method for class 'Uniform'
plot(x, asp = NA, xlab = "x", ylab = "y", zlab = "z", ...)
## S3 method for class 'Uniform'
plot(x, asp = NA, xlab = "x", ylab = "y", zlab = "z", ...)

Arguments

`x`	Object of class `Uniform`.
`asp`	A `numeric` value, giving the aspect ratio for $y$ -axis to $x$ -axis $y/x$ for the 2D case, it is redundant in the 3D case (default is `NA`), see the official help for `asp` by typing "`? asp`".
`xlab`, `ylab`, `zlab`	Titles for the $x$ and $y$ axes in the 2D case, and $x$ , $y$ , and $z$ axes in the 3D case, respectively (default is `xlab="x"`, `ylab="y"`, and `zlab="z"`).
`...`	Additional parameters for `plot`.

Value

None

Examples

## Not run: 
n<-10 #try also 20, 100, and 1000
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)

Xdt<-runif.tri(n,Tr)
Xdt
plot(Xdt,asp=1)

## End(Not run)

## Not run: 
n<-10 #try also 20, 100, and 1000
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)

Xdt<-runif.tri(n,Tr)
Xdt
plot(Xdt,asp=1)

## End(Not run)

The plot of the arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for a 2D data set - multiple triangle case

Description

Plots the arcs of AS-PCD whose vertices are the data points in Xp and Delaunay triangles based on Yp points.

AS proximity regions are constructed with respect to the Delaunay triangles based on Yp points, i.e., AS proximity regions are defined only for Xp points inside the convex hull of Yp points. That is, arcs may exist for Xp points only inside the convex hull of Yp points.

See (Ceyhan (2005, 2010)) for more on AS-PCDs. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

plotASarcs(
  Xp,
  Yp,
  M = "CC",
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)
plotASarcs(
  Xp,
  Yp,
  M = "CC",
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the AS-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangulation. The Delaunay triangles partition the convex hull of `Yp` points.
`M`	The center of the triangle. `"CC"` stands for circumcenter of each Delaunay triangle or 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle; default is `M="CC"` i.e., the circumcenter of each triangle.
`asp`	A `numeric` value, giving the aspect ratio for $y$ axis to $x$ -axis $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`".
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`...`	Additional `plot` parameters.

Value

A plot of the arcs of the AS-PCD for a 2D data set Xp where AS proximity regions are defined with respect to the Delaunay triangles based on Yp points; also plots the Delaunay triangles based on Yp points.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

#plotASarcs(Xp,Yp,M,xlab="",ylab="")
plotASarcs(Xp,Yp,M,asp=1,xlab="",ylab="")

plotASarcs(Xp,Yp[1:3,],M,xlab="",ylab="")

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

#plotASarcs(Xp,Yp,M,xlab="",ylab="")
plotASarcs(Xp,Yp,M,asp=1,xlab="",ylab="")

plotASarcs(Xp,Yp[1:3,],M,xlab="",ylab="")

## End(Not run)

The plot of the arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for a 2D data set - one triangle case

Description

Plots the arcs of AS-PCD whose vertices are the data points, Xp and the triangle tri. AS proximity regions are constructed with respect to the triangle tri, i.e., only for Xp points inside the triangle tri.

Vertex regions are based on the center M="CC" for circumcenter of tri; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is M="CC" the circumcenter of tri. When the center is the circumcenter, CC, the vertex regions are constructed based on the orthogonal projections to the edges, while with any interior center M, the vertex regions are constructed using the extensions of the lines combining vertices with M.

See also (Ceyhan (2005, 2010)).

Usage

plotASarcs.tri(
  Xp,
  tri,
  M = "CC",
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  vert.reg = FALSE,
  ...
)
plotASarcs.tri(
  Xp,
  tri,
  M = "CC",
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  vert.reg = FALSE,
  ...
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the AS-PCD.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$ ; default is `M="CC"` i.e., the circumcenter of `tri`.
`asp`	A `numeric` value, giving the aspect ratio for $y$ axis to $x$ -axis $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`".
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`vert.reg`	A logical argument to add vertex regions to the plot, default is `vert.reg=FALSE`.
`...`	Additional `plot` parameters.

Value

A plot of the arcs of the AS-PCD for a 2D data set Xp where AS proximity regions are defined with respect to the triangle tri; also plots the triangle tri

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g  #try also Xp<-cbind(runif(n,1,2),runif(n,0,2))

M<-as.numeric(runif.tri(1,Tr)$g)  #try also  #M<-c(1.6,1.2)

plotASarcs.tri(Xp,Tr,M,main="Arcs of AS-PCD",xlab="",ylab="")

plotASarcs.tri(Xp,Tr,M,main="Arcs of AS-PCD",xlab="",ylab="",vert.reg = TRUE)

# or try the default center
#plotASarcs.tri(Xp,Tr,asp=1,main="arcs of AS-PCD",xlab="",ylab="",vert.reg = TRUE);
#M = (arcsAStri(Xp,Tr)$param)$c #the part "M = as.numeric(arcsAStri(Xp,Tr)$param)" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with vertex regions)
#but first we need to determine whether the center used for vertex regions is CC or not
#see the description for more detail.
CC<-circumcenter.tri(Tr)

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}

#now we add the vertex names and annotation
txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]+c(-.02,.02,.02,.01,.05,-0.03,-.01)
yc<-txt[,2]+c(.02,.02,.02,.07,.02,.05,-.06)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$g  #try also Xp<-cbind(runif(n,1,2),runif(n,0,2))

M<-as.numeric(runif.tri(1,Tr)$g)  #try also  #M<-c(1.6,1.2)

plotASarcs.tri(Xp,Tr,M,main="Arcs of AS-PCD",xlab="",ylab="")

plotASarcs.tri(Xp,Tr,M,main="Arcs of AS-PCD",xlab="",ylab="",vert.reg = TRUE)

# or try the default center
#plotASarcs.tri(Xp,Tr,asp=1,main="arcs of AS-PCD",xlab="",ylab="",vert.reg = TRUE);
#M = (arcsAStri(Xp,Tr)$param)$c #the part "M = as.numeric(arcsAStri(Xp,Tr)$param)" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with vertex regions)
#but first we need to determine whether the center used for vertex regions is CC or not
#see the description for more detail.
CC<-circumcenter.tri(Tr)

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}

#now we add the vertex names and annotation
txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]+c(-.02,.02,.02,.01,.05,-0.03,-.01)
yc<-txt[,2]+c(.02,.02,.02,.07,.02,.05,-.06)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The plot of the Arc Slice (AS) Proximity Regions for a 2D data set - multiple triangle case

Description

Plots the Xp points in and outside of the convex hull of Yp points and also plots the AS proximity regions for Xp points and Delaunay triangles based on Yp points.

AS proximity regions are constructed with respect to the Delaunay triangles based on Yp points (these triangles partition the convex hull of Yp points), i.e., AS proximity regions are only defined for Xp points inside the convex hull of Yp points.

See (Ceyhan (2005, 2010)) for more on AS-PCDs. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

plotASregs(
  Xp,
  Yp,
  M = "CC",
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)
plotASregs(
  Xp,
  Yp,
  M = "CC",
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

`Xp`	A set of 2D points for which AS proximity regions are constructed.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangulation. The Delaunay triangles partition the convex hull of `Yp` points.
`M`	The center of the triangle. `"CC"` stands for circumcenter of each Delaunay triangle or 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle; default is `M="CC"` i.e., the circumcenter of each triangle.
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`...`	Additional `plot` parameters.

Value

Plot of the Xp points, Delaunay triangles based on Yp and also the AS proximity regions for Xp points inside the convex hull of Yp points

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
nx<-10 ; ny<-5

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3) #or M="CC"

plotASregs(Xp,Yp,M,xlab="",ylab="")

plotASregs(Xp,Yp[1:3,],M,xlab="",ylab="")

Xp<-c(.5,.5)
plotASregs(Xp,Yp,M,xlab="",ylab="")

## End(Not run)

## Not run: 
nx<-10 ; ny<-5

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3) #or M="CC"

plotASregs(Xp,Yp,M,xlab="",ylab="")

plotASregs(Xp,Yp[1:3,],M,xlab="",ylab="")

Xp<-c(.5,.5)
plotASregs(Xp,Yp,M,xlab="",ylab="")

## End(Not run)

The plot of the Arc Slice (AS) Proximity Regions for a 2D data set - one triangle case

Description

Plots the points in and outside of the triangle tri and also the AS proximity regions for points in data set Xp.

AS proximity regions are defined with respect to the triangle tri, so AS proximity regions are defined only for points inside the triangle tri and vertex regions are based on the center M="CC" for circumcenter of tri; or $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri; default is M="CC" the circumcenter of tri. When vertex regions are constructed with circumcenter, CC, the vertex regions are constructed based on the orthogonal projections to the edges, while with any interior center M, the vertex regions are constructed using the extensions of the lines combining vertices with M.

See also (Ceyhan (2005, 2010)).

Usage

plotASregs.tri(
  Xp,
  tri,
  M = "CC",
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  vert.reg = FALSE,
  ...
)
plotASregs.tri(
  Xp,
  tri,
  M = "CC",
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  vert.reg = FALSE,
  ...
)

Arguments

`Xp`	A set of 2D points for which AS proximity regions are constructed.
`tri`	Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
`M`	The center of the triangle. `"CC"` stands for circumcenter of the triangle `tri` or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle $T_b$ ; default is `M="CC"` i.e., the circumcenter of `tri`.
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`vert.reg`	A logical argument to add vertex regions to the plot, default is `vert.reg=FALSE`.
`...`	Additional `plot` parameters.

Value

Plot of the AS proximity regions for points inside the triangle tri (and only the points outside tri)

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp0<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)  #try also  #M<-c(1.6,1.2);

plotASregs.tri(Xp0,Tr,M,main="Proximity Regions for AS-PCD", xlab="",ylab="")
Xp = Xp0[1,]
plotASregs.tri(Xp,Tr,M,main="Proximity Regions for AS-PCD", xlab="",ylab="")

#can plot the arcs of the AS-PCD
#plotASarcs.tri(Xp,Tr,M,main="Arcs of AS-PCD",xlab="",ylab="")

plotASregs.tri(Xp,Tr,M,main="Proximity Regions for AS-PCD", xlab="",ylab="",vert.reg=TRUE)

# or try the default center
#plotASregs.tri(Xp,Tr,main="Proximity Regions for AS-PCD", xlab="",ylab="",vert.reg=TRUE);
M = (arcsAStri(Xp,Tr)$param)$c #the part "M = as.numeric(arcsAStri(Xp,Tr)$param)" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with vertex regions)
#but first we need to determine whether the center used for vertex regions is CC or not
#see the description for more detail.
CC<-circumcenter.tri(Tr)
#Arcs<-arcsAStri(Xp,Tr,M)
#M = as.numeric(Arcs$parameters)
if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}

#now we add the vertex names and annotation
txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]+c(-.02,.03,.03,.03,.05,-0.03,-.01)
yc<-txt[,2]+c(.02,.02,.02,.07,.02,.05,-.06)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp0<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)  #try also  #M<-c(1.6,1.2);

plotASregs.tri(Xp0,Tr,M,main="Proximity Regions for AS-PCD", xlab="",ylab="")
Xp = Xp0[1,]
plotASregs.tri(Xp,Tr,M,main="Proximity Regions for AS-PCD", xlab="",ylab="")

#can plot the arcs of the AS-PCD
#plotASarcs.tri(Xp,Tr,M,main="Arcs of AS-PCD",xlab="",ylab="")

plotASregs.tri(Xp,Tr,M,main="Proximity Regions for AS-PCD", xlab="",ylab="",vert.reg=TRUE)

# or try the default center
#plotASregs.tri(Xp,Tr,main="Proximity Regions for AS-PCD", xlab="",ylab="",vert.reg=TRUE);
M = (arcsAStri(Xp,Tr)$param)$c #the part "M = as.numeric(arcsAStri(Xp,Tr)$param)" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with vertex regions)
#but first we need to determine whether the center used for vertex regions is CC or not
#see the description for more detail.
CC<-circumcenter.tri(Tr)
#Arcs<-arcsAStri(Xp,Tr,M)
#M = as.numeric(Arcs$parameters)
if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}

#now we add the vertex names and annotation
txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]+c(-.02,.03,.03,.03,.05,-0.03,-.01)
yc<-txt[,2]+c(.02,.02,.02,.07,.02,.05,-.06)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The plot of the arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for a 2D data set - multiple triangle case

Description

Plots the arcs of Central Similarity Proximity Catch Digraph (CS-PCD) whose vertices are the data points in Xp in the multiple triangle case and the Delaunay triangles based on Yp points.

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points). Loops are not allowed so arcs are only possible for points inside the convex hull of Yp points.

See (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)) more on the CS-PCDs. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

plotCSarcs(
  Xp,
  Yp,
  t,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)
plotCSarcs(
  Xp,
  Yp,
  t,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle, default for $M=(1,1,1)$ which is the center of mass of each triangle.
`asp`	A `numeric` value, giving the aspect ratio $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`"
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both)
`...`	Additional `plot` parameters.

Value

A plot of the arcs of the CS-PCD whose vertices are the points in data set Xp and the Delaunay triangles based on Yp points

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2014). “Comparison of Relative Density of Two Random Geometric Digraph Families in Testing Spatial Clustering.” TEST, 23(1), 100-134.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)
t<-1.5  #try also t<-2

plotCSarcs(Xp,Yp,t,M,xlab="",ylab="")

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)
t<-1.5  #try also t<-2

plotCSarcs(Xp,Yp,t,M,xlab="",ylab="")

## End(Not run)

The plot of the arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) for 1D data (vertices jittered along $y$ -coordinate) - one interval case

Description

Plots the arcs of CS-PCD whose vertices are the 1D points, Xp. CS proximity regions are constructed with expansion parameter $t > 0$ and centrality parameter $c \in (0,1)$ and the intervals are based on the interval int $=(a,b)$ That is, data set Xp constitutes the vertices of the digraph and int determines the end points of the interval.

For better visualization, a uniform jitter from $U(-Jit,Jit)$ (default for $Jit=.1$ ) is added to the $y$ -direction where Jit equals to the range of $\{$ Xp, int $\}$ multiplied by Jit with default for $Jit=.1$ ). center is a logical argument, if TRUE, plot includes the center of the interval int as a vertical line in the plot, else center of the interval is not plotted.

Usage

plotCSarcs.int(
  Xp,
  int,
  t,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  center = FALSE,
  ...
)
plotCSarcs.int(
  Xp,
  int,
  t,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  center = FALSE,
  ...
)

Arguments

`Xp`	A `vector` of 1D points constituting the vertices of the CS-PCD.
`int`	A `vector` of two 1D points constituting the end points of the interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center of the interval with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default=`0.1` and `Xp` points are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of range of $\{$ `Xp`, `int` $\}$ multiplied by `Jit`).
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles of the $x$ and $y$ axes in the plot (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`center`	A logical argument, if `TRUE`, plot includes the center of the interval `int` as a vertical line in the plot, else center of the interval is not plotted.
`...`	Additional `plot` parameters.

Value

A plot of the arcs of CS-PCD whose vertices are the 1D data set Xp in which vertices are jittered along $y$ -axis for better visualization.

Author(s)

Elvan Ceyhan

References

There are no references for Rd macro ⁠\insertAllCites⁠ on this help page.

Examples

tau<-2
c<-.4
a<-0; b<-10; int<-c(a,b)

#n is number of X points
n<-10;  #try also n<-20;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(n,a-xf,b+xf)

Xlim=range(Xp,int)
Ylim=3*c(-1,1)

jit<-.1
plotCSarcs.int(Xp,int,t=tau,c,jit,xlab="",ylab="",xlim=Xlim,ylim=Ylim)

set.seed(1)
plotCSarcs.int(Xp,int,t=1.5,c=.3,jit,xlab="",ylab="",center=TRUE)
set.seed(1)
plotCSarcs.int(Xp,int,t=2,c=.4,jit,xlab="",ylab="",center=TRUE)

tau<-2
c<-.4
a<-0; b<-10; int<-c(a,b)

#n is number of X points
n<-10;  #try also n<-20;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(n,a-xf,b+xf)

Xlim=range(Xp,int)
Ylim=3*c(-1,1)

jit<-.1
plotCSarcs.int(Xp,int,t=tau,c,jit,xlab="",ylab="",xlim=Xlim,ylim=Ylim)

set.seed(1)
plotCSarcs.int(Xp,int,t=1.5,c=.3,jit,xlab="",ylab="",center=TRUE)
set.seed(1)
plotCSarcs.int(Xp,int,t=2,c=.4,jit,xlab="",ylab="",center=TRUE)

The plot of the arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for a 2D data set - one triangle case

Description

Plots the arcs of CS-PCD whose vertices are the data points, Xp and the triangle tri. CS proximity regions are constructed with respect to the triangle tri with expansion parameter $t>0$ , i.e., arcs may exist only for Xp points inside the triangle tri.

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

plotCSarcs.tri(
  Xp,
  tri,
  t,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  edge.reg = FALSE,
  ...
)
plotCSarcs.tri(
  Xp,
  tri,
  t,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  edge.reg = FALSE,
  ...
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the CS-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`.
`asp`	A `numeric` value, giving the aspect ratio $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`".
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`edge.reg`	A logical argument to add edge regions to the plot, default is `edge.reg=FALSE`.
`...`	Additional `plot` parameters.

Value

A plot of the arcs of the CS-PCD whose vertices are the points in data set Xp and the triangle tri

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-1.5  #try also t<-2

plotCSarcs.tri(Xp,Tr,t,M,main="Arcs of CS-PCD with t=1.5",xlab="",ylab="",edge.reg = TRUE)

# or try the default center
#plotCSarcs.tri(Xp,Tr,t,main="Arcs of CS-PCD with t=1.5",xlab="",ylab="",edge.reg = TRUE);
#M=(arcsCStri(Xp,Tr,r)$param)$c #the part "M=(arcsPEtri(Xp,Tr,r)$param)$cent" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with edge regions)
txt<-rbind(Tr,M)
xc<-txt[,1]+c(-.02,.02,.02,.03)
yc<-txt[,2]+c(.02,.02,.02,.03)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-1.5  #try also t<-2

plotCSarcs.tri(Xp,Tr,t,M,main="Arcs of CS-PCD with t=1.5",xlab="",ylab="",edge.reg = TRUE)

# or try the default center
#plotCSarcs.tri(Xp,Tr,t,main="Arcs of CS-PCD with t=1.5",xlab="",ylab="",edge.reg = TRUE);
#M=(arcsCStri(Xp,Tr,r)$param)$c #the part "M=(arcsPEtri(Xp,Tr,r)$param)$cent" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with edge regions)
txt<-rbind(Tr,M)
xc<-txt[,1]+c(-.02,.02,.02,.03)
yc<-txt[,2]+c(.02,.02,.02,.03)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

## End(Not run)

The plot of the arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) for 1D data (vertices jittered along $y$ -coordinate) - multiple interval case

Description

Plots the arcs of CS-PCD whose vertices are the 1D points, Xp. CS proximity regions are constructed with expansion parameter $t>0$ and centrality parameter $c \in (0,1)$ and the intervals are based on Yp points (i.e. the intervalization is based on Yp points). That is, data set Xp constitutes the vertices of the digraph and Yp determines the end points of the intervals.

centers is a logical argument, if TRUE, plot includes the centers of the intervals as vertical lines in the plot, else centers of the intervals are not plotted.

See also (Ceyhan (2016)).

Usage

plotCSarcs1D(
  Xp,
  Yp,
  t,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  centers = FALSE,
  ...
)
plotCSarcs1D(
  Xp,
  Yp,
  t,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  centers = FALSE,
  ...
)

Arguments

`Xp`	A `vector` of 1D points constituting the vertices of the CS-PCD.
`Yp`	A `vector` of 1D points constituting the end points of the intervals.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default=`0.1` and `Xp` points are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of `Xp` and `Yp` multiplied by `Jit`).
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles of the $x$ and $y$ axes in the plot (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`centers`	A logical argument, if `TRUE`, plot includes the centers of the intervals as vertical lines in the plot, else centers of the intervals are not plotted.
`...`	Additional `plot` parameters.

Value

A plot of the arcs of CS-PCD whose vertices are the 1D data set Xp in which vertices are jittered along $y$ -axis for better visualization.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

t<-1.5
c<-.4
a<-0; b<-10; int<-c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xr<-range(a,b)
xf<-(xr[2]-xr[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Xlim=range(Xp,Yp)
Ylim=c(-.2,.2)

jit<-.1

plotCSarcs1D(Xp,Yp,t,c,jit,xlab="",ylab="",xlim=Xlim,ylim=Ylim)

set.seed(1)
plotCSarcs1D(Xp,Yp,t=1.5,c=.3,jit,main="t=1.5, c=.3",xlab="",ylab="",centers=TRUE)
set.seed(1)
plotCSarcs1D(Xp,Yp,t=2,c=.3,jit,main="t=2, c=.3",xlab="",ylab="",centers=TRUE)
set.seed(1)
plotCSarcs1D(Xp,Yp,t=1.5,c=.5,jit,main="t=1.5, c=.5",xlab="",ylab="",centers=TRUE)
set.seed(1)
plotCSarcs1D(Xp,Yp,t=2,c=.5,jit,main="t=2, c=.5",xlab="",ylab="",centers=TRUE)

t<-1.5
c<-.4
a<-0; b<-10; int<-c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xr<-range(a,b)
xf<-(xr[2]-xr[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Xlim=range(Xp,Yp)
Ylim=c(-.2,.2)

jit<-.1

plotCSarcs1D(Xp,Yp,t,c,jit,xlab="",ylab="",xlim=Xlim,ylim=Ylim)

set.seed(1)
plotCSarcs1D(Xp,Yp,t=1.5,c=.3,jit,main="t=1.5, c=.3",xlab="",ylab="",centers=TRUE)
set.seed(1)
plotCSarcs1D(Xp,Yp,t=2,c=.3,jit,main="t=2, c=.3",xlab="",ylab="",centers=TRUE)
set.seed(1)
plotCSarcs1D(Xp,Yp,t=1.5,c=.5,jit,main="t=1.5, c=.5",xlab="",ylab="",centers=TRUE)
set.seed(1)
plotCSarcs1D(Xp,Yp,t=2,c=.5,jit,main="t=2, c=.5",xlab="",ylab="",centers=TRUE)

The plot of the Central Similarity (CS) Proximity Regions for a 2D data set - multiple triangle case

Description

Plots the points in and outside of the Delaunay triangles based on Yp points which partition the convex hull of Yp points and also plots the CS proximity regions for Xp points and the Delaunay triangles based on Yp points.

CS proximity regions are constructed with respect to the Delaunay triangles with the expansion parameter $t>0$ .

Edge regions in each triangle is based on the center $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle (default for $M=(1,1,1)$ which is the center of mass of the triangle).

See (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)) more on the CS proximity regions. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

plotCSregs(
  Xp,
  Yp,
  t,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)
plotCSregs(
  Xp,
  Yp,
  t,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

`Xp`	A set of 2D points for which CS proximity regions are constructed.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri`.
`asp`	A `numeric` value, giving the aspect ratio $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`".
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`...`	Additional `plot` parameters.

Value

Plot of the Xp points, Delaunay triangles based on Yp and also the CS proximity regions for Xp points inside the convex hull of Yp points

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2014). “Comparison of Relative Density of Two Random Geometric Digraph Families in Testing Spatial Clustering.” TEST, 23(1), 100-134.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)
tau<-1.5  #try also tau<-2

plotCSregs(Xp,Yp,tau,M,xlab="",ylab="")

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)
tau<-1.5  #try also tau<-2

plotCSregs(Xp,Yp,tau,M,xlab="",ylab="")

## End(Not run)

The plot of the Central Similarity (CS) Proximity Regions for a general interval (vertices jittered along $y$ -coordinate) - one interval case

Description

Plots the points in and outside of the interval int and also the CS proximity regions (which are also intervals). CS proximity regions are constructed with expansion parameter $t > 0$ and centrality parameter $c \in (0,1)$ .

For better visualization, a uniform jitter from $U(-Jit,Jit)$ (default is $Jit=.1$ ) times range of proximity regions and Xp) is added to the $y$ -direction. center is a logical argument, if TRUE, plot includes the center of the interval as a vertical line in the plot, else center of the interval is not plotted.

Usage

plotCSregs.int(
  Xp,
  int,
  t,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  center = FALSE,
  ...
)
plotCSregs.int(
  Xp,
  int,
  t,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  center = FALSE,
  ...
)

Arguments

`Xp`	A set of 1D points for which CS proximity regions are to be constructed.
`int`	A `vector` of two real numbers representing an interval.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default=`0.1` and `Xp` points are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of `Xp` and proximity region intervals multiplied by `Jit`).
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges.
`center`	A logical argument, if `TRUE`, plot includes the center of the interval as a vertical line in the plot, else center of the interval is not plotted.
`...`	Additional `plot` parameters.

Value

Plot of the CS proximity regions for 1D points in or outside the interval int

Author(s)

Elvan Ceyhan

References

There are no references for Rd macro ⁠\insertAllCites⁠ on this help page.

Examples

c<-.4
tau<-2
a<-0; b<-10; int<-c(a,b)

n<-10
xf<-(int[2]-int[1])*.1

Xp<-runif(n,a-xf,b+xf)  #try also Xp<-runif(n,a-5,b+5)

plotCSregs.int(7,int,tau,c,xlab="x",ylab="")

plotCSregs.int(Xp,int,tau,c,xlab="x",ylab="")

plotCSregs.int(17,int,tau,c,xlab="x",ylab="")
plotCSregs.int(1,int,tau,c,xlab="x",ylab="")
plotCSregs.int(4,int,tau,c,xlab="x",ylab="")

plotCSregs.int(-7,int,tau,c,xlab="x",ylab="")

c<-.4
tau<-2
a<-0; b<-10; int<-c(a,b)

n<-10
xf<-(int[2]-int[1])*.1

Xp<-runif(n,a-xf,b+xf)  #try also Xp<-runif(n,a-5,b+5)

plotCSregs.int(7,int,tau,c,xlab="x",ylab="")

plotCSregs.int(Xp,int,tau,c,xlab="x",ylab="")

plotCSregs.int(17,int,tau,c,xlab="x",ylab="")
plotCSregs.int(1,int,tau,c,xlab="x",ylab="")
plotCSregs.int(4,int,tau,c,xlab="x",ylab="")

plotCSregs.int(-7,int,tau,c,xlab="x",ylab="")

The plot of the Central Similarity (CS) Proximity Regions for a 2D data set - one triangle case

Description

Plots the points in and outside of the triangle tri and also the CS proximity regions which are also triangular for points inside the triangle tri with edge regions are based on the center of mass CM.

CS proximity regions are defined with respect to the triangle tri with expansion parameter $t>0$ , so CS proximity regions are defined only for points inside the triangle tri.

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

plotCSregs.tri(
  Xp,
  tri,
  t,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  edge.reg = FALSE,
  ...
)
plotCSregs.tri(
  Xp,
  tri,
  t,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  edge.reg = FALSE,
  ...
)

Arguments

`Xp`	A set of 2D points for which CS proximity regions are constructed.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`.
`asp`	A `numeric` value, giving the aspect ratio $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`".
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`edge.reg`	A logical argument to add edge regions to the plot, default is `edge.reg=FALSE`.
`...`	Additional `plot` parameters.

Value

Plot of the CS proximity regions for points inside the triangle tri (and just the points outside tri)

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp0<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-.5  #try also t<-2

plotCSregs.tri(Xp0,Tr,t,M,main="Proximity Regions for CS-PCD", xlab="",ylab="")

Xp = Xp0[1,]
plotCSregs.tri(Xp,Tr,t,M,main="CS Proximity Regions with t=.5", xlab="",ylab="",edge.reg=TRUE)

# or try the default center
plotCSregs.tri(Xp,Tr,t,main="CS Proximity Regions with t=.5", xlab="",ylab="",edge.reg=TRUE);
#M=(arcsCStri(Xp,Tr,r)$param)$c #the part "M=(arcsPEtri(Xp,Tr,r)$param)$cent" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with edge regions)
txt<-rbind(Tr,M)
xc<-txt[,1]+c(-.02,.02,.02,.02)
yc<-txt[,2]+c(.02,.02,.02,.03)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp0<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-.5  #try also t<-2

plotCSregs.tri(Xp0,Tr,t,M,main="Proximity Regions for CS-PCD", xlab="",ylab="")

Xp = Xp0[1,]
plotCSregs.tri(Xp,Tr,t,M,main="CS Proximity Regions with t=.5", xlab="",ylab="",edge.reg=TRUE)

# or try the default center
plotCSregs.tri(Xp,Tr,t,main="CS Proximity Regions with t=.5", xlab="",ylab="",edge.reg=TRUE);
#M=(arcsCStri(Xp,Tr,r)$param)$c #the part "M=(arcsPEtri(Xp,Tr,r)$param)$cent" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with edge regions)
txt<-rbind(Tr,M)
xc<-txt[,1]+c(-.02,.02,.02,.02)
yc<-txt[,2]+c(.02,.02,.02,.03)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

## End(Not run)

The plot of the Central Similarity (CS) Proximity Regions (vertices jittered along $y$ -coordinate) - multiple interval case

Description

Plots the points in and outside of the intervals based on Yp points and also the CS proximity regions (which are also intervals).

CS proximity region is constructed with expansion parameter $t>0$ and centrality parameter $c \in (0,1)$ . For better visualization, a uniform jitter from $U(-Jit,Jit)$ (default is $Jit=.1$ ) times range of Xp and Yp and the proximity regions (intervals)) is added to the $y$ -direction.

centers is a logical argument, if TRUE, plot includes the centers of the intervals as vertical lines in the plot, else centers of the intervals are not plotted.

See also (Ceyhan (2016)).

Usage

plotCSregs1D(
  Xp,
  Yp,
  t,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  centers = FALSE,
  ...
)
plotCSregs1D(
  Xp,
  Yp,
  t,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  centers = FALSE,
  ...
)

Arguments

`Xp`	A set of 1D points for which CS proximity regions are plotted.
`Yp`	A set of 1D points which constitute the end points of the intervals which partition the real line.
`t`	A positive real number which serves as the expansion parameter in CS proximity region.
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default=`0.1` and `Xp` points are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of `Xp` and `Yp` and the proximity regions (intervals) multiplied by `Jit`).
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles of the $x$ and $y$ axes in the plot (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`centers`	A logical argument, if `TRUE`, plot includes the centers of the intervals as vertical lines in the plot, else centers of the intervals are not plotted.
`...`	Additional `plot` parameters.

Value

Plot of the CS proximity regions for 1D points located in the middle or end intervals based on Yp points

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

Examples

t<-2
c<-.4
a<-0; b<-10;

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xr<-range(a,b)
xf<-(xr[2]-xr[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

plotCSregs1D(Xp,Yp,t,c,xlab="",ylab="")

plotCSregs1D(Xp,Yp+10,t,c,xlab="",ylab="")

t<-2
c<-.4
a<-0; b<-10;

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xr<-range(a,b)
xf<-(xr[2]-xr[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

plotCSregs1D(Xp,Yp,t,c,xlab="",ylab="")

plotCSregs1D(Xp,Yp+10,t,c,xlab="",ylab="")

The scatterplot of points from one class and plot of the Delaunay triangulation of the other class

Description

Plots the scatter plot of Xp points together with the Delaunay triangles based on the Yp points. Both sets of points are of 2D.

See (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

plotDelaunay.tri(
  Xp,
  Yp,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)
plotDelaunay.tri(
  Xp,
  Yp,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

`Xp`	A set of 2D points whose scatterplot is to be plotted.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both)
`...`	Additional `plot` parameters.

Value

A scatterplot of Xp points and the Delaunay triangulation of Yp points.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab="",main="X points and Delaunay Triangulation of Y points")

## End(Not run)

## Not run: 
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab="",main="X points and Delaunay Triangulation of Y points")

## End(Not run)

The plot of the subintervals based on `Yp` points together with `Xp` points

Description

Plots the Xp points and the intervals based on Yp points.

Usage

plotIntervals(
  Xp,
  Yp,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)
plotIntervals(
  Xp,
  Yp,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

`Xp`	A set of 1D points whose scatter-plot is provided.
`Yp`	A set of 1D points which constitute the end points of the intervals which partition the real line.
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`...`	Additional `plot` parameters.

Value

Plot of the intervals based on Yp points and also scatter plot of Xp points

Author(s)

Elvan Ceyhan

Examples

## Not run: 
a<-0; b<-10;

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

plotIntervals(Xp,Yp,xlab="",ylab="")

## End(Not run)

## Not run: 
a<-0; b<-10;

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

plotIntervals(Xp,Yp,xlab="",ylab="")

## End(Not run)

The plot of the arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for a 2D data set - multiple triangle case

Description

Plots the arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) whose vertices are the data points in Xp in the multiple triangle case and the Delaunay triangles based on Yp points.

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points). Loops are not allowed so arcs are only possible for points inside the convex hull of Yp points.

Usage

plotPEarcs(
  Xp,
  Yp,
  r,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)
plotPEarcs(
  Xp,
  Yp,
  r,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as `M="CC"`), default for $M=(1,1,1)$ which is the center of mass of each triangle.
`asp`	A `numeric` value, giving the aspect ratio $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`".
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`...`	Additional `plot` parameters.

Value

A plot of the arcs of the PE-PCD whose vertices are the points in data set Xp and the Delaunay triangles based on Yp points

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

r<-1.5  #try also r<-2

plotPEarcs(Xp,Yp,r,M,xlab="",ylab="")

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

r<-1.5  #try also r<-2

plotPEarcs(Xp,Yp,r,M,xlab="",ylab="")

## End(Not run)

The plot of the arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) for 1D data (vertices jittered along $y$ -coordinate) - one interval case

Description

Plots the arcs of PE-PCD whose vertices are the 1D points, Xp. PE proximity regions are constructed with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ and the intervals are based on the interval int $=(a,b)$ That is, data set Xp constitutes the vertices of the digraph and int determines the end points of the interval.

See also (Ceyhan (2012)).

Usage

plotPEarcs.int(
  Xp,
  int,
  r,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  center = FALSE,
  ...
)
plotPEarcs.int(
  Xp,
  int,
  r,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  center = FALSE,
  ...
)

Arguments

`Xp`	A `vector` of 1D points constituting the vertices of the PE-PCD.
`int`	A `vector` of two 1D points constituting the end points of the interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center of the interval with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default=`0.1` and `Xp` points are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of range of $\{$ `Xp`, `int` $\}$ multiplied by `Jit`).
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles of the $x$ and $y$ axes in the plot (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`center`	A logical argument, if `TRUE`, plot includes the center of the interval `int` as a vertical line in the plot, else center of the interval is not plotted.
`...`	Additional `plot` parameters.

Value

A plot of the arcs of PE-PCD whose vertices are the 1D data set Xp in which vertices are jittered along $y$ -axis for better visualization.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b)

#n is number of X points
n<-10;  #try also n<-20;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(n,a-xf,b+xf)

Xlim=range(Xp,int)
Ylim=.1*c(-1,1)

jit<-.1
set.seed(1)
plotPEarcs.int(Xp,int,r=1.5,c=.3,jit,xlab="",ylab="",center=TRUE)
set.seed(1)
plotPEarcs.int(Xp,int,r=2,c=.3,jit,xlab="",ylab="",center=TRUE)

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b)

#n is number of X points
n<-10;  #try also n<-20;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(n,a-xf,b+xf)

Xlim=range(Xp,int)
Ylim=.1*c(-1,1)

jit<-.1
set.seed(1)
plotPEarcs.int(Xp,int,r=1.5,c=.3,jit,xlab="",ylab="",center=TRUE)
set.seed(1)
plotPEarcs.int(Xp,int,r=2,c=.3,jit,xlab="",ylab="",center=TRUE)

## End(Not run)

The plot of the arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for a 2D data set - one triangle case

Description

Plots the arcs of PE-PCD whose vertices are the data points, Xp and the triangle tri. PE proximity regions are constructed with respect to the triangle tri with expansion parameter $r \ge 1$ , i.e., arcs may exist only for Xp points inside the triangle tri.

See also (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)).

Usage

plotPEarcs.tri(
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  vert.reg = FALSE,
  ...
)
plotPEarcs.tri(
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  vert.reg = FALSE,
  ...
)

Arguments

`Xp`	A set of 2D points which constitute the vertices of the PE-PCD.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.
`asp`	A `numeric` value, giving the aspect ratio $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`".
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`vert.reg`	A logical argument to add vertex regions to the plot, default is `vert.reg=FALSE`.
`...`	Additional `plot` parameters.

Value

A plot of the arcs of the PE-PCD whose vertices are the points in data set Xp and the triangle tri

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)
#try also M<-c(1.6,1.0) or M<-circumcenter.tri(Tr)
r<-1.5  #try also r<-2
plotPEarcs.tri(Xp,Tr,r,M,main="Arcs of PE-PCD with r = 1.5",
xlab="",ylab="",vert.reg = TRUE)

# or try the default center
#plotPEarcs.tri(Xp,Tr,r,main="Arcs of PE-PCD with r = 1.5",
#xlab="",ylab="",vert.reg = TRUE);
#M=(arcsPEtri(Xp,Tr,r)$param)$cent
#the part "M=(arcsPEtri(Xp,Tr,r)$param)$cent" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with vertex regions)
ifelse(isTRUE(all.equal(M,circumcenter.tri(Tr))),
{Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2); cent.name="CC"},
{Ds<-prj.cent2edges(Tr,M); cent.name="M"})

txt<-rbind(Tr,M,Ds)
xc<-txt[,1]+c(-.02,.02,.02,.02,.04,-0.03,-.01)
yc<-txt[,2]+c(.02,.02,.02,.07,.02,.04,-.06)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)
#try also M<-c(1.6,1.0) or M<-circumcenter.tri(Tr)
r<-1.5  #try also r<-2
plotPEarcs.tri(Xp,Tr,r,M,main="Arcs of PE-PCD with r = 1.5",
xlab="",ylab="",vert.reg = TRUE)

# or try the default center
#plotPEarcs.tri(Xp,Tr,r,main="Arcs of PE-PCD with r = 1.5",
#xlab="",ylab="",vert.reg = TRUE);
#M=(arcsPEtri(Xp,Tr,r)$param)$cent
#the part "M=(arcsPEtri(Xp,Tr,r)$param)$cent" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with vertex regions)
ifelse(isTRUE(all.equal(M,circumcenter.tri(Tr))),
{Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2); cent.name="CC"},
{Ds<-prj.cent2edges(Tr,M); cent.name="M"})

txt<-rbind(Tr,M,Ds)
xc<-txt[,1]+c(-.02,.02,.02,.02,.04,-0.03,-.01)
yc<-txt[,2]+c(.02,.02,.02,.07,.02,.04,-.06)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The plot of the arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) for 1D data (vertices jittered along $y$ -coordinate) - multiple interval case

Description

Plots the arcs of PE-PCD whose vertices are the 1D points, Xp. PE proximity regions are constructed with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ and the intervals are based on Yp points (i.e. the intervalization is based on Yp points). That is, data set Xp constitutes the vertices of the digraph and Yp determines the end points of the intervals.

For better visualization, a uniform jitter from $U(-Jit,Jit)$ (default for $Jit=.1$ ) is added to the $y$ -direction where Jit equals to the range of Xp and Yp multiplied by Jit with default for $Jit=.1$ ). centers is a logical argument, if TRUE, plot includes the centers of the intervals as vertical lines in the plot, else centers of the intervals are not plotted.

See also (Ceyhan (2012)).

Usage

plotPEarcs1D(
  Xp,
  Yp,
  r,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  centers = FALSE,
  ...
)
plotPEarcs1D(
  Xp,
  Yp,
  r,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  centers = FALSE,
  ...
)

Arguments

`Xp`	A `vector` of 1D points constituting the vertices of the PE-PCD.
`Yp`	A `vector` of 1D points constituting the end points of the intervals.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default=`0.1` and `Xp` points are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of the union of `Xp` and `Yp` points multiplied by `Jit`).
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles of the $x$ and $y$ axes in the plot (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`centers`	A logical argument, if `TRUE`, plot includes the centers of the intervals as vertical lines in the plot, else centers of the intervals are not plotted.
`...`	Additional `plot` parameters.

Value

A plot of the arcs of PE-PCD whose vertices are the 1D data set Xp in which vertices are jittered along $y$ -axis for better visualization.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Xlim=range(Xp,Yp)
Ylim=.1*c(-1,1)

jit<-.1

set.seed(1)
plotPEarcs1D(Xp,Yp,r=1.5,c=.3,jit,xlab="",ylab="",centers=TRUE)
set.seed(1)
plotPEarcs1D(Xp,Yp,r=2,c=.3,jit,xlab="",ylab="",centers=TRUE)

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

Xlim=range(Xp,Yp)
Ylim=.1*c(-1,1)

jit<-.1

set.seed(1)
plotPEarcs1D(Xp,Yp,r=1.5,c=.3,jit,xlab="",ylab="",centers=TRUE)
set.seed(1)
plotPEarcs1D(Xp,Yp,r=2,c=.3,jit,xlab="",ylab="",centers=TRUE)

## End(Not run)

The plot of the Proportional Edge (PE) Proximity Regions for a 2D data set - multiple triangle case

Description

Plots the points in and outside of the Delaunay triangles based on Yp points which partition the convex hull of Yp points and also plots the PE proximity regions for Xp points and the Delaunay triangles based on Yp points.

PE proximity regions are constructed with respect to the Delaunay triangles with the expansion parameter $r \ge 1$ .

Vertex regions in each triangle is based on the center $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle or based on circumcenter of each Delaunay triangle (default for $M=(1,1,1)$ which is the center of mass of the triangle).

See (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)) for more on the PE proximity regions. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

plotPEregs(
  Xp,
  Yp,
  r,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)
plotPEregs(
  Xp,
  Yp,
  r,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

`Xp`	A set of 2D points for which PE proximity regions are constructed.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as `M="CC"`), default for $M=(1,1,1)$ which is the center of mass of each triangle.
`asp`	A `numeric` value, giving the aspect ratio $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`".
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both)
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`...`	Additional `plot` parameters.

Value

Plot of the Xp points, Delaunay triangles based on Yp points and also the PE proximity regions for Xp points inside the convex hull of Yp points

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)
r<-1.5  #try also r<-2

plotPEregs(Xp,Yp,r,M,xlab="",ylab="")

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)
r<-1.5  #try also r<-2

plotPEregs(Xp,Yp,r,M,xlab="",ylab="")

## End(Not run)

The plot of the Proportional Edge (PE) Proximity Regions for a general interval (vertices jittered along $y$ -coordinate) - one interval case

Description

Plots the points in and outside of the interval int and also the PE proximity regions (which are also intervals). PE proximity regions are constructed with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ .

See also (Ceyhan (2012)).

Usage

plotPEregs.int(
  Xp,
  int,
  r,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  center = FALSE,
  ...
)
plotPEregs.int(
  Xp,
  int,
  r,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  center = FALSE,
  ...
)

Arguments

`Xp`	A set of 1D points for which PE proximity regions are to be constructed.
`int`	A `vector` of two real numbers representing an interval.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default=`0.1` and `Xp` points are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of the union of `Xp` and `Yp` points multiplied by `Jit`).
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges.
`center`	A logical argument, if `TRUE`, plot includes the center of the interval as a vertical line in the plot, else center of the interval is not plotted.
`...`	Additional `plot` parameters.

Value

Plot of the PE proximity regions for 1D points in or outside the interval int

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

n<-10
xf<-(int[2]-int[1])*.1
Xp<-runif(n,a-xf,b+xf)  #try also Xp<-runif(n,a-5,b+5)
plotPEregs.int(Xp,int,r,c,xlab="x",ylab="")

plotPEregs.int(7,int,r,c,xlab="x",ylab="")

## End(Not run)

## Not run: 
c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

n<-10
xf<-(int[2]-int[1])*.1
Xp<-runif(n,a-xf,b+xf)  #try also Xp<-runif(n,a-5,b+5)
plotPEregs.int(Xp,int,r,c,xlab="x",ylab="")

plotPEregs.int(7,int,r,c,xlab="x",ylab="")

## End(Not run)

The plot of the Proportional Edge (PE) Proximity Regions for a 3D data set - standard regular tetrahedron case

Description

Plots the points in and outside of the standard regular tetrahedron $T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))$ and also the PE proximity regions for points in data set Xp.

PE proximity regions are defined with respect to the standard regular tetrahedron $T_h$ with expansion parameter $r \ge 1$ , so PE proximity regions are defined only for points inside $T_h$ .

Vertex regions are based on circumcenter (which is equivalent to the center of mass for the standard regular tetrahedron) of $T_h$ .

See also (Ceyhan (2005, 2010)).

Usage

plotPEregs.std.tetra(
  Xp,
  r,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  zlab = NULL,
  xlim = NULL,
  ylim = NULL,
  zlim = NULL,
  ...
)
plotPEregs.std.tetra(
  Xp,
  r,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  zlab = NULL,
  xlim = NULL,
  ylim = NULL,
  zlim = NULL,
  ...
)

Arguments

`Xp`	A set of 3D points for which PE proximity regions are constructed.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`, `zlab`	titles for the $x$ , $y$ , and $z$ axes, respectively (default=`NULL` for all).
`xlim`, `ylim`, `zlim`	Two `numeric` vectors of length 2, giving the $x$ -, $y$ -, and $z$ -coordinate ranges (default=`NULL` for all).
`...`	Additional `scatter3D` parameters.

Value

Plot of the PE proximity regions for points inside the standard regular tetrahedron $T_h$ (and just the points outside $T_h$ )

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
r<-1.5

n<-3  #try also n<-20
Xp<-runif.std.tetra(n)$g  #try also Xp[,1]<-Xp[,1]+1

plotPEregs.std.tetra(Xp[1:3,],r)

P1<-c(.1,.1,.1)
plotPEregs.std.tetra(rbind(P1,P1),r)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
r<-1.5

n<-3  #try also n<-20
Xp<-runif.std.tetra(n)$g  #try also Xp[,1]<-Xp[,1]+1

plotPEregs.std.tetra(Xp[1:3,],r)

P1<-c(.1,.1,.1)
plotPEregs.std.tetra(rbind(P1,P1),r)

## End(Not run)

The plot of the Proportional Edge (PE) Proximity Regions for a 3D data set - one tetrahedron case

Description

Plots the points in and outside of the tetrahedron th and also the PE proximity regions (which are also tetrahedrons) for points inside the tetrahedron th.

PE proximity regions are constructed with respect to tetrahedron th with expansion parameter $r \ge 1$ and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM", so PE proximity regions are defined only for points inside the tetrahedron th.

See also (Ceyhan (2005, 2010)).

Usage

plotPEregs.tetra(
  Xp,
  th,
  r,
  M = "CM",
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  zlab = NULL,
  xlim = NULL,
  ylim = NULL,
  zlim = NULL,
  ...
)
plotPEregs.tetra(
  Xp,
  th,
  r,
  M = "CM",
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  zlab = NULL,
  xlim = NULL,
  ylim = NULL,
  zlim = NULL,
  ...
)

Arguments

`Xp`	A set of 3D points for which PE proximity regions are constructed.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`.
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`, `zlab`	Titles for the $x$ , $y$ , and $z$ axes, respectively (default=`NULL` for all).
`xlim`, `ylim`, `zlim`	Two `numeric` vectors of length 2, giving the $x$ -, $y$ -, and $z$ -coordinate ranges (default=`NULL` for all).
`...`	Additional `scatter3D` parameters.

Value

Plot of the PE proximity regions for points inside the tetrahedron th (and just the points outside th)

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
set.seed(1)
tetra<-rbind(A,B,C,D)+matrix(runif(12,-.25,.25),ncol=3) #adding jitter to make it non-regular

n<-5  #try also n<-20
Xp<-runif.tetra(n,tetra)$g  #try also Xp[,1]<-Xp[,1]+1

M<-"CM"  #try also M<-"CC"
r<-1.5

plotPEregs.tetra(Xp,tetra,r) #uses the default M="CM"
plotPEregs.tetra(Xp,tetra,r,M="CC")

plotPEregs.tetra(Xp[1,],tetra,r)  #uses the default M="CM"
plotPEregs.tetra(Xp[1,],tetra,r,M)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
set.seed(1)
tetra<-rbind(A,B,C,D)+matrix(runif(12,-.25,.25),ncol=3) #adding jitter to make it non-regular

n<-5  #try also n<-20
Xp<-runif.tetra(n,tetra)$g  #try also Xp[,1]<-Xp[,1]+1

M<-"CM"  #try also M<-"CC"
r<-1.5

plotPEregs.tetra(Xp,tetra,r) #uses the default M="CM"
plotPEregs.tetra(Xp,tetra,r,M="CC")

plotPEregs.tetra(Xp[1,],tetra,r)  #uses the default M="CM"
plotPEregs.tetra(Xp[1,],tetra,r,M)

## End(Not run)

The plot of the Proportional Edge (PE) Proximity Regions for a 2D data set - one triangle case

Description

Plots the points in and outside of the triangle tri and also the PE proximity regions for points in data set Xp.

PE proximity regions are defined with respect to the triangle tri with expansion parameter $r \ge 1$ , so PE proximity regions are defined only for points inside the triangle tri.

See also (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)).

Usage

plotPEregs.tri(
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  vert.reg = FALSE,
  ...
)
plotPEregs.tri(
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  asp = NA,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  vert.reg = FALSE,
  ...
)

Arguments

`Xp`	A set of 2D points for which PE proximity regions are constructed.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$ , i.e., the center of mass of `tri`.
`asp`	A `numeric` value, giving the aspect ratio $y/x$ (default is `NA`), see the official help page for `asp` by typing "`? asp`".
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`vert.reg`	A logical argument to add vertex regions to the plot, default is `vert.reg=FALSE`.
`...`	Additional `plot` parameters.

Value

Plot of the PE proximity regions for points inside the triangle tri (and just the points outside tri)

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp0<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)
#try also M<-c(1.6,1.0) or M = circumcenter.tri(Tr)
r<-1.5  #try also r<-2

plotPEregs.tri(Xp0,Tr,r,M)
Xp = Xp0[1,]
plotPEregs.tri(Xp,Tr,r,M)

plotPEregs.tri(Xp,Tr,r,M,
main="PE Proximity Regions with r = 1.5",
xlab="",ylab="",vert.reg = TRUE)

# or try the default center
#plotPEregs.tri(Xp,Tr,r,main="PE Proximity Regions with r = 1.5",xlab="",ylab="",vert.reg = TRUE);
#M=(arcsPEtri(Xp,Tr,r)$param)$c
#the part "M=(arcsPEtri(Xp,Tr,r)$param)$cent" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with vertex regions)
ifelse(isTRUE(all.equal(M,circumcenter.tri(Tr))),
       {Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2); cent.name="CC"},
       {Ds<-prj.cent2edges(Tr,M); cent.name<-"M"})

txt<-rbind(Tr,M,Ds)
xc<-txt[,1]+c(-.02,.02,.02,.02,.03,-0.03,-.01)
yc<-txt[,2]+c(.02,.02,.02,.07,.02,.05,-.06)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp0<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)
#try also M<-c(1.6,1.0) or M = circumcenter.tri(Tr)
r<-1.5  #try also r<-2

plotPEregs.tri(Xp0,Tr,r,M)
Xp = Xp0[1,]
plotPEregs.tri(Xp,Tr,r,M)

plotPEregs.tri(Xp,Tr,r,M,
main="PE Proximity Regions with r = 1.5",
xlab="",ylab="",vert.reg = TRUE)

# or try the default center
#plotPEregs.tri(Xp,Tr,r,main="PE Proximity Regions with r = 1.5",xlab="",ylab="",vert.reg = TRUE);
#M=(arcsPEtri(Xp,Tr,r)$param)$c
#the part "M=(arcsPEtri(Xp,Tr,r)$param)$cent" is optional,
#for the below annotation of the plot

#can add vertex labels and text to the figure (with vertex regions)
ifelse(isTRUE(all.equal(M,circumcenter.tri(Tr))),
       {Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2); cent.name="CC"},
       {Ds<-prj.cent2edges(Tr,M); cent.name<-"M"})

txt<-rbind(Tr,M,Ds)
xc<-txt[,1]+c(-.02,.02,.02,.02,.03,-0.03,-.01)
yc<-txt[,2]+c(.02,.02,.02,.07,.02,.05,-.06)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The plot of the Proportional Edge (PE) Proximity Regions (vertices jittered along $y$ -coordinate) - multiple interval case

Description

Plots the points in and outside of the intervals based on Yp points and also the PE proximity regions (i.e., intervals). PE proximity region is constructed with expansion parameter $r \ge 1$ and centrality parameter $c \in (0,1)$ .

For better visualization, a uniform jitter from $U(-Jit,Jit)$ (default is $Jit=.1$ ) times range of Xp and Yp and the proximity regions (intervals)) is added to the $y$ -direction.

centers is a logical argument, if TRUE, plot includes the centers of the intervals as vertical lines in the plot, else centers of the intervals are not plotted.

See also (Ceyhan (2012)).

Usage

plotPEregs1D(
  Xp,
  Yp,
  r,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  centers = FALSE,
  ...
)
plotPEregs1D(
  Xp,
  Yp,
  r,
  c = 0.5,
  Jit = 0.1,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  centers = FALSE,
  ...
)

Arguments

`Xp`	A set of 1D points for which PE proximity regions are plotted.
`Yp`	A set of 1D points which constitute the end points of the intervals which partition the real line.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$ .
`c`	A positive real number in $(0,1)$ parameterizing the center inside middle intervals with the default `c=.5`. For the interval, $(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .
`Jit`	A positive real number that determines the amount of jitter along the $y$ -axis, default=`0.1` and `Xp` points are jittered according to $U(-Jit,Jit)$ distribution along the $y$ -axis where `Jit` equals to the range of the union of `Xp` and `Yp` points multiplied by `Jit`).
`main`	An overall title for the plot (default=`NULL`).
`xlab`, `ylab`	Titles for the $x$ and $y$ axes, respectively (default=`NULL` for both).
`xlim`, `ylim`	Two `numeric` vectors of length 2, giving the $x$ - and $y$ -coordinate ranges (default=`NULL` for both).
`centers`	A logical argument, if `TRUE`, plot includes the centers of the intervals as vertical lines in the plot, else centers of the intervals are not plotted (default is `FALSE`).
`...`	Additional `plot` parameters.

Value

Plot of the PE proximity regions for 1D points located in the middle or end intervals based on Yp points

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Examples

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

plotPEregs1D(Xp,Yp,r,c,xlab="x",ylab="")

## End(Not run)

## Not run: 
r<-2
c<-.4
a<-0; b<-10; int<-c(a,b);

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

plotPEregs1D(Xp,Yp,r,c,xlab="x",ylab="")

## End(Not run)

Print a `Extrema` `object`

Description

Prints the call of the object of class "Extrema" and also the type (i.e. a brief description) of the extrema).

Usage

## S3 method for class 'Extrema'
print(x, ...)
## S3 method for class 'Extrema'
print(x, ...)

Arguments

`x`	A `Extrema` `object`.
`...`	Additional arguments for the S3 method `'print'`.

Value

The call of the object of class "Extrema" and also the type (i.e. a brief description) of the extrema).

Examples

## Not run: 
n<-10
Xp<-runif.std.tri(n)$gen.points
Ext<-cl2edges.std.tri(Xp)
Ext
print(Ext)

typeof(Ext))
attributes(Ext)

## End(Not run)

## Not run: 
n<-10
Xp<-runif.std.tri(n)$gen.points
Ext<-cl2edges.std.tri(Xp)
Ext
print(Ext)

typeof(Ext))
attributes(Ext)

## End(Not run)

Print a `Lines` `object`

Description

Prints the call of the object of class "Lines" and also the coefficients of the line (in the form: y = slope * x + intercept).

Usage

## S3 method for class 'Lines'
print(x, ...)
## S3 method for class 'Lines'
print(x, ...)

Arguments

`x`	A `Lines` `object`.
`...`	Additional arguments for the S3 method `'print'`.

Value

The call of the object of class "Lines" and the coefficients of the line (in the form: y = slope * x + intercept).

Examples

A<-c(-1.22,-2.33); B<-c(2.55,3.75)
xr<-range(A,B);
xf<-(xr[2]-xr[1])*.1 #how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=3) #try also l=10, 20 or 100

lnAB<-Line(A,B,x)
lnAB
print(lnAB)

typeof(lnAB)
attributes(lnAB)

A<-c(-1.22,-2.33); B<-c(2.55,3.75)
xr<-range(A,B);
xf<-(xr[2]-xr[1])*.1 #how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=3) #try also l=10, 20 or 100

lnAB<-Line(A,B,x)
lnAB
print(lnAB)

typeof(lnAB)
attributes(lnAB)

Print a `Lines3D` `object`

Description

Prints the call of the object of class "Lines3D", the coefficients of the line (in the form: x=x0 + A*t, y=y0 + B*t, and z=z0 + C*t), and the initial point together with the direction vector.

Usage

## S3 method for class 'Lines3D'
print(x, ...)
## S3 method for class 'Lines3D'
print(x, ...)

Arguments

`x`	A `Lines3D` `object`.
`...`	Additional arguments for the S3 method `'print'`.

Value

The call of the object of class "Lines3D", the coefficients of the line (in the form: x=x0 + A*t, y=y0 + B*t, and z=z0 + C*t), and the initial point together with the direction vector.

Examples

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3);
vecs<-rbind(P,Q)
Line3D(P,Q,.1)
Line3D(P,Q,.1,dir.vec=FALSE)

tr<-range(vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=3) #try also l=10, 20 or 100

lnPQ3D<-Line3D(P,Q,tsq)
lnPQ3D
print(lnPQ3D)

typeof(lnPQ3D)
attributes(lnPQ3D)

## End(Not run)

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3);
vecs<-rbind(P,Q)
Line3D(P,Q,.1)
Line3D(P,Q,.1,dir.vec=FALSE)

tr<-range(vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=3) #try also l=10, 20 or 100

lnPQ3D<-Line3D(P,Q,tsq)
lnPQ3D
print(lnPQ3D)

typeof(lnPQ3D)
attributes(lnPQ3D)

## End(Not run)

Print a `NumArcs` `object`

Description

Prints the call of the object of class "NumArcs" and also the desc (i.e. a brief description) of the output.

Usage

## S3 method for class 'NumArcs'
print(x, ...)
## S3 method for class 'NumArcs'
print(x, ...)

Arguments

`x`	A `NumArcs` `object`.
`...`	Additional arguments for the S3 method `'print'`.

Value

The call of the object of class "NumArcs" and also the desc (i.e. a brief description) of the output: number of arcs in the proximity catch digraph (PCD) and related quantities in the induced subdigraphs for points in the Delaunay cells.

Examples

## Not run: 
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-"CC"  #try also M<-c(1,1,1)

Narcs<-num.arcsAS(Xp,Yp,M)
Narcs
print(Narcs)

typeof(Narcs)
attributes(Narcs)

## End(Not run)

## Not run: 
nx<-15; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-"CC"  #try also M<-c(1,1,1)

Narcs<-num.arcsAS(Xp,Yp,M)
Narcs
print(Narcs)

typeof(Narcs)
attributes(Narcs)

## End(Not run)

Print a `Patterns` `object`

Description

Prints the call of the object of class "Patterns" and also the type (or description) of the pattern).

Usage

## S3 method for class 'Patterns'
print(x, ...)
## S3 method for class 'Patterns'
print(x, ...)

Arguments

`x`	A `Patterns` `object`.
`...`	Additional arguments for the S3 method `'print'`.

Value

The call of the object of class "Patterns" and also the type (or description) of the pattern).

Examples

## Not run: 
nx<-10; #try also 20, 100, and 1000
ny<-5; #try also 1
e<-.15;
Y<-cbind(runif(ny),runif(ny))
#with default bounding box (i.e., unit square)

Xdt<-rseg.circular(nx,Y,e)
Xdt
print(Xdt)

typeof(Xdt))
attributes(Xdt)

## End(Not run)

## Not run: 
nx<-10; #try also 20, 100, and 1000
ny<-5; #try also 1
e<-.15;
Y<-cbind(runif(ny),runif(ny))
#with default bounding box (i.e., unit square)

Xdt<-rseg.circular(nx,Y,e)
Xdt
print(Xdt)

typeof(Xdt))
attributes(Xdt)

## End(Not run)

Print a `PCDs` `object`

Description

Prints the call of the object of class "PCDs" and also the type (i.e. a brief description) of the proximity catch digraph (PCD.

Usage

## S3 method for class 'PCDs'
print(x, ...)
## S3 method for class 'PCDs'
print(x, ...)

Arguments

`x`	A `PCDs` `object`.
`...`	Additional arguments for the S3 method `'print'`.

Value

The call of the object of class "PCDs" and also the type (i.e. a brief description) of the proximity catch digraph (PCD.

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
print(Arcs)

typeof(Arcs))
attributes(Arcs)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
print(Arcs)

typeof(Arcs))
attributes(Arcs)

## End(Not run)

Print a `Planes` `object`

Description

Prints the call of the object of class "Planes" and also the coefficients of the plane (in the form: z = A*x + B*y + C).

Usage

## S3 method for class 'Planes'
print(x, ...)
## S3 method for class 'Planes'
print(x, ...)

Arguments

`x`	A `Planes` `object`.
`...`	Additional arguments for the S3 method `'print'`.

Value

The call of the object of class "Planes" and the coefficients of the plane (in the form: z = A*x + B*y + C).

Examples

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3); C<-c(3,9,12)
pts<-rbind(P,Q,C)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5) #try also l=10, 20 or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5) #try also l=10, 20 or 100

plPQC<-Plane(P,Q,C,x,y)
plPQC
print(plPQC)

typeof(plPQC)
attributes(plPQC)

## End(Not run)

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3); C<-c(3,9,12)
pts<-rbind(P,Q,C)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5) #try also l=10, 20 or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5) #try also l=10, 20 or 100

plPQC<-Plane(P,Q,C,x,y)
plPQC
print(plPQC)

typeof(plPQC)
attributes(plPQC)

## End(Not run)

Print a summary of a `Extrema` `object`

Description

Prints some information about the object.

Usage

## S3 method for class 'summary.Extrema'
print(x, ...)
## S3 method for class 'summary.Extrema'
print(x, ...)

Arguments

`x`	An `object` of class `"summary.Extrema"`, generated by `summary.Extrema`.
`...`	Additional parameters for `print`.

Value

None

Print a summary of a `Lines` `object`

Description

Prints some information about the object.

Usage

## S3 method for class 'summary.Lines'
print(x, ...)
## S3 method for class 'summary.Lines'
print(x, ...)

Arguments

`x`	An `object` of class `"summary.Lines"`, generated by `summary.Lines`.
`...`	Additional parameters for `print`.

Value

None

Print a summary of a `Lines3D` `object`

Description

Prints some information about the object.

Usage

## S3 method for class 'summary.Lines3D'
print(x, ...)
## S3 method for class 'summary.Lines3D'
print(x, ...)

Arguments

`x`	An `object` of class `"summary.Lines3D"`, generated by `summary.Lines3D`.
`...`	Additional parameters for `print`.

Value

None

Print a summary of a `NumArcs` `object`

Description

Prints some information about the object.

Usage

## S3 method for class 'summary.NumArcs'
print(x, ...)
## S3 method for class 'summary.NumArcs'
print(x, ...)

Arguments

`x`	An `object` of class `"summary.NumArcs"`, generated by `summary.NumArcs`.
`...`	Additional parameters for `print`.

Value

None

Print a summary of a `Patterns` `object`

Description

Prints some information about the object.

Usage

## S3 method for class 'summary.Patterns'
print(x, ...)
## S3 method for class 'summary.Patterns'
print(x, ...)

Arguments

`x`	An `object` of class `"summary.Patterns"`, generated by `summary.Patterns`.
`...`	Additional parameters for `print`.

Value

None

Print a summary of a `PCDs` `object`

Description

Prints some information about the object.

Usage

## S3 method for class 'summary.PCDs'
print(x, ...)
## S3 method for class 'summary.PCDs'
print(x, ...)

Arguments

`x`	An `object` of class `"summary.PCDs"`, generated by `summary.PCDs`.
`...`	Additional parameters for `print`.

Value

None

Print a summary of a `Planes` `object`

Description

Prints some information about the object.

Usage

## S3 method for class 'summary.Planes'
print(x, ...)
## S3 method for class 'summary.Planes'
print(x, ...)

Arguments

`x`	An `object` of class `"summary.Planes"`, generated by `summary.Planes`.
`...`	Additional parameters for `print`.

Value

None

Print a summary of a `TriLines` `object`

Description

Prints some information about the object

Usage

## S3 method for class 'summary.TriLines'
print(x, ...)
## S3 method for class 'summary.TriLines'
print(x, ...)

Arguments

`x`	An `object` of class `"summary.TriLines"`, generated by `summary.TriLines`.
`...`	Additional parameters for `print`.

Value

None

Print a summary of a `Uniform` `object`

Description

Prints some information about the object.

Usage

## S3 method for class 'summary.Uniform'
print(x, ...)
## S3 method for class 'summary.Uniform'
print(x, ...)

Arguments

`x`	An `object` of class `"summary.Uniform"`, generated by `summary.Uniform`.
`...`	Additional parameters for `print`.

Value

None

Print a `TriLines` `object`

Description

Prints the call of the object of class "TriLines" and also the coefficients of the line (in the form: y = slope * x + intercept), and the vertices of the triangle with respect to which the line is defined.

Usage

## S3 method for class 'TriLines'
print(x, ...)
## S3 method for class 'TriLines'
print(x, ...)

Arguments

`x`	A `TriLines` `object`.
`...`	Additional arguments for the S3 method `'print'`.

Value

The call of the object of class "TriLines" , the coefficients of the line (in the form: y = slope * x + intercept), and the vertices of the triangle with respect to which the line is defined.

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,l=3)

lnACM<-lineA2CMinTe(x)
lnACM
print(lnACM)

typeof(lnACM)
attributes(lnACM)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,l=3)

lnACM<-lineA2CMinTe(x)
lnACM
print(lnACM)

typeof(lnACM)
attributes(lnACM)

## End(Not run)

Print a `Uniform` `object`

Description

Prints the call of the object of class "Uniform" and also the type (i.e. a brief description) of the uniform distribution).

Usage

## S3 method for class 'Uniform'
print(x, ...)
## S3 method for class 'Uniform'
print(x, ...)

Arguments

`x`	A `Uniform` `object`.
`...`	Additional arguments for the S3 method `'print'`.

Value

The call of the object of class "Uniform" and also the type (i.e. a brief description) of the uniform distribution).

Examples

## Not run: 
n<-10 #try also 20, 100, and 1000
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)

Xdt<-runif.tri(n,Tr)
Xdt
print(Xdt)

typeof(Xdt))
attributes(Xdt)

## End(Not run)

## Not run: 
n<-10 #try also 20, 100, and 1000
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)

Xdt<-runif.tri(n,Tr)
Xdt
print(Xdt)

typeof(Xdt))
attributes(Xdt)

## End(Not run)

Projections of a point inside a triangle to its edges

Description

Returns the projections from a general center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of a triangle to the edges on the extension of the lines joining M to the vertices (see the examples for an illustration).

See also (Ceyhan (2005, 2010)).

Usage

prj.cent2edges(tri, M)
prj.cent2edges(tri, M)

Arguments

`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`.

Value

Three projection points (stacked row-wise) from a general center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of a triangle to the edges on the extension of the lines joining M to the vertices; row $i$ is the projection point into edge $i$ , for $i=1,2,3$ .

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Ds<-prj.cent2edges(Tr,M)  #try also prj.cent2edges(Tr,M=c(1,1))
Ds

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Projection of Center M on the edges of a triangle",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]
yc<-Tr[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.04,-.02)
yc<-txt[,2]+c(-.02,.04,.04,-.06)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Ds<-prj.cent2edges(Tr,M)  #try also prj.cent2edges(Tr,M=c(1,1))
Ds

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Projection of Center M on the edges of a triangle",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]
yc<-Tr[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.04,-.02)
yc<-txt[,2]+c(-.02,.04,.04,-.06)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

Projections of a point inside the standard basic triangle form to its edges

Description

Returns the projections from a general center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the standard basic triangle form $T_b=T((0,0),(1,0),(c_1,c_2))$ to the edges on the extension of the lines joining M to the vertices (see the examples for an illustration). In the standard basic triangle form $T_b$ , $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

See also (Ceyhan (2005, 2010)).

Usage

prj.cent2edges.basic.tri(c1, c2, M)
prj.cent2edges.basic.tri(c1, c2, M)

Arguments

`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle form adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle form.

Value

Three projection points (stacked row-wise) from a general center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of a standard basic triangle form to the edges on the extension of the lines joining M to the vertices; row $i$ is the projection point into edge $i$ , for $i=1,2,3$ .

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Ds<-prj.cent2edges.basic.tri(c1,c2,M)
Ds

Xlim<-range(Tb[,1])
Ylim<-range(Tb[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.1,.1),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
L<-rbind(M,M,M); R<-Tb
segments(L[,1], L[,2], R[,1], R[,2], lty = 3,col=2)

xc<-Tb[,1]+c(-.04,.05,.04)
yc<-Tb[,2]+c(.02,.02,.03)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.03,-.03,0)
yc<-txt[,2]+c(-.02,.02,.02,-.03)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Ds<-prj.cent2edges.basic.tri(c1,c2,M)
Ds

Xlim<-range(Tb[,1])
Ylim<-range(Tb[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.1,.1),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
L<-rbind(M,M,M); R<-Tb
segments(L[,1], L[,2], R[,1], R[,2], lty = 3,col=2)

xc<-Tb[,1]+c(-.04,.05,.04)
yc<-Tb[,2]+c(.02,.02,.03)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.03,-.03,0)
yc<-txt[,2]+c(-.02,.02,.02,-.03)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

Projections of Centers for non-degenerate asymptotic distribution of domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs) to its edges

Description

Returns the projections from center cent to the edges on the extension of the lines joining cent to the vertices in the triangle, tri. Here M is one of the three centers which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in tri for a given expansion parameter r in $(1,1.5]$ . The center label cent values 1,2,3 correspond to the vertices $M_1$ , $M_2$ , and $M_3$ (i.e., row numbers in the output of center.nondegPE(tri,r)); default for cent is 1. cent becomes center of mass $CM$ for $r=1.5$ .

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011)).

Usage

prj.nondegPEcent2edges(tri, r, cent = 1)
prj.nondegPEcent2edges(tri, r, cent = 1)

Arguments

`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,1.5]$ for this function.
`cent`	Index of the center (as $1,2,3$ corresponding to $M_1,\,M_2,\,M_3$ ) which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in `tri` for expansion parameter `r` in $(1,1.5]$ ; default `cent=1`.

Value

Three projection points (stacked row-wise) from one of the centers (as $1,2,3$ corresponding to $M_1,\,M_2,\,M_3$ ) which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in tri for expansion parameter r in $(1,1.5]$ .

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
r<-1.35

prj.nondegPEcent2edges(Tr,r,cent=2)

Ms<-center.nondegPE(Tr,r)
M1=Ms[1,]

Ds<-prj.nondegPEcent2edges(Tr,r,cent=1)

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Projections from a non-degeneracy center\n to the edges of the triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Ms,pch=".",col=1)
polygon(Ms,lty = 2)

xc<-Tr[,1]+c(-.02,.03,.02)
yc<-Tr[,2]+c(-.02,.04,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-Ms
xc<-txt[,1]+c(-.02,.04,-.04)
yc<-txt[,2]+c(-.02,.04,.04)
txt.str<-c("M1","M2","M3")
text(xc,yc,txt.str)

points(Ds,pch=4,col=2)
L<-rbind(M1,M1,M1); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2,lwd=2,col=4)
txt<-Ds
xc<-txt[,1]+c(-.02,.04,-.04)
yc<-txt[,2]+c(-.02,.04,.04)
txt.str<-c("D1","D2","D3")
text(xc,yc,txt.str)

prj.nondegPEcent2edges(Tr,r,cent=3)
#gives an error message if center index, cent, is different from 1, 2 or 3
prj.nondegPEcent2edges(Tr,r=1.49,cent=2)
#gives an error message if r>1.5

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
r<-1.35

prj.nondegPEcent2edges(Tr,r,cent=2)

Ms<-center.nondegPE(Tr,r)
M1=Ms[1,]

Ds<-prj.nondegPEcent2edges(Tr,r,cent=1)

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Projections from a non-degeneracy center\n to the edges of the triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Ms,pch=".",col=1)
polygon(Ms,lty = 2)

xc<-Tr[,1]+c(-.02,.03,.02)
yc<-Tr[,2]+c(-.02,.04,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-Ms
xc<-txt[,1]+c(-.02,.04,-.04)
yc<-txt[,2]+c(-.02,.04,.04)
txt.str<-c("M1","M2","M3")
text(xc,yc,txt.str)

points(Ds,pch=4,col=2)
L<-rbind(M1,M1,M1); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2,lwd=2,col=4)
txt<-Ds
xc<-txt[,1]+c(-.02,.04,-.04)
yc<-txt[,2]+c(-.02,.04,.04)
txt.str<-c("D1","D2","D3")
text(xc,yc,txt.str)

prj.nondegPEcent2edges(Tr,r,cent=3)
#gives an error message if center index, cent, is different from 1, 2 or 3
prj.nondegPEcent2edges(Tr,r=1.49,cent=2)
#gives an error message if r>1.5

## End(Not run)

The radii of points from one class with respect to points from the other class

Description

Returns the radii of the balls centered at x points where radius of an x point equals to the minimum distance to y points (i.e., distance to the closest y point). That is, for each x point $radius= \min_{y \in Y}(d(x,y))$ . x and y points must be of the same dimension.

Usage

radii(x, y)
radii(x, y)

Arguments

`x`	A set of $d$ -dimensional points for which the radii are computed. Radius of an `x` point equals to the distance to the closest `y` point.
`y`	A set of $d$ -dimensional points representing the reference points for the balls. That is, radius of an `x` point is defined as the minimum distance to the `y` points.

Value

A list with three elements

`rad`	A `vector` whose entries are the radius values for the `x` points. Radius of an `x` point equals to the distance to the closest `y` point
`index.of.clYp`	A `vector` of indices of the closest `y` points to the `x` points. The $i$ -th entry in this `vector` is the index of the closest `y` point to $i$ -th `x` point.
`closest.Yp`	A `vector` of the closest `y` points to the `x` points. The $i$ -th entry in this `vector` or $i$ -th row in the matrix is the closest `y` point to $i$ -th `x` point.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
nx<-10
ny<-5
X<-cbind(runif(nx),runif(nx))
Y<-cbind(runif(ny),runif(ny))
Rad<-radii(X,Y)
Rad
rd<-Rad$rad

Xlim<-range(X[,1]-rd,X[,1]+rd,Y[,1])
Ylim<-range(X[,2]-rd,X[,2]+rd,Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(rbind(Y),asp=1,pch=16,col=2,xlab="",ylab="",
main="Circles Centered at Class X Points with \n Radius Equal to the Distance to Closest Y Point",
axes=TRUE, xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(rbind(X))
interp::circles(X[,1],X[,2],Rad$rad,lty=1,lwd=1,col=4)

#For 1D data
nx<-10
ny<-5
Xm<-as.matrix(X)
Ym<-as.matrix(Y)
radii(Xm,Ym)  #this works as Xm and Ym are treated as 1D data sets
#but will give error if radii(X,Y) is used
#as X and Y are treated as vectors (i.e., points)

#For 3D data
nx<-10
ny<-5
X<-cbind(runif(nx),runif(nx),runif(nx))
Y<-cbind(runif(ny),runif(ny),runif(ny))
radii(X,Y)

## End(Not run)

## Not run: 
nx<-10
ny<-5
X<-cbind(runif(nx),runif(nx))
Y<-cbind(runif(ny),runif(ny))
Rad<-radii(X,Y)
Rad
rd<-Rad$rad

Xlim<-range(X[,1]-rd,X[,1]+rd,Y[,1])
Ylim<-range(X[,2]-rd,X[,2]+rd,Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(rbind(Y),asp=1,pch=16,col=2,xlab="",ylab="",
main="Circles Centered at Class X Points with \n Radius Equal to the Distance to Closest Y Point",
axes=TRUE, xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(rbind(X))
interp::circles(X[,1],X[,2],Rad$rad,lty=1,lwd=1,col=4)

#For 1D data
nx<-10
ny<-5
Xm<-as.matrix(X)
Ym<-as.matrix(Y)
radii(Xm,Ym)  #this works as Xm and Ym are treated as 1D data sets
#but will give error if radii(X,Y) is used
#as X and Y are treated as vectors (i.e., points)

#For 3D data
nx<-10
ny<-5
X<-cbind(runif(nx),runif(nx),runif(nx))
Y<-cbind(runif(ny),runif(ny),runif(ny))
radii(X,Y)

## End(Not run)

The radius of a point from one class with respect to points from the other class

Description

Returns the radius for the ball centered at point p with radius=min distance to Y points. That is, for the point p $radius= \min_{y \in Y}d(p,y)$ (i.e., distance from p to the closest Y point). The point p and Y points must be of same dimension.

Usage

radius(p, Y)
radius(p, Y)

Arguments

`p`	A $d$ -dimensional point for which radius is computed. Radius of `p` equals to the distance to the closest `Y` point to `p`.
`Y`	A set of $d$ -dimensional points representing the reference points for the balls. That is, radius of the point `p` is defined as the minimum distance to the `Y` points.

Value

A list with three elements

`rad`	Radius value for the point, `p` defined as $\min_{y in Y} d(p,y)$
`index.of.clYpnt`	Index of the closest `Y` points to the point `p`
`closest.Ypnt`	The closest `Y` point to the point `p`

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

ny<-10
Y<-cbind(runif(ny),runif(ny))
radius(A,Y)

nx<-10
X<-cbind(runif(nx),runif(nx))
rad<-rep(0,nx)
for (i in 1:nx)
rad[i]<-radius(X[i,],Y)$rad

Xlim<-range(X[,1]-rad,X[,1]+rad,Y[,1])
Ylim<-range(X[,2]-rad,X[,2]+rad,Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(rbind(Y),asp=1,pch=16,col=2,xlab="",ylab="",
main="Circles Centered at Class X Points with \n Radius Equal to the Distance to Closest Y Point",
axes=TRUE, xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(rbind(X))
interp::circles(X[,1],X[,2],rad,lty=1,lwd=1,col=4)

#For 1D data
ny<-5
Y<-runif(ny)
Ym = as.matrix(Y)
radius(1,Ym)  #this works as Y is treated as 1D data sets
#but will give error if radius(1,Y) is used
#as Y is treated as a vector (i.e., points)

#For 3D data
ny<-5
X<-runif(3)
Y<-cbind(runif(ny),runif(ny),runif(ny))
radius(X,Y)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

ny<-10
Y<-cbind(runif(ny),runif(ny))
radius(A,Y)

nx<-10
X<-cbind(runif(nx),runif(nx))
rad<-rep(0,nx)
for (i in 1:nx)
rad[i]<-radius(X[i,],Y)$rad

Xlim<-range(X[,1]-rad,X[,1]+rad,Y[,1])
Ylim<-range(X[,2]-rad,X[,2]+rad,Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(rbind(Y),asp=1,pch=16,col=2,xlab="",ylab="",
main="Circles Centered at Class X Points with \n Radius Equal to the Distance to Closest Y Point",
axes=TRUE, xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(rbind(X))
interp::circles(X[,1],X[,2],rad,lty=1,lwd=1,col=4)

#For 1D data
ny<-5
Y<-runif(ny)
Ym = as.matrix(Y)
radius(1,Ym)  #this works as Y is treated as 1D data sets
#but will give error if radius(1,Y) is used
#as Y is treated as a vector (i.e., points)

#For 3D data
ny<-5
X<-runif(3)
Y<-cbind(runif(ny),runif(ny),runif(ny))
radius(X,Y)

## End(Not run)

Generation of points associated (in a radial or circular fashion) with a given set of points

Description

An object of class "Patterns". Generates n 2D points uniformly in $(a_1-e,a_1+e) \times (a_1-e,a_1+e) \cap U_i B(y_i,e)$ ( $a_1$ and $b1$ are denoted as a1 and b1 as arguments) where $Y_p=(y_1,y_2,\ldots,y_{n_y})$ with $n_y$ being number of Yp points for various values of e under the association pattern and $B(y_i,e)$ is the ball centered at $y_i$ with radius e.

e must be positive and very large values of e provide patterns close to CSR. a1 is defaulted to the minimum of the $x$ -coordinates of the Yp points, a2 is defaulted to the maximum of the $x$ -coordinates of the Yp points, b1 is defaulted to the minimum of the $y$ -coordinates of the Yp points, b2 is defaulted to the maximum of the $y$ -coordinates of the Yp points. This function is also very similar to rassoc.matern, where rassoc.circular needs the study window to be specified, while rassoc.matern does not.

Usage

rassoc.circular(
  n,
  Yp,
  e,
  a1 = min(Yp[, 1]),
  a2 = max(Yp[, 1]),
  b1 = min(Yp[, 2]),
  b2 = max(Yp[, 2])
)
rassoc.circular(
  n,
  Yp,
  e,
  a1 = min(Yp[, 1]),
  a2 = max(Yp[, 1]),
  b1 = min(Yp[, 2]),
  b2 = max(Yp[, 2])
)

Arguments

`n`	A positive integer representing the number of points to be generated.
`Yp`	A set of 2D points representing the reference points. The generated points are associated (in a circular or radial fashion) with these points.
`e`	A positive real number representing the radius of the balls centered at `Yp` points. Only these balls are allowed for the generated points (i.e., generated points would be in the union of these balls).
`a1`, `a2`	Real numbers representing the range of $x$ -coordinates in the region (default is the range of $x$ -coordinates of the `Yp` points).
`b1`, `b2`	Real numbers representing the range of $y$ -coordinates in the region (default is the range of $y$ -coordinates of the `Yp` points).

Value

A list with the elements

`type`	The type of the point pattern
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	Radial attraction parameter of the association pattern
`ref.points`	The input set of attraction points `Yp`, i.e., points with which generated points are associated.
`gen.points`	The output set of generated points associated with `Yp` points
`tri.Yp`	Logical output for triangulation based on `Yp` points should be implemented or not. if `TRUE` triangulation based on `Yp` points is to be implemented (default is set to `FALSE`).
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of attraction (i.e., `Yp`) points.
`xlimit`, `ylimit`	The possible range of the $x$ - and $y$ -coordinates of the generated points.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
nx<-100; ny<-4;  #try also nx<-1000; ny<-10;

e<-.15;
#with default bounding box (i.e., unit square)
Y<-cbind(runif(ny),runif(ny))

Xdt<-rassoc.circular(nx,Y,e)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xdt<-Xdt$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Circular Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),
     pch=16,col=2,lwd=2)
points(Xdt)

#with default bounding box (i.e., unit square)
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Circular Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),pch=16,
     col=2,lwd=2)
points(Xdt)

#with a rectangular bounding box
a1<-0; a2<-10;
b1<-0; b2<-5;
e<-1.1;  #try also e<-5; #pattern very close to CSR!

Y<-cbind(runif(ny,a1,a2),runif(ny,b1,b2))
#try also Y<-cbind(runif(ny,a1,a2/2),runif(ny,b1,b2/2))

Xdt<-rassoc.circular(nx,Y,e,a1,a2,b1,b2)$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Circular Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),
     pch=16,col=2,lwd=2)
points(Xdt)

## End(Not run)

## Not run: 
nx<-100; ny<-4;  #try also nx<-1000; ny<-10;

e<-.15;
#with default bounding box (i.e., unit square)
Y<-cbind(runif(ny),runif(ny))

Xdt<-rassoc.circular(nx,Y,e)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xdt<-Xdt$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Circular Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),
     pch=16,col=2,lwd=2)
points(Xdt)

#with default bounding box (i.e., unit square)
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Circular Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),pch=16,
     col=2,lwd=2)
points(Xdt)

#with a rectangular bounding box
a1<-0; a2<-10;
b1<-0; b2<-5;
e<-1.1;  #try also e<-5; #pattern very close to CSR!

Y<-cbind(runif(ny,a1,a2),runif(ny,b1,b2))
#try also Y<-cbind(runif(ny,a1,a2/2),runif(ny,b1,b2/2))

Xdt<-rassoc.circular(nx,Y,e,a1,a2,b1,b2)$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Circular Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),
     pch=16,col=2,lwd=2)
points(Xdt)

## End(Not run)

Generation of points associated (in a Matern-like fashion) to a given set of points

Description

An object of class "Patterns". Generates n 2D points uniformly in $\cup B(y_i,e)$ where $Y_p=(y_1,y_2,\ldots,y_{n_y})$ with $n_y$ being number of Yp points for various values of e under the association pattern and $B(y_i,e)$ is the ball centered at $y_i$ with radius e.

The pattern resembles the Matern cluster pattern (see rMatClust in the spatstat.random package for further information (Baddeley and Turner (2005)). rMatClust(kappa, scale, mu, win) in the simplest case generates a uniform Poisson point process of "parent" points with intensity kappa. Then each parent point is replaced by a random cluster of "offspring" points, the number of points per cluster being Poisson(mu) distributed, and their positions being placed and uniformly inside a disc of radius scale centered on the parent point. The resulting point pattern is a realization of the classical "stationary Matern cluster process" generated inside the window win.

The main difference of rassoc.matern and rMatClust is that the parent points are Yp points which are given beforehand and we do not discard them in the end in rassoc.matern and the offspring points are the points associated with the reference points, Yp; e must be positive and very large values of e provide patterns close to CSR.

This function is also very similar to rassoc.circular, where rassoc.circular needs the study window to be specified, while rassoc.matern does not.

Usage

rassoc.matern(n, Yp, e)
rassoc.matern(n, Yp, e)

Arguments

`n`	A positive integer representing the number of points to be generated.
`Yp`	A set of 2D points representing the reference points. The generated points are associated (in a Matern-cluster like fashion) with these points.
`e`	A positive real number representing the radius of the balls centered at `Yp` points. Only these balls are allowed for the generated points (i.e., generated points would be in the union of these balls).

Value

A list with the elements

`type`	The type of the point pattern
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	Radial (i.e., circular) attraction parameter of the association pattern.
`ref.points`	The input set of attraction points `Yp`, i.e., points with which generated points are associated.
`gen.points`	The output set of generated points associated with `Yp` points.
`tri.Yp`	Logical output for triangulation based on `Yp` points should be implemented or not. if `TRUE` triangulation based on `Yp` points is to be implemented (default is set to `FALSE`).
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of attraction (i.e., `Yp`) points.
`xlimit`, `ylimit`	The possible ranges of the $x$ - and $y$ -coordinates of the generated points.

Author(s)

Elvan Ceyhan

References

Baddeley AJ, Turner R (2005). “spatstat: An R Package for Analyzing Spatial Point Patterns.” Journal of Statistical Software, 12(6), 1-42.

Examples

## Not run: 
nx<-100; ny<-4;  #try also nx<-1000; ny<-10;

e<-.15;
 #try also e<-1.1; #closer to CSR than association, as e is large

#Y points uniform in unit square
Y<-cbind(runif(ny),runif(ny))

Xdt<-rassoc.matern(nx,Y,e)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xdt<-Xdt$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Matern-like Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),
     pch=16,col=2,lwd=2)
points(Xdt)

a1<-0; a2<-10;
b1<-0; b2<-5;
e<-1.1;

#Y points uniform in a rectangle
Y<-cbind(runif(ny,a1,a2),runif(ny,b1,b2))
#try also Y<-cbind(runif(ny,a1,a2/2),runif(ny,b1,b2/2))

Xdt<-rassoc.matern(nx,Y,e)$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Matern-like Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),pch=16,col=2,lwd=2)
points(Xdt)

## End(Not run)

## Not run: 
nx<-100; ny<-4;  #try also nx<-1000; ny<-10;

e<-.15;
 #try also e<-1.1; #closer to CSR than association, as e is large

#Y points uniform in unit square
Y<-cbind(runif(ny),runif(ny))

Xdt<-rassoc.matern(nx,Y,e)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xdt<-Xdt$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Matern-like Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),
     pch=16,col=2,lwd=2)
points(Xdt)

a1<-0; a2<-10;
b1<-0; b2<-5;
e<-1.1;

#Y points uniform in a rectangle
Y<-cbind(runif(ny,a1,a2),runif(ny,b1,b2))
#try also Y<-cbind(runif(ny,a1,a2/2),runif(ny,b1,b2/2))

Xdt<-rassoc.matern(nx,Y,e)$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,xlab="x",ylab="y",
main="Matern-like Association of X points with Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),pch=16,col=2,lwd=2)
points(Xdt)

## End(Not run)

Generation of points associated (in a Type I fashion) with a given set of points

Description

An object of class "Patterns". Generates n points uniformly in the support for Type I association in the convex hull of set of points, Yp. delta is the parameter of association (that is, only $\delta 100$ % area around each vertex in each Delaunay triangle is allowed for point generation).

delta corresponds to eps in the standard equilateral triangle $T_e$ as $delta=4eps^2/3$ (see rseg.std.tri function).

If Yp consists only of 3 points, then the function behaves like the function rassoc.tri.

DTmesh must be the Delaunay triangulation of Yp and DTr must be the corresponding Delaunay triangles (both DTmesh and DTr are NULL by default). If NULL, DTmesh is computed via tri.mesh and DTr is computed via triangles function in interp package.

tri.mesh function yields the triangulation nodes with their neighbours, and creates a triangulation object, and triangles function yields a triangulation data structure from the triangulation object created by tri.mesh (the first three columns are the vertex indices of the Delaunay triangles).

See (Ceyhan et al. (2006); Ceyhan et al. (2007); Ceyhan (2011)) for more on the association pattern. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

rassoc.multi.tri(n, Yp, delta, DTmesh = NULL, DTr = NULL)
rassoc.multi.tri(n, Yp, delta, DTmesh = NULL, DTr = NULL)

Arguments

`n`	A positive integer representing the number of points to be generated.
`Yp`	A set of 2D points from which Delaunay triangulation is constructed.
`delta`	A positive real number in $(0,4/9)$ . `delta` is the parameter of association (that is, only $\delta 100$ % area around each vertex in each Delaunay triangle is allowed for point generation).
`DTmesh`	Delaunay triangulation of `Yp`, default is `NULL`, which is computed via `tri.mesh` function in `interp` package. `tri.mesh` function yields the triangulation nodes with their neighbours, and creates a triangulation object.
`DTr`	Delaunay triangles based on `Yp`, default is `NULL`, which is computed via `tri.mesh` function in `interp` package. `triangles` function yields a triangulation data structure from the triangulation object created by `tri.mesh`.

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	Attraction parameter, `delta`, of the Type I association pattern. `delta` is in $(0,4/9)$ only $\delta 100$ % of the area around each vertex in each Delaunay triangle is allowed for point generation.
`ref.points`	The input set of points `Yp`; reference points, i.e., points with which generated points are associated.
`gen.points`	The output set of generated points associated with `Yp` points.
`tri.Y`	Logical output, `TRUE` if triangulation based on `Yp` points should be implemented.
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of reference (i.e., `Yp`) points.
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the reference points, which are the `Yp` points

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
del<-.4

Xdt<-rassoc.multi.tri(nx,Yp,del)
Xdt
summary(Xdt)
plot(Xdt)

#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points
TRY<-interp::triangles(DTY)[,1:3];
Xp<-rassoc.multi.tri(nx,Yp,del,DTY,TRY)$g
#data under CSR in the convex hull of Ypoints

Xlim<-range(Yp[,1])
Ylim<-range(Yp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot of the data in the convex hull of Y points together with the Delaunay triangulation
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points

plot(Xp,main="Points from Type I Association \n in Multipe Triangles",
xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05),type="n")
interp::plot.triSht(DTY, add=TRUE,
do.points=TRUE,col="blue")
points(Xp,pch=".",cex=3)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
del<-.4

Xdt<-rassoc.multi.tri(nx,Yp,del)
Xdt
summary(Xdt)
plot(Xdt)

#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points
TRY<-interp::triangles(DTY)[,1:3];
Xp<-rassoc.multi.tri(nx,Yp,del,DTY,TRY)$g
#data under CSR in the convex hull of Ypoints

Xlim<-range(Yp[,1])
Ylim<-range(Yp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot of the data in the convex hull of Y points together with the Delaunay triangulation
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points

plot(Xp,main="Points from Type I Association \n in Multipe Triangles",
xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05),type="n")
interp::plot.triSht(DTY, add=TRUE,
do.points=TRUE,col="blue")
points(Xp,pch=".",cex=3)

## End(Not run)

Generation of points associated (in a Type I fashion) with the vertices of $T_e$

Description

An object of class "Patterns". Generates n points uniformly in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ under the type I association alternative for eps in $(0,\sqrt{3}/3=0.5773503]$ . The allowed triangular regions around the vertices are determined by the parameter eps.

In the type I association, the triangular support regions around the vertices are determined by the parameter eps where $\sqrt{3}/3$ -eps serves as the height of these triangles (see examples for a sample plot.)

See also (Ceyhan et al. (2006); Ceyhan et al. (2007); Ceyhan (2011)).

Usage

rassoc.std.tri(n, eps)
rassoc.std.tri(n, eps)

Arguments

`n`	A positive integer representing the number of points to be generated.
`eps`	A positive real number representing the parameter of type I association (where $\sqrt{3}/3$ -`eps` serves as the height of the triangular support regions around the vertices).

Value

A list with the elements

`type`	The type of the point pattern
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	The attraction parameter of the association pattern, `eps`, where $\sqrt{3}/3$ -`eps` serves as the height of the triangular support regions around the vertices
`ref.points`	The input set of points `Y`; reference points, i.e., points with which generated points are associated (i.e., vertices of $T_e$ ).
`gen.points`	The output set of generated points associated with `Y` points (i.e., vertices of $T_e$ ).
`tri.Y`	Logical output for triangulation based on `Y` points should be implemented or not. if `TRUE` triangulation based on `Y` points is to be implemented (default is set to `FALSE`).
`desc.pat`	Description of the point pattern.
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of reference (i.e., `Y`) points.
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the reference points, which are the vertices of $T_e$ here

Author(s)

Elvan Ceyhan

References

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-100  #try also n<-20 or n<-100 or 1000
eps<-.25  #try also .15, .5, .75

set.seed(1)
Xdt<-rassoc.std.tri(n,eps)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-Xdt$gen.points
plot(Te,pch=".",xlab="",ylab="",
main="Type I association in the \n standard equilateral triangle",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

#The support for the Type I association alternative
sr<-(sqrt(3)/3-eps)/(sqrt(3)/2)
C1<-C+sr*(A-C); C2<-C+sr*(B-C)
A1<-A+sr*(B-A); A2<-A+sr*(C-A)
B1<-B+sr*(A-B); B2<-B+sr*(C-B)
supp<-rbind(A1,B1,B2,C2,C1,A2)

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Support of the Type I Association",
xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
if (sr<=.5)
{
  polygon(Te,col=5)
  polygon(supp,col=0)
} else
{
  polygon(Te,col=0,lwd=2.5)
  polygon(rbind(A,A1,A2),col=5,border=NA)
  polygon(rbind(B,B1,B2),col=5,border=NA)
  polygon(rbind(C,C1,C2),col=5,border=NA)
}
points(Xp)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-100  #try also n<-20 or n<-100 or 1000
eps<-.25  #try also .15, .5, .75

set.seed(1)
Xdt<-rassoc.std.tri(n,eps)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-Xdt$gen.points
plot(Te,pch=".",xlab="",ylab="",
main="Type I association in the \n standard equilateral triangle",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

#The support for the Type I association alternative
sr<-(sqrt(3)/3-eps)/(sqrt(3)/2)
C1<-C+sr*(A-C); C2<-C+sr*(B-C)
A1<-A+sr*(B-A); A2<-A+sr*(C-A)
B1<-B+sr*(A-B); B2<-B+sr*(C-B)
supp<-rbind(A1,B1,B2,C2,C1,A2)

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Support of the Type I Association",
xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
if (sr<=.5)
{
  polygon(Te,col=5)
  polygon(supp,col=0)
} else
{
  polygon(Te,col=0,lwd=2.5)
  polygon(rbind(A,A1,A2),col=5,border=NA)
  polygon(rbind(B,B1,B2),col=5,border=NA)
  polygon(rbind(C,C1,C2),col=5,border=NA)
}
points(Xp)

## End(Not run)

Generation of points associated (in a Type I fashion) with the vertices of a triangle

Description

An object of class "Patterns". Generates n points uniformly in the support for Type I association in a given triangle, tri. delta is the parameter of association (that is, only $\delta 100$ % area around each vertex in the triangle is allowed for point generation). delta corresponds to eps in the standard equilateral triangle $T_e$ as $delta=4eps^2/3$ (see rseg.std.tri function).

See (Ceyhan et al. (2006); Ceyhan et al. (2007); Ceyhan (2011)) for more on the association pattern.

Usage

rassoc.tri(n, tri, delta)
rassoc.tri(n, tri, delta)

Arguments

`n`	A positive integer representing the number of points to be generated from the association pattern in the triangle, `tri`.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`delta`	A positive real number in $(0,4/9)$ . `delta` is the parameter of association (that is, only $\delta 100$ % area around each vertex in the triangle is allowed for point generation).

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	Attraction parameter, `delta`, of the Type I association pattern. `delta` is in $(0,4/9)$ only $\delta 100$ % of the area around each vertex in the triangle `tri` is allowed for point generation.
`ref.points`	The input set of points, i.e., vertices of `tri`; reference points, i.e., points with which generated points are associated.
`gen.points`	The output set of generated points associated with the vertices of `tri`.
`tri.Y`	Logical output, `TRUE` if triangulation based on `Yp` points should be implemented.
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of reference (i.e., `Yp`) points.
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the reference points, which are the `Yp` points

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
n<-100
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)
del<-.4

Xdt<-rassoc.tri(n,Tr,del)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g
Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Points from Type I Association \n in one Triangle",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.03)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
n<-100
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)
del<-.4

Xdt<-rassoc.tri(n,Tr,del)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g
Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Points from Type I Association \n in one Triangle",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.03)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

## End(Not run)

Generation of points associated (in a Type II fashion) with the edges of $T_e$

Description

An object of class "Patterns". Generates n points uniformly in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ under the type II association alternative for eps in $(0,\sqrt{3}/6=0.2886751]$ .

In the type II association, the annular allowed regions around the edges are determined by the parameter eps where $\sqrt{3}/6$ -eps is the distance from the interior triangle (i.e., forbidden region for association) to $T_e$ (see examples for a sample plot.)

Usage

rassocII.std.tri(n, eps)
rassocII.std.tri(n, eps)

Arguments

`n`	A positive integer representing the number of points to be generated.
`eps`	A positive real number representing the parameter of type II association (where $\sqrt{3}/6$ -`eps` is the distance from the interior triangle distance from the interior triangle to $T_e$ ).

Value

A list with the elements

`type`	The type of the point pattern
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	The attraction parameter, `eps`, of the association pattern, where $\sqrt{3}/6$ -`eps` is the distance from the interior triangle to $T_e$
`ref.points`	The input set of points `Y`; reference points, i.e., points with which generated points are associated (i.e., vertices of $T_e$ ).
`gen.points`	The output set of generated points associated with `Y` points (i.e., edges of $T_e$ ).
`tri.Y`	Logical output for triangulation based on `Y` points should be implemented or not. if `TRUE` triangulation based on `Y` points is to be implemented (default is set to `FALSE`).
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of reference (i.e., `Y`) points, which is 3 here.
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the reference points, which are the vertices of $T_e$ here.

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-100  #try also n<-20 or n<-100 or 1000
eps<-.2  #try also .25, .1

set.seed(1)
Xdt<-rassocII.std.tri(n,eps)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-Xdt$gen.points
plot(Te,pch=".",xlab="",ylab="",
main="Type II association in the \n standard equilateral triangle",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

#The support for the Type II association alternative
A1<-c(1/2-eps*sqrt(3),sqrt(3)/6-eps);
B1<-c(1/2+eps*sqrt(3),sqrt(3)/6-eps);
C1<-c(1/2,sqrt(3)/6+2*eps);
supp<-rbind(A1,B1,C1)

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Support of the Type II Association",
xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te,col=5)
polygon(supp,col=0)
points(Xp)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-100  #try also n<-20 or n<-100 or 1000
eps<-.2  #try also .25, .1

set.seed(1)
Xdt<-rassocII.std.tri(n,eps)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-Xdt$gen.points
plot(Te,pch=".",xlab="",ylab="",
main="Type II association in the \n standard equilateral triangle",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

#The support for the Type II association alternative
A1<-c(1/2-eps*sqrt(3),sqrt(3)/6-eps);
B1<-c(1/2+eps*sqrt(3),sqrt(3)/6-eps);
C1<-c(1/2,sqrt(3)/6+2*eps);
supp<-rbind(A1,B1,C1)

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Support of the Type II Association",
xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te,col=5)
polygon(supp,col=0)
points(Xp)

## End(Not run)

The index of the edge region in a standard basic triangle form that contains a point

Description

Returns the index of the edge whose region contains point, p, in the standard basic triangle form $T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2))$ and edge regions based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the standard basic triangle form $T_b$ .

Edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ . If the point, p, is not inside tri, then the function yields NA as output. Edge region 1 is the triangle $T(B,C,M)$ , edge region 2 is $T(A,C,M)$ , and edge region 3 is $T(A,B,M)$ . In the standard basic triangle form $T_b$ $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

rel.edge.basic.tri(p, c1, c2, M)
rel.edge.basic.tri(p, c1, c2, M)

Arguments

`p`	A 2D point for which `M`-edge region it resides in is to be determined in the standard basic triangle form $T_b$ .
`c1`, `c2`	Positive real numbers which constitute the upper vertex of the standard basic triangle form (i.e., the vertex adjacent to the shorter edges of $T_b$ ); $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle form $T_b$ .

Value

A list with three elements

`re`	Index of the `M`-edge region that contains point, `p` in the standard basic triangle form $T_b$ .
`tri`	The vertices of the triangle, where row labels are $A$ , $B$ , and $C$ with edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ .
`desc`	Description of the edge labels

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);
M<-c(.6,.2)

P<-c(.4,.2)
rel.edge.basic.tri(P,c1,c2,M)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)

n<-20  #try also n<-40
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.basic.tri(Xp[i,],c1,c2,M)$re)
re

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tb)
L<-Tb; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Tb,M)
xc<-txt[,1]+c(-.03,.03,.02,0)
yc<-txt[,2]+c(.02,.02,.02,-.03)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);
M<-c(.6,.2)

P<-c(.4,.2)
rel.edge.basic.tri(P,c1,c2,M)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)

n<-20  #try also n<-40
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.basic.tri(Xp[i,],c1,c2,M)$re)
re

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tb)
L<-Tb; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Tb,M)
xc<-txt[,1]+c(-.03,.03,.02,0)
yc<-txt[,2]+c(.02,.02,.02,-.03)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

## End(Not run)

The index of the $CM$ -edge region in a standard basic triangle form that contains a point

Description

Returns the index of the edge whose region contains point, p, in the standard basic triangle form $T_b=T(A=(0,0),B=(1,0),C=(c_1,c_2)$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ with edge regions based on center of mass $CM=(A+B+C)/3$ .

Edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ . If the point, p, is not inside tri, then the function yields NA as output. Edge region 1 is the triangle $T(B,C,CM)$ , edge region 2 is $T(A,C,CM)$ , and edge region 3 is $T(A,B,CM)$ .

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

rel.edge.basic.triCM(p, c1, c2)
rel.edge.basic.triCM(p, c1, c2)

Arguments

`p`	A 2D point for which $CM$ -edge region it resides in is to be determined in the standard basic triangle form $T_b$ .
`c1`, `c2`	Positive real numbers which constitute the upper vertex of the standard basic triangle form (i.e., the vertex adjacent to the shorter edges of $T_b$ ); $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Value

A list with three elements

`re`	Index of the $CM$ -edge region that contains point, `p` in the standard basic triangle form $T_b$
`tri`	The vertices of the triangle, where row labels are $A=(0,0)$ , $B=(1,0)$ , and $C=(c_1,c_2)$ with edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ .
`desc`	Description of the edge labels

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6
P<-c(.4,.2)
rel.edge.basic.triCM(P,c1,c2)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)
CM<-(A+B+C)/3

rel.edge.basic.triCM(A,c1,c2)
rel.edge.basic.triCM(B,c1,c2)
rel.edge.basic.triCM(C,c1,c2)
rel.edge.basic.triCM(CM,c1,c2)

n<-20  #try also n<-40
Xp<-runif.basic.tri(n,c1,c2)$g

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.basic.triCM(Xp[i,],c1,c2)$re)
re

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tb)
L<-Tb; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Tb,CM)
xc<-txt[,1]+c(-.03,.03,.02,0)
yc<-txt[,2]+c(.02,.02,.02,-.04)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
P<-c(.4,.2)
rel.edge.basic.triCM(P,c1,c2)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)
CM<-(A+B+C)/3

rel.edge.basic.triCM(A,c1,c2)
rel.edge.basic.triCM(B,c1,c2)
rel.edge.basic.triCM(C,c1,c2)
rel.edge.basic.triCM(CM,c1,c2)

n<-20  #try also n<-40
Xp<-runif.basic.tri(n,c1,c2)$g

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.basic.triCM(Xp[i,],c1,c2)$re)
re

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tb)
L<-Tb; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Tb,CM)
xc<-txt[,1]+c(-.03,.03,.02,0)
yc<-txt[,2]+c(.02,.02,.02,-.04)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

## End(Not run)

The index of the edge region in the standard equilateral triangle that contains a point

Description

Returns the index of the edge whose region contains point, p, in the standard equilateral triangle $T_e=T(A=(0,0),B=(1,0),C=(1/2,\sqrt{3}/2))$ with edge regions based on center of mass $CM=(A+B+C)/3$ .

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

rel.edge.std.triCM(p)
rel.edge.std.triCM(p)

Arguments

`p`	A 2D point for which $CM$ -edge region it resides in is to be determined in the the standard equilateral triangle $T_e$ .

Value

A list with three elements

`re`	Index of the $CM$ -edge region that contains point, `p` in the standard equilateral triangle $T_e$
`tri`	The vertices of the standard equilateral triangle $T_e$ , where row labels are $A$ , $B$ , and $C$ with edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ .
`desc`	Description of the edge labels

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
P<-c(.4,.2)
rel.edge.std.triCM(P)

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
CM<-(A+B+C)/3

n<-20  #try also n<-40
Xp<-runif.std.tri(n)$gen.points

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.std.triCM(Xp[i,])$re)
re

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,asp=1,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
points(Xp,pch=".")
polygon(Te)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.03,.03,.03,-.06)
yc<-txt[,2]+c(.02,.02,.02,.03)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

p1<-(A+B+CM)/3
p2<-(B+C+CM)/3
p3<-(A+C+CM)/3

plot(Te,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Te,CM,p1,p2,p3)
xc<-txt[,1]+c(-.03,.03,.03,-.06,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.03,0,0,0)
txt.str<-c("A","B","C","CM","re=3","re=1","re=2")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
P<-c(.4,.2)
rel.edge.std.triCM(P)

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
CM<-(A+B+C)/3

n<-20  #try also n<-40
Xp<-runif.std.tri(n)$gen.points

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.std.triCM(Xp[i,])$re)
re

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,asp=1,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
points(Xp,pch=".")
polygon(Te)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.03,.03,.03,-.06)
yc<-txt[,2]+c(.02,.02,.02,.03)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

p1<-(A+B+CM)/3
p2<-(B+C+CM)/3
p3<-(A+C+CM)/3

plot(Te,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Te,CM,p1,p2,p3)
xc<-txt[,1]+c(-.03,.03,.03,-.06,0,0,0)
yc<-txt[,2]+c(.02,.02,.02,.03,0,0,0)
txt.str<-c("A","B","C","CM","re=3","re=1","re=2")
text(xc,yc,txt.str)

## End(Not run)

The index of the edge region in a triangle that contains the point

Description

Returns the index of the edge whose region contains point, p, in the triangle tri $=T(A,B,C)$ with edge regions based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri.

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

rel.edge.tri(p, tri, M)
rel.edge.tri(p, tri, M)

Arguments

`p`	A 2D point for which `M`-edge region it resides in is to be determined in the triangle `tri`.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`.

Value

A list with three elements

`re`	Index of the `M`-edge region that contains point, `p` in the triangle `tri`.
`tri`	The vertices of the triangle, where row labels are $A$ , $B$ , and $C$ with edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ .
`desc`	Description of the edge labels

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(1.4,1.2)
M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

rel.edge.tri(P,Tr,M)

n<-20  #try also n<-40
Xp<-runif.tri(n,Tr)$g

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.tri(Xp[i,],Tr,M)$re)
re

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}

plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".")
L<-Tr; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Tr,M)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

p1<-(A+B+M)/3
p2<-(B+C+M)/3
p3<-(A+C+M)/3

plot(Tr,xlab="",ylab="", main="Illustration of M-edge regions in a triangle",
axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-Tr; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,M,p1,p2,p3)
xc<-txt[,1]+c(-.02,.02,.02,.02,.02,.02,.02)
yc<-txt[,2]+c(.02,.02,.04,.05,.02,.02,.02)
txt.str<-c("A","B","C","M","re=3","re=1","re=2")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(1.4,1.2)
M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

rel.edge.tri(P,Tr,M)

n<-20  #try also n<-40
Xp<-runif.tri(n,Tr)$g

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.tri(Xp[i,],Tr,M)$re)
re

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}

plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".")
L<-Tr; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Tr,M)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)

p1<-(A+B+M)/3
p2<-(B+C+M)/3
p3<-(A+C+M)/3

plot(Tr,xlab="",ylab="", main="Illustration of M-edge regions in a triangle",
axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-Tr; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,M,p1,p2,p3)
xc<-txt[,1]+c(-.02,.02,.02,.02,.02,.02,.02)
yc<-txt[,2]+c(.02,.02,.04,.05,.02,.02,.02)
txt.str<-c("A","B","C","M","re=3","re=1","re=2")
text(xc,yc,txt.str)

## End(Not run)

The index of the $CM$ -edge region in a triangle that contains the point

Description

Returns the index of the edge whose region contains point, p, in the triangle tri $=T(A,B,C)$ with edge regions based on center of mass $CM=(A+B+C)/3$ .

Edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ . If the point, p, is not inside tri, then the function yields NA as output. Edge region 1 is the triangle $T(B,C,CM)$ , edge region 2 is $T(A,C,CM)$ , and edge region 3 is $T(A,B,CM)$ .

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

rel.edge.triCM(p, tri)
rel.edge.triCM(p, tri)

Arguments

`p`	A 2D point for which $CM$ -edge region it resides in is to be determined in the triangle `tri`.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with three elements

`re`	Index of the $CM$ -edge region that contains point, `p` in the triangle `tri`.
`tri`	The vertices of the triangle, where row labels are $A$ , $B$ , and $C$ with edges are labeled as 3 for edge $AB$ , 1 for edge $BC$ , and 2 for edge $AC$ .
`desc`	Description of the edge labels

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
P<-c(1.4,1.2)
rel.edge.triCM(P,Tr)

P<-c(1.5,1.61)
rel.edge.triCM(P,Tr)

CM<-(A+B+C)/3

n<-20  #try also n<-40
Xp<-runif.tri(n,Tr)$g

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.triCM(Xp[i,],Tr)$re)
re

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tr)
L<-Tr; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Tr,CM)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

p1<-(A+B+CM)/3
p2<-(B+C+CM)/3
p3<-(A+C+CM)/3

plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-Tr; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,CM,p1,p2,p3)
xc<-txt[,1]+c(-.02,.02,.02,.02,.02,.02,.02)
yc<-txt[,2]+c(.02,.02,.04,.05,.02,.02,.02)
txt.str<-c("A","B","C","CM","re=3","re=1","re=2")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
P<-c(1.4,1.2)
rel.edge.triCM(P,Tr)

P<-c(1.5,1.61)
rel.edge.triCM(P,Tr)

CM<-(A+B+C)/3

n<-20  #try also n<-40
Xp<-runif.tri(n,Tr)$g

re<-vector()
for (i in 1:n)
  re<-c(re,rel.edge.triCM(Xp[i,],Tr)$re)
re

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tr)
L<-Tr; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(re))

txt<-rbind(Tr,CM)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

p1<-(A+B+CM)/3
p2<-(B+C+CM)/3
p3<-(A+C+CM)/3

plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-Tr; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,CM,p1,p2,p3)
xc<-txt[,1]+c(-.02,.02,.02,.02,.02,.02,.02)
yc<-txt[,2]+c(.02,.02,.04,.05,.02,.02,.02)
txt.str<-c("A","B","C","CM","re=3","re=1","re=2")
text(xc,yc,txt.str)

## End(Not run)

The indices of the `M`-edge regions in a triangle that contains the points in a give data set

Description

Returns the indices of the edges whose regions contain the points in data set Xp in a triangle tri $=T(A,B,C)$ and edge regions are based on the center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri (see the plots in the example for illustrations).

The vertices of the triangle tri are labeled as $1=A$ , $2=B$ , and $3=C$ also according to the row number the vertex is recorded in tri and the corresponding edges are $1=BC$ , $2=AC$ , and $3=AB$ .

If a point in Xp is not inside tri, then the function yields NA as output for that entry. The corresponding edge region is the polygon with the vertex, M, and vertices other than the non-adjacent vertex, i.e., edge region 1 is the triangle $T(B,M,C)$ , edge region 2 is $T(A,M,C)$ and edge region 3 is $T(A,B,M)$ .

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

rel.edges.tri(Xp, tri, M)
rel.edges.tri(Xp, tri, M)

Arguments

`Xp`	A set of 2D points representing the set of data points for which indices of the edge regions containing them are to be determined.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`.

Value

A list with the elements

`re`	Indices (i.e., a `vector` of indices) of the edges whose region contains points in `Xp` in the triangle `tri`
`tri`	The vertices of the triangle, where row number corresponds to the vertex index opposite to edge whose index is given in re.
`desc`	Description of the edge labels as `"Edge labels are AB=3, BC=1, and AC=2"`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-c(1.6,1.2)

P<-c(.4,.2)
rel.edges.tri(P,Tr,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

(re<-rel.edges.tri(Xp,Tr,M))

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Scatterplot of data points \n and the M-edge regions",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-Tr; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.02,.03,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.05,.06,-.05,-.02)
yc<-txt[,2]+c(.03,.03,.05,-.08)
txt.str<-c("M","re=2","re=3","re=1")
text(xc,yc,txt.str)
text(Xp,labels=factor(re$re))

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

M<-c(1.6,1.2)

P<-c(.4,.2)
rel.edges.tri(P,Tr,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

(re<-rel.edges.tri(Xp,Tr,M))

D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Scatterplot of data points \n and the M-edge regions",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-Tr; R<-rbind(M,M,M)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.02,.03,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.05,.06,-.05,-.02)
yc<-txt[,2]+c(.03,.03,.05,-.08)
txt.str<-c("M","re=2","re=3","re=1")
text(xc,yc,txt.str)
text(Xp,labels=factor(re$re))

## End(Not run)

The indices of the $CM$ -edge regions in a triangle that contains the points in a give data set

Description

Returns the indices of the edges whose regions contain the points in data set Xp in a triangle tri $=(A,B,C)$ and edge regions are based on the center of mass $CM$ of tri. (see the plots in the example for illustrations).

If a point in Xp is not inside tri, then the function yields NA as output for that entry. The corresponding edge region is the polygon with the vertex, $CM$ , and vertices other than the non-adjacent vertex, i.e., edge region 1 is the triangle $T(B,CM,C)$ , edge region 2 is $T(A,CM,C)$ and edge region 3 is $T(A,B,CM)$ .

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

rel.edges.triCM(Xp, tri)
rel.edges.triCM(Xp, tri)

Arguments

`Xp`	A set of 2D points representing the set of data points for which indices of the edge regions containing them are to be determined.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with the elements

`re`	Indices (i.e., a `vector` of indices) of the edges whose region contains points in `Xp` in the triangle `tri`
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv`.
`desc`	Description of the edge labels as `"Edge labels are AB=3, BC=1, and AC=2"`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(.4,.2)
rel.edges.triCM(P,Tr)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

re<-rel.edges.triCM(Xp,Tr)
re
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-Tr; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.02,.03,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-rbind(CM,Ds)
xc<-txt[,1]+c(.05,.06,-.05,-.02)
yc<-txt[,2]+c(.03,.03,.05,-.08)
txt.str<-c("CM","re=2","re=3","re=1")
text(xc,yc,txt.str)
text(Xp,labels=factor(re$re))

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(.4,.2)
rel.edges.triCM(P,Tr)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

re<-rel.edges.triCM(Xp,Tr)
re
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-Tr; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.02,.03,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

txt<-rbind(CM,Ds)
xc<-txt[,1]+c(.05,.06,-.05,-.02)
yc<-txt[,2]+c(.03,.03,.05,-.08)
txt.str<-c("CM","re=2","re=3","re=1")
text(xc,yc,txt.str)
text(Xp,labels=factor(re$re))

## End(Not run)

The index of the vertex region in a standard basic triangle form that contains a given point

Description

Returns the index of the related vertex in the standard basic triangle form whose region contains point p. The standard basic triangle form is $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ ..

Vertex regions are based on the general center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the standard basic triangle form $T_b$ . Vertices of the standard basic triangle form $T_b$ are labeled according to the row number the vertex is recorded, i.e., as 1=(0,0), 2=(1,0),and $3=(c_1,c_2)$ .

If the point, p, is not inside $T_b$ , then the function yields NA as output. The corresponding vertex region is the polygon with the vertex, M, and projections from M to the edges on the lines joining vertices and M. That is, rv=1 has vertices $(0,0),D_3,M,D_2$ ; rv=2 has vertices $(1,0),D_1,M,D_3$ ; and $rv=3$ has vertices $(c_1,c_2),D_2,M,D_1$ (see the illustration in the examples).

See also (Ceyhan (2005, 2010)).

Usage

rel.vert.basic.tri(p, c1, c2, M)
rel.vert.basic.tri(p, c1, c2, M)

Arguments

`p`	A 2D point for which `M`-vertex region it resides in is to be determined in the standard basic triangle form $T_b$ .
`c1`, `c2`	Positive real numbers which constitute the vertex of the standard basic triangle form adjacent to the shorter edges; $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard basic triangle form.

Value

A list with two elements

`rv`	Index of the vertex whose region contains point, `p`; index of the vertex is the same as the row number in the standard basic triangle form, $T_b$
`tri`	The vertices of the standard basic triangle form, $T_b$ , where row number corresponds to the vertex index `rv` with `rv=1` is row $1=(0,0)$ , `rv=2` is row $2=(1,0)$ , and $rv=3$ is row $3=(c_1,c_2)$ .

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);
M<-c(.6,.2)

P<-c(.4,.2)
rel.vert.basic.tri(P,c1,c2,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Rv<-vector()
for (i in 1:n)
{ Rv<-c(Rv,rel.vert.basic.tri(Xp[i,],c1,c2,M)$rv)}
Rv

Ds<-prj.cent2edges.basic.tri(c1,c2,M)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.1,.1),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tb[,1]+c(-.04,.05,.04)
yc<-Tb[,2]+c(.02,.02,.03)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.03,0)
yc<-txt[,2]+c(-.02,.02,.02,-.03)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);
M<-c(.6,.2)

P<-c(.4,.2)
rel.vert.basic.tri(P,c1,c2,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Rv<-vector()
for (i in 1:n)
{ Rv<-c(Rv,rel.vert.basic.tri(Xp[i,],c1,c2,M)$rv)}
Rv

Ds<-prj.cent2edges.basic.tri(c1,c2,M)

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.1,.1),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tb[,1]+c(-.04,.05,.04)
yc<-Tb[,2]+c(.02,.02,.03)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.03,0)
yc<-txt[,2]+c(-.02,.02,.02,-.03)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))

## End(Not run)

The index of the $CC$ -vertex region in a standard basic triangle form that contains a point

Description

Returns the index of the vertex whose region contains point p in the standard basic triangle form $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ and vertex regions are based on the circumcenter $CC$ of $T_b$ . (see the plots in the example for illustrations).

The vertices of the standard basic triangle form $T_b$ are labeled as $1=(0,0)$ , $2=(1,0)$ ,and $3=(c_1,c_2)$ also according to the row number the vertex is recorded in $T_b$ . If the point, p, is not inside $T_b$ , then the function yields NA as output. The corresponding vertex region is the polygon whose interior points are closest to that vertex.

See also (Ceyhan (2005, 2010)).

Usage

rel.vert.basic.triCC(p, c1, c2)
rel.vert.basic.triCC(p, c1, c2)

Arguments

`p`	A 2D point for which $CC$ -vertex region it resides in is to be determined in the standard basic triangle form $T_b$ .
`c1`, `c2`	Positive real numbers which constitute the upper vertex of the standard basic triangle form (i.e., the vertex adjacent to the shorter edges of $T_b$ ); $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Value

A list with two elements

`rv`	Index of the $CC$ -vertex region that contains point, `p` in the standard basic triangle form $T_b$
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv` with row $1=(0,0)$ , row $2=(1,0)$ , and row $3=(c_1,c_2)$ .

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6;  #try also c1<-.5; c2<-.5;

P<-c(.3,.2)
rel.vert.basic.triCC(P,c1,c2)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)
CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1])
Ylim<-range(Tb[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,asp=1,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,0.02,-.01,.05,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.03,.03,-.03)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

RV1<-(A+D3+CC+D2)/4
RV2<-(B+D3+CC+D1)/4
RV3<-(C+D2+CC+D1)/4

txt<-rbind(RV1,RV2,RV3)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

n<-20  #try also n<-40
Xp<-runif.basic.tri(n,c1,c2)$g

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.basic.triCC(Xp[i,],c1,c2)$rv)
Rv

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,asp=1,xlab="",pch=".",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(Rv))

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,0.02,-.01,.05,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.03,.03,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6;  #try also c1<-.5; c2<-.5;

P<-c(.3,.2)
rel.vert.basic.triCC(P,c1,c2)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)
CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tb[,1])
Ylim<-range(Tb[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,asp=1,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,0.02,-.01,.05,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.03,.03,-.03)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

RV1<-(A+D3+CC+D2)/4
RV2<-(B+D3+CC+D1)/4
RV3<-(C+D2+CC+D1)/4

txt<-rbind(RV1,RV2,RV3)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

n<-20  #try also n<-40
Xp<-runif.basic.tri(n,c1,c2)$g

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.basic.triCC(Xp[i,],c1,c2)$rv)
Rv

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,asp=1,xlab="",pch=".",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tb)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(Rv))

txt<-rbind(Tb,CC,Ds)
xc<-txt[,1]+c(-.03,.03,0.02,-.01,.05,-.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.03,.03,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The index of the $CM$ -vertex region in a standard basic triangle form that contains a point

Description

Returns the index of the vertex whose region contains point p in the standard basic triangle form $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ and vertex regions are based on the center of mass CM=((1+c1)/3,c2/3) of $T_b$ . (see the plots in the example for illustrations).

See also (Ceyhan (2005, 2010); Ceyhan et al. (2006))

Usage

rel.vert.basic.triCM(p, c1, c2)
rel.vert.basic.triCM(p, c1, c2)

Arguments

`p`	A 2D point for which $CM$ -vertex region it resides in is to be determined in the standard basic triangle form $T_b$ .
`c1`, `c2`	Positive real numbers which constitute the upper vertex of the standard basic triangle form (i.e., the vertex adjacent to the shorter edges of $T_b$ ); $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Value

A list with two elements

`rv`	Index of the $CM$ -vertex region that contains point, `p` in the standard basic triangle form $T_b$
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv` with row $1=(0,0)$ , row $2=(1,0)$ , and row $3=(c_1,c_2)$ .

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

#' @author Elvan Ceyhan

Examples

## Not run: 
c1<-.4; c2<-.6
P<-c(.4,.2)
rel.vert.basic.triCM(P,c1,c2)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

n<-20  #try also n<-40
Xp<-runif.basic.tri(n,c1,c2)$g

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.basic.triCM(Xp[i,],c1,c2)$rv)
Rv

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,xlab="",ylab="",axes="T",pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tb)
L<-Ds; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(Rv))

txt<-rbind(Tb,CM,Ds)
xc<-txt[,1]+c(-.03,.03,.02,-.01,.06,-.05,.0)
yc<-txt[,2]+c(.02,.02,.02,.04,.02,.02,-.03)
txt.str<-c("A","B","C","CM","D1","D2","D3")
text(xc,yc,txt.str)

plot(Tb,xlab="",ylab="",axes="T",pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-Ds; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

RV1<-(A+D3+CM+D2)/4
RV2<-(B+D3+CM+D1)/4
RV3<-(C+D2+CM+D1)/4

txt<-rbind(RV1,RV2,RV3)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(Tb,CM,Ds)
xc<-txt[,1]+c(-.03,.03,.02,-.01,.04,-.03,.0)
yc<-txt[,2]+c(.02,.02,.02,.04,.02,.02,-.03)
txt.str<-c("A","B","C","CM","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
P<-c(.4,.2)
rel.vert.basic.triCM(P,c1,c2)

A<-c(0,0);B<-c(1,0);C<-c(c1,c2);
Tb<-rbind(A,B,C)
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

n<-20  #try also n<-40
Xp<-runif.basic.tri(n,c1,c2)$g

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.basic.triCM(Xp[i,],c1,c2)$rv)
Rv

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,xlab="",ylab="",axes="T",pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xp,pch=".")
polygon(Tb)
L<-Ds; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(Rv))

txt<-rbind(Tb,CM,Ds)
xc<-txt[,1]+c(-.03,.03,.02,-.01,.06,-.05,.0)
yc<-txt[,2]+c(.02,.02,.02,.04,.02,.02,-.03)
txt.str<-c("A","B","C","CM","D1","D2","D3")
text(xc,yc,txt.str)

plot(Tb,xlab="",ylab="",axes="T",pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-Ds; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

RV1<-(A+D3+CM+D2)/4
RV2<-(B+D3+CM+D1)/4
RV3<-(C+D2+CM+D1)/4

txt<-rbind(RV1,RV2,RV3)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(Tb,CM,Ds)
xc<-txt[,1]+c(-.03,.03,.02,-.01,.04,-.03,.0)
yc<-txt[,2]+c(.02,.02,.02,.04,.02,.02,-.03)
txt.str<-c("A","B","C","CM","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The index of the vertex region in an end-interval that contains a given point

Description

Returns the index of the vertex in the interval, int, whose end interval contains the 1D point p, that is, it finds the index of the vertex for the point, p, outside the interval int $=(a,b)=$ (vertex 1,vertex 2); vertices of interval are labeled as 1 and 2 according to their order in the interval.

If the point, p, is inside int, then the function yields NA as output. The corresponding vertex region is an interval as $(-\infty,a)$ or $(b,\infty)$ for the interval $(a,b)$ . Then if $p<a$ , then rv=1 and if $p>b$ , then rv=2. Unlike rel.vert.mid.int, centrality parameter (i.e., center of the interval is not relevant for rel.vert.end.int.)

See also (Ceyhan (2012, 2016)).

Usage

rel.vert.end.int(p, int)
rel.vert.end.int(p, int)

Arguments

`p`	A 1D point whose end interval region is provided by the function.
`int`	A `vector` of two real numbers representing an interval.

Value

A list with two elements

`rv`	Index of the end vertex whose region contains point, `p`.
`int`	The vertices of the interval as a `vector` where position of the vertex corresponds to the vertex index as `int=(rv=1,rv=2)`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
a<-0; b<-10; int<-c(a,b)

rel.vert.end.int(-6,int)
rel.vert.end.int(16,int)

n<-10
xf<-(int[2]-int[1])*.5
XpL<-runif(n,a-xf,a)
XpR<-runif(n,b,b+xf)
Xp<-c(XpL,XpR)
rel.vert.end.int(Xp[1],int)

Rv<-vector()
for (i in 1:length(Xp))
  Rv<-c(Rv,rel.vert.end.int(Xp[i],int)$rv)
Rv

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",xlim=Xlim+xd*c(-.05,.05))
abline(h=0)
abline(v=c(a,b),col=1,lty = 2)
points(cbind(Xp,0))
text(cbind(Xp,0.1),labels=factor(Rv))
text(cbind(c(a,b),-0.1),c("rv=1","rv=2"))

jit<-.1
yjit<-runif(length(Xp),-jit,jit)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),
main="vertex region indices for the points\n in the end intervals",
     xlab=" ", ylab=" ",pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=3*range(yjit))
points(Xp, yjit,xlim=Xlim+xd*c(-.05,.05),pch=".",cex=3)
abline(h=0)
abline(v=c(a,b),lty = 2)
text(Xp,yjit,labels=factor(Rv))
text(cbind(c(a,b),-.01),c("rv=1","rv=2"))

## End(Not run)

## Not run: 
a<-0; b<-10; int<-c(a,b)

rel.vert.end.int(-6,int)
rel.vert.end.int(16,int)

n<-10
xf<-(int[2]-int[1])*.5
XpL<-runif(n,a-xf,a)
XpR<-runif(n,b,b+xf)
Xp<-c(XpL,XpR)
rel.vert.end.int(Xp[1],int)

Rv<-vector()
for (i in 1:length(Xp))
  Rv<-c(Rv,rel.vert.end.int(Xp[i],int)$rv)
Rv

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",xlim=Xlim+xd*c(-.05,.05))
abline(h=0)
abline(v=c(a,b),col=1,lty = 2)
points(cbind(Xp,0))
text(cbind(Xp,0.1),labels=factor(Rv))
text(cbind(c(a,b),-0.1),c("rv=1","rv=2"))

jit<-.1
yjit<-runif(length(Xp),-jit,jit)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),
main="vertex region indices for the points\n in the end intervals",
     xlab=" ", ylab=" ",pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=3*range(yjit))
points(Xp, yjit,xlim=Xlim+xd*c(-.05,.05),pch=".",cex=3)
abline(h=0)
abline(v=c(a,b),lty = 2)
text(Xp,yjit,labels=factor(Rv))
text(cbind(c(a,b),-.01),c("rv=1","rv=2"))

## End(Not run)

The index of the vertex region in a middle interval that contains a given point

Description

Returns the index of the vertex whose region contains point p in the interval int $=(a,b)=$ (vertex 1,vertex 2) with (parameterized) center $M_c$ associated with the centrality parameter $c \in (0,1)$ ; vertices of interval are labeled as 1 and 2 according to their order in the interval int. If the point, p, is not inside int, then the function yields NA as output. The corresponding vertex region is the interval $(a,b)$ as $(a,M_c)$ and $(M_c,b)$ where $M_c=a+c(b-a)$ .

See also (Ceyhan (2012, 2016)).

Usage

rel.vert.mid.int(p, int, c = 0.5)
rel.vert.mid.int(p, int, c = 0.5)

Arguments

`p`	A 1D point. The vertex region `p` resides is to be found.
`int`	A `vector` of two real numbers representing an interval.
`c`	A positive real number in $(0,1)$ parameterizing the center inside `int` $=(a,b)$ with the default `c=.5`. For the interval, `int` $=(a,b)$ , the parameterized center is $M_c=a+c(b-a)$ .

Value

A list with two elements

`rv`	Index of the vertex in the interval `int` whose region contains point, `p`.
`int`	The vertices of the interval as a `vector` where position of the vertex corresponds to the vertex index as `int=(rv=1,rv=2)`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c<-.4
a<-0; b<-10; int = c(a,b)

Mc<-centerMc(int,c)

rel.vert.mid.int(6,int,c)

n<-20  #try also n<-40
xr<-range(a,b,Mc)
xf<-(int[2]-int[1])*.5
Xp<-runif(n,a,b)

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.mid.int(Xp[i],int,c)$rv)
Rv

jit<-.1
yjit<-runif(n,-jit,jit)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(Mc,0),main="vertex region indices for the points", xlab=" ",
ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*range(yjit),pch=".",cex=3)
abline(h=0)
points(Xp,yjit)
abline(v=c(a,b,Mc),lty = 2,col=c(1,1,2))
text(Xp,yjit,labels=factor(Rv))
text(cbind(c(a,b,Mc),.02),c("rv=1","rv=2","Mc"))

## End(Not run)

## Not run: 
c<-.4
a<-0; b<-10; int = c(a,b)

Mc<-centerMc(int,c)

rel.vert.mid.int(6,int,c)

n<-20  #try also n<-40
xr<-range(a,b,Mc)
xf<-(int[2]-int[1])*.5
Xp<-runif(n,a,b)

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.mid.int(Xp[i],int,c)$rv)
Rv

jit<-.1
yjit<-runif(n,-jit,jit)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(Mc,0),main="vertex region indices for the points", xlab=" ",
ylab=" ", xlim=Xlim+xd*c(-.05,.05),ylim=3*range(yjit),pch=".",cex=3)
abline(h=0)
points(Xp,yjit)
abline(v=c(a,b,Mc),lty = 2,col=c(1,1,2))
text(Xp,yjit,labels=factor(Rv))
text(cbind(c(a,b,Mc),.02),c("rv=1","rv=2","Mc"))

## End(Not run)

The index of the vertex region in the standard equilateral triangle that contains a given point

Description

Returns the index of the vertex whose region contains point p in standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with vertex regions are constructed with center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of $T_e$ . (see the plots in the example for illustrations).

The vertices of triangle, $T_e$ , are labeled as $1,2,3$ according to the row number the vertex is recorded in $T_e$ . If the point, p, is not inside $T_e$ , then the function yields NA as output. The corresponding vertex region is the polygon with the vertex, M, and projections from M to the edges on the lines joining vertices and M.

See also (Ceyhan (2005, 2010)).

Usage

rel.vert.std.tri(p, M)
rel.vert.std.tri(p, M)

Arguments

`p`	A 2D point for which `M`-vertex region it resides in is to be determined in the standard equilateral triangle $T_e$ .
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$ .

Value

A list with two elements

`rv`	Index of the vertex whose region contains point, `p`.
`tri`	The vertices of the triangle, $T_e$ , where row number corresponds to the vertex index in `rv` with row $1=(0,0)$ , row $2=(1,0)$ , and row $3=(1/2,\sqrt{3}/2)$ .

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-20  #try also n<-40

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

rel.vert.std.tri(Xp[1,],M)

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.std.tri(Xp[i,],M)$rv)
Rv

Ds<-prj.cent2edges(Te,M)

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Te)}
#need to run this when M is given in barycentric coordinates

plot(Te,asp=1,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Te,M)
xc<-txt[,1]+c(-.02,.03,.02,0)
yc<-txt[,2]+c(.02,.02,.03,.05)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-20  #try also n<-40

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

rel.vert.std.tri(Xp[1,],M)

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.std.tri(Xp[i,],M)$rv)
Rv

Ds<-prj.cent2edges(Te,M)

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Te)}
#need to run this when M is given in barycentric coordinates

plot(Te,asp=1,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Te,M)
xc<-txt[,1]+c(-.02,.03,.02,0)
yc<-txt[,2]+c(.02,.02,.03,.05)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))

## End(Not run)

The index of the $CM$ -vertex region in the standard equilateral triangle that contains a given point

Description

The vertices of triangle, $T_e$ , are labeled as $1,2,3$ according to the row number the vertex is recorded in $T_e$ . If the point, p, is not inside $T_e$ , then the function yields NA as output. The corresponding vertex region is the polygon with the vertex, $CM$ , and midpoints of the edges adjacent to the vertex.

See also (Ceyhan (2005, 2010)).

Usage

rel.vert.std.triCM(p)
rel.vert.std.triCM(p)

Arguments

`p`	A 2D point for which $CM$ -vertex region it resides in is to be determined in the standard equilateral triangle $T_e$ .

Value

A list with two elements

`rv`	Index of the vertex whose region contains point, `p`.
`tri`	The vertices of the triangle, $T_e$ , where row number corresponds to the vertex index in `rv` with row $1=(0,0)$ , row $2=(1,0)$ , and row $3=(1/2,\sqrt{3}/2)$ .

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)

n<-20  #try also n<-40

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

rel.vert.std.triCM(Xp[1,])

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.std.triCM(Xp[i,])$rv)
Rv

CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,asp=1,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,pch=".",col=1)
L<-matrix(rep(CM,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.03,.02,0)
yc<-txt[,2]+c(.02,.02,.03,.05)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)

n<-20  #try also n<-40

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

rel.vert.std.triCM(Xp[1,])

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.std.triCM(Xp[i,])$rv)
Rv

CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,asp=1,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp,pch=".",col=1)
L<-matrix(rep(CM,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.03,.02,0)
yc<-txt[,2]+c(.02,.02,.03,.05)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))

## End(Not run)

The index of the $CC$ -vertex region in a tetrahedron that contains a point

Description

Returns the index of the vertex whose region contains point p in a tetrahedron $th=T(A,B,C,D)$ and vertex regions are based on the circumcenter $CC$ of th. (see the plots in the example for illustrations).

The vertices of the tetrahedron th are labeled as $1=A$ , $2=B$ , $3=C$ , and $4=C$ also according to the row number the vertex is recorded in th.

If the point, p, is not inside th, then the function yields NA as output. The corresponding vertex region is the polygon whose interior points are closest to that vertex. If th is regular tetrahedron, then $CC$ and $CM$ (center of mass) coincide.

See also (Ceyhan (2005, 2010)).

Usage

rel.vert.tetraCC(p, th)
rel.vert.tetraCC(p, th)

Arguments

`p`	A 3D point for which $CC$ -vertex region it resides in is to be determined in the tetrahedron `th`.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.

Value

A list with two elements

`rv`	Index of the $CC$ -vertex region that contains point, `p` in the tetrahedron `th`
`tri`	The vertices of the tetrahedron, where row number corresponds to the vertex index in `rv`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
set.seed(123)
A<-c(0,0,0)+runif(3,-.2,.2);
B<-c(1,0,0)+runif(3,-.2,.2);
C<-c(1/2,sqrt(3)/2,0)+runif(3,-.2,.2);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3)+runif(3,-.2,.2);
tetra<-rbind(A,B,C,D)

n<-20  #try also n<-40

Xp<-runif.tetra(n,tetra)$g

rel.vert.tetraCC(Xp[1,],tetra)

Rv<-vector()
for (i in 1:n)
 Rv<-c(Rv,rel.vert.tetraCC(Xp[i,],tetra)$rv)
Rv

CC<-circumcenter.tetra(tetra)
CC

Xlim<-range(tetra[,1],Xp[,1],CC[1])
Ylim<-range(tetra[,2],Xp[,2],CC[2])
Zlim<-range(tetra[,3],Xp[,3],CC[3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(tetra[,1],tetra[,2],tetra[,3],
phi =0,theta=40, bty = "g",
main="Scatterplot of data points \n and CC-vertex regions",
xlim=Xlim+xd*c(-.05,.05), ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],
add=TRUE,lwd=2)
#add the data points
plot3D::points3D(Xp[,1],Xp[,2],Xp[,3],pch=".",cex=3, add=TRUE)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)
plot3D::text3D(CC[1],CC[2],CC[3], labels=c("CC"), add=TRUE)

D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2;
D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CC,6),ncol=3,byrow=TRUE)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],
add=TRUE,lty = 2)

F1<-intersect.line.plane(A,CC,B,C,D)
L<-matrix(rep(F1,4),ncol=3,byrow=TRUE); R<-rbind(D4,D5,D6,CC)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=2,
add=TRUE,lty = 2)

F2<-intersect.line.plane(B,CC,A,C,D)
L<-matrix(rep(F2,4),ncol=3,byrow=TRUE); R<-rbind(D2,D3,D6,CC)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=3,
add=TRUE,lty = 2)

F3<-intersect.line.plane(C,CC,A,B,D)
L<-matrix(rep(F3,4),ncol=3,byrow=TRUE); R<-rbind(D3,D5,D6,CC)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=4,
add=TRUE,lty = 2)

F4<-intersect.line.plane(D,CC,A,B,C)
L<-matrix(rep(F4,4),ncol=3,byrow=TRUE); R<-rbind(D1,D2,D4,CC)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=5,
add=TRUE,lty = 2)

plot3D::text3D(Xp[,1],Xp[,2],Xp[,3], labels=factor(Rv), add=TRUE)

## End(Not run)

## Not run: 
set.seed(123)
A<-c(0,0,0)+runif(3,-.2,.2);
B<-c(1,0,0)+runif(3,-.2,.2);
C<-c(1/2,sqrt(3)/2,0)+runif(3,-.2,.2);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3)+runif(3,-.2,.2);
tetra<-rbind(A,B,C,D)

n<-20  #try also n<-40

Xp<-runif.tetra(n,tetra)$g

rel.vert.tetraCC(Xp[1,],tetra)

Rv<-vector()
for (i in 1:n)
 Rv<-c(Rv,rel.vert.tetraCC(Xp[i,],tetra)$rv)
Rv

CC<-circumcenter.tetra(tetra)
CC

Xlim<-range(tetra[,1],Xp[,1],CC[1])
Ylim<-range(tetra[,2],Xp[,2],CC[2])
Zlim<-range(tetra[,3],Xp[,3],CC[3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(tetra[,1],tetra[,2],tetra[,3],
phi =0,theta=40, bty = "g",
main="Scatterplot of data points \n and CC-vertex regions",
xlim=Xlim+xd*c(-.05,.05), ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],
add=TRUE,lwd=2)
#add the data points
plot3D::points3D(Xp[,1],Xp[,2],Xp[,3],pch=".",cex=3, add=TRUE)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)
plot3D::text3D(CC[1],CC[2],CC[3], labels=c("CC"), add=TRUE)

D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2;
D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CC,6),ncol=3,byrow=TRUE)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],
add=TRUE,lty = 2)

F1<-intersect.line.plane(A,CC,B,C,D)
L<-matrix(rep(F1,4),ncol=3,byrow=TRUE); R<-rbind(D4,D5,D6,CC)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=2,
add=TRUE,lty = 2)

F2<-intersect.line.plane(B,CC,A,C,D)
L<-matrix(rep(F2,4),ncol=3,byrow=TRUE); R<-rbind(D2,D3,D6,CC)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=3,
add=TRUE,lty = 2)

F3<-intersect.line.plane(C,CC,A,B,D)
L<-matrix(rep(F3,4),ncol=3,byrow=TRUE); R<-rbind(D3,D5,D6,CC)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=4,
add=TRUE,lty = 2)

F4<-intersect.line.plane(D,CC,A,B,C)
L<-matrix(rep(F4,4),ncol=3,byrow=TRUE); R<-rbind(D1,D2,D4,CC)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=5,
add=TRUE,lty = 2)

plot3D::text3D(Xp[,1],Xp[,2],Xp[,3], labels=factor(Rv), add=TRUE)

## End(Not run)

The index of the $CM$ -vertex region in a tetrahedron that contains a point

Description

Returns the index of the vertex whose region contains point p in a tetrahedron $th=T(A,B,C,D)$ and vertex regions are based on the center of mass $CM=(A+B+C+D)/4$ of th. (see the plots in the example for illustrations).

The vertices of the tetrahedron th are labeled as $1=A$ , $2=B$ , $3=C$ , and $4=C$ also according to the row number the vertex is recorded in th.

If the point, p, is not inside th, then the function yields NA as output. The corresponding vertex region is the simplex with the vertex, $CM$ , and midpoints of the edges adjacent to the vertex.

See also (Ceyhan (2005, 2010)).

Usage

rel.vert.tetraCM(p, th)
rel.vert.tetraCM(p, th)

Arguments

`p`	A 3D point for which $CM$ -vertex region it resides in is to be determined in the tetrahedron `th`.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.

Value

A list with two elements

`rv`	Index of the $CM$ -vertex region that contains point, `p` in the tetrahedron `th`
`th`	The vertices of the tetrahedron, where row number corresponds to the vertex index in `rv`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-20  #try also n<-40

Xp<-runif.std.tetra(n)$g

rel.vert.tetraCM(Xp[1,],tetra)

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv, rel.vert.tetraCM(Xp[i,],tetra)$rv )
Rv

Xlim<-range(tetra[,1],Xp[,1])
Ylim<-range(tetra[,2],Xp[,2])
Zlim<-range(tetra[,3],Xp[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

CM<-apply(tetra,2,mean)

plot3D::scatter3D(tetra[,1],tetra[,2],tetra[,3], phi =0,theta=40, bty = "g",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
#add the data points
plot3D::points3D(Xp[,1],Xp[,2],Xp[,3],pch=".",cex=3, add=TRUE)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)
plot3D::text3D(CM[1],CM[2],CM[3], labels=c("CM"), add=TRUE)

D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CM,6),ncol=3,byrow=TRUE)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty = 2)

F1<-intersect.line.plane(A,CM,B,C,D)
L<-matrix(rep(F1,4),ncol=3,byrow=TRUE); R<-rbind(D4,D5,D6,CM)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=2,
add=TRUE,lty = 2)

F2<-intersect.line.plane(B,CM,A,C,D)
L<-matrix(rep(F2,4),ncol=3,byrow=TRUE); R<-rbind(D2,D3,D6,CM)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=3,
add=TRUE,lty = 2)

F3<-intersect.line.plane(C,CM,A,B,D)
L<-matrix(rep(F3,4),ncol=3,byrow=TRUE); R<-rbind(D3,D5,D6,CM)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=4,
add=TRUE,lty = 2)

F4<-intersect.line.plane(D,CM,A,B,C)
L<-matrix(rep(F4,4),ncol=3,byrow=TRUE); R<-rbind(D1,D2,D4,CM)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=5,
add=TRUE,lty = 2)

plot3D::text3D(Xp[,1],Xp[,2],Xp[,3], labels=factor(Rv), add=TRUE)

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0);
D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-20  #try also n<-40

Xp<-runif.std.tetra(n)$g

rel.vert.tetraCM(Xp[1,],tetra)

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv, rel.vert.tetraCM(Xp[i,],tetra)$rv )
Rv

Xlim<-range(tetra[,1],Xp[,1])
Ylim<-range(tetra[,2],Xp[,2])
Zlim<-range(tetra[,3],Xp[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

CM<-apply(tetra,2,mean)

plot3D::scatter3D(tetra[,1],tetra[,2],tetra[,3], phi =0,theta=40, bty = "g",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
#add the data points
plot3D::points3D(Xp[,1],Xp[,2],Xp[,3],pch=".",cex=3, add=TRUE)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)
plot3D::text3D(CM[1],CM[2],CM[3], labels=c("CM"), add=TRUE)

D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CM,6),ncol=3,byrow=TRUE)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty = 2)

F1<-intersect.line.plane(A,CM,B,C,D)
L<-matrix(rep(F1,4),ncol=3,byrow=TRUE); R<-rbind(D4,D5,D6,CM)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=2,
add=TRUE,lty = 2)

F2<-intersect.line.plane(B,CM,A,C,D)
L<-matrix(rep(F2,4),ncol=3,byrow=TRUE); R<-rbind(D2,D3,D6,CM)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=3,
add=TRUE,lty = 2)

F3<-intersect.line.plane(C,CM,A,B,D)
L<-matrix(rep(F3,4),ncol=3,byrow=TRUE); R<-rbind(D3,D5,D6,CM)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=4,
add=TRUE,lty = 2)

F4<-intersect.line.plane(D,CM,A,B,C)
L<-matrix(rep(F4,4),ncol=3,byrow=TRUE); R<-rbind(D1,D2,D4,CM)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],col=5,
add=TRUE,lty = 2)

plot3D::text3D(Xp[,1],Xp[,2],Xp[,3], labels=factor(Rv), add=TRUE)

## End(Not run)

The index of the vertex region in a triangle that contains a given point

Description

Returns the index of the related vertex in the triangle, tri, whose region contains point p.

Vertex regions are based on the general center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle tri. Vertices of the triangle tri are labeled according to the row number the vertex is recorded.

If the point, p, is not inside tri, then the function yields NA as output. The corresponding vertex region is the polygon with the vertex, M, and projections from M to the edges on the lines joining vertices and M (see the illustration in the examples).

See also (Ceyhan (2005, 2010)).

Usage

rel.vert.tri(p, tri, M)
rel.vert.tri(p, tri, M)

Arguments

`p`	A 2D point for which `M`-vertex region it resides in is to be determined in the triangle `tri`.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`.

Value

A list with two elements

`rv`	Index of the vertex whose region contains point, `p`; index of the vertex is the same as the row number in the triangle, `tri`
`tri`	The vertices of the triangle, `tri`, where row number corresponds to the vertex index `rv` with `rv=1` is row 1, `rv=2` is row 2, and $rv=3$ is is row 3.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
M<-c(1.6,1.0)

P<-c(1.5,1.6)
rel.vert.tri(P,Tr,M)
#try also rel.vert.tri(P,Tr,M=c(2,2))
#center is not in the interior of the triangle

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Rv<-vector()
for (i in 1:n)
{Rv<-c(Rv,rel.vert.tri(Xp[i,],Tr,M)$rv)}
Rv

Ds<-prj.cent2edges(Tr,M)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Illustration of M-Vertex Regions\n in a Triangle",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]
yc<-Tr[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.04,0)
yc<-txt[,2]+c(-.02,.04,.05,-.08)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
M<-c(1.6,1.0)

P<-c(1.5,1.6)
rel.vert.tri(P,Tr,M)
#try also rel.vert.tri(P,Tr,M=c(2,2))
#center is not in the interior of the triangle

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Rv<-vector()
for (i in 1:n)
{Rv<-c(Rv,rel.vert.tri(Xp[i,],Tr,M)$rv)}
Rv

Ds<-prj.cent2edges(Tr,M)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Illustration of M-Vertex Regions\n in a Triangle",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]
yc<-Tr[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(-.02,.04,-.04,0)
yc<-txt[,2]+c(-.02,.04,.05,-.08)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(Rv))

## End(Not run)

The index of the $CC$ -vertex region in a triangle that contains a point

Description

Returns the index of the vertex whose region contains point p in a triangle tri $=(A,B,C)$ and vertex regions are based on the circumcenter $CC$ of tri. (see the plots in the example for illustrations).

The vertices of the triangle tri are labeled as $1=A$ , $2=B$ , and $3=C$ also according to the row number the vertex is recorded in tri. If the point, p, is not inside tri, then the function yields NA as output. The corresponding vertex region is the polygon whose interior points are closest to that vertex. If tri is equilateral triangle, then $CC$ and $CM$ (center of mass) coincide.

See also (Ceyhan (2005, 2010)).

Usage

rel.vert.triCC(p, tri)
rel.vert.triCC(p, tri)

Arguments

`p`	A 2D point for which $CC$ -vertex region it resides in is to be determined in the triangle `tri`.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with two elements

`rv`	Index of the $CC$ -vertex region that contains point, `p` in the triangle `tri`
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(1.3,1.2)
rel.vert.triCC(P,Tr)

CC<-circumcenter.tri(Tr)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],CC[1])
Ylim<-range(Tr[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,asp=1,xlab="",ylab="",pch=".",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.07,.08,.06,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.03,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

RV1<-(A+.5*(D3-A)+A+.5*(D2-A))/2
RV2<-(B+.5*(D3-B)+B+.5*(D1-B))/2
RV3<-(C+.5*(D2-C)+C+.5*(D1-C))/2

txt<-rbind(RV1,RV2,RV3)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

n<-20  #try also n<-40
Xp<-runif.tri(n,Tr)$g

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.triCC(Xp[i,],Tr)$rv)
Rv

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,asp=1,xlab="",ylab="",
main="Illustration of CC-Vertex Regions\n in a Triangle",
pch=".",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".")
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(Rv))

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.07,.08,.06,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.03,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(1.3,1.2)
rel.vert.triCC(P,Tr)

CC<-circumcenter.tri(Tr)  #the circumcenter
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],CC[1])
Ylim<-range(Tr[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,asp=1,xlab="",ylab="",pch=".",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.07,.08,.06,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.03,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

RV1<-(A+.5*(D3-A)+A+.5*(D2-A))/2
RV2<-(B+.5*(D3-B)+B+.5*(D1-B))/2
RV3<-(C+.5*(D2-C)+C+.5*(D1-C))/2

txt<-rbind(RV1,RV2,RV3)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

n<-20  #try also n<-40
Xp<-runif.tri(n,Tr)$g

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.triCC(Xp[i,],Tr)$rv)
Rv

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,asp=1,xlab="",ylab="",
main="Illustration of CC-Vertex Regions\n in a Triangle",
pch=".",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".")
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(Rv))

txt<-rbind(Tr,CC,Ds)
xc<-txt[,1]+c(-.07,.08,.06,.12,-.1,-.1,-.09)
yc<-txt[,2]+c(.02,-.02,.03,.0,.02,.06,-.04)
txt.str<-c("A","B","C","CC","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The index of the $CM$ -vertex region in a triangle that contains a given point

Description

Returns the index of the vertex whose region contains point p in the triangle tri $=(y_1,y_2,y_3)$ with vertex regions are constructed with center of mass $CM=(y_1+y_2+y_3)/3$ (see the plots in the example for illustrations).

The vertices of triangle, tri, are labeled as $1,2,3$ according to the row number the vertex is recorded in tri. If the point, p, is not inside tri, then the function yields NA as output. The corresponding vertex region is the polygon with the vertex, $CM$ , and midpoints of the edges adjacent to the vertex.

See (Ceyhan (2005, 2010))

Usage

rel.vert.triCM(p, tri)
rel.vert.triCM(p, tri)

Arguments

`p`	A 2D point for which $CM$ -vertex region it resides in is to be determined in the triangle `tri`.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with two elements

`rv`	Index of the $CM$ -vertex region that contains point, `p` in the triangle `tri`.
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.6,2);
Tr<-rbind(A,B,C);
P<-c(1.4,1.2)
rel.vert.triCM(P,Tr)

n<-20  #try also n<-40
Xp<-runif.tri(n,Tr)$g

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.triCM(Xp[i,],Tr)$rv)
Rv

CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".")
L<-Ds; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(Rv))

txt<-rbind(Tr,CM,D1,D2,D3)
xc<-txt[,1]+c(-.02,.02,.02,-.02,.02,-.01,-.01)
yc<-txt[,2]+c(-.02,-.04,.06,-.02,.02,.06,-.06)
txt.str<-c("rv=1","rv=2","rv=3","CM","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.6,2);
Tr<-rbind(A,B,C);
P<-c(1.4,1.2)
rel.vert.triCM(P,Tr)

n<-20  #try also n<-40
Xp<-runif.tri(n,Tr)$g

Rv<-vector()
for (i in 1:n)
  Rv<-c(Rv,rel.vert.triCM(Xp[i,],Tr)$rv)
Rv

CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,xlab="",ylab="",axes=TRUE,pch=".",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".")
L<-Ds; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE)
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)
text(Xp,labels=factor(Rv))

txt<-rbind(Tr,CM,D1,D2,D3)
xc<-txt[,1]+c(-.02,.02,.02,-.02,.02,-.01,-.01)
yc<-txt[,2]+c(-.02,-.04,.06,-.02,.02,.06,-.06)
txt.str<-c("rv=1","rv=2","rv=3","CM","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

The indices of the vertex regions in a triangle that contains the points in a give data set

Description

Returns the indices of the vertices whose regions contain the points in data set Xp in a triangle tri $=T(A,B,C)$ .

Vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle to the edges on the extension of the lines joining M to the vertices or based on the circumcenter of tri. Vertices of triangle tri are labeled as $1,2,3$ according to the row number the vertex is recorded.

If a point in Xp is not inside tri, then the function yields NA as output for that entry. The corresponding vertex region is the polygon with the vertex, M, and projection points from M to the edges crossing the vertex (as the output of prj.cent2edges(Tr,M)) or $CC$ -vertex region (see the examples for an illustration).

See also (Ceyhan (2005, 2011)).

Usage

rel.verts.tri(Xp, tri, M)
rel.verts.tri(Xp, tri, M)

Arguments

`Xp`	A set of 2D points representing the set of data points for which indices of the vertex regions containing them are to be determined.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri`.

Value

A list with two elements

`rv`	Indices of the vertices whose regions contains points in `Xp`.
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
M<-c(1.6,1.0)

P<-c(.4,.2)
rel.verts.tri(P,Tr,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also  #M<-c(1.6,1.0)

rel.verts.tri(Xp,Tr,M)
rel.verts.tri(rbind(Xp,c(2,2)),Tr,M)

rv<-rel.verts.tri(Xp,Tr,M)
rv

ifelse(identical(M,circumcenter.tri(Tr)),
Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2),Ds<-prj.cent2edges(Tr,M))

Xlim<-range(Tr[,1],M[1],Xp[,1])
Ylim<-range(Tr[,2],M[2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Scatterplot of data points \n and M-vertex regions in a triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]
yc<-Tr[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.02,.04,-.03,0)
yc<-txt[,2]+c(.07,.04,.05,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
M<-c(1.6,1.0)

P<-c(.4,.2)
rel.verts.tri(P,Tr,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also  #M<-c(1.6,1.0)

rel.verts.tri(Xp,Tr,M)
rel.verts.tri(rbind(Xp,c(2,2)),Tr,M)

rv<-rel.verts.tri(Xp,Tr,M)
rv

ifelse(identical(M,circumcenter.tri(Tr)),
Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2),Ds<-prj.cent2edges(Tr,M))

Xlim<-range(Tr[,1],M[1],Xp[,1])
Ylim<-range(Tr[,2],M[2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

plot(Tr,pch=".",xlab="",ylab="",
main="Scatterplot of data points \n and M-vertex regions in a triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]
yc<-Tr[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.02,.04,-.03,0)
yc<-txt[,2]+c(.07,.04,.05,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

The indices of the vertex regions in a triangle that contains the points in a give data set

Description

Returns the indices of the vertices whose regions contain the points in data set Xp in a triangle tri $=(A,B,C)$ and vertex regions are based on the center cent which yields nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in tri for expansion parameter r in $(1,1.5]$ .

Vertices of triangle tri are labeled as $1,2,3$ according to the row number the vertex is recorded if a point in Xp is not inside tri, then the function yields NA as output for that entry. The corresponding vertex region is the polygon with the vertex, cent, and projection points on the edges. The center label cent values 1,2,3 correspond to the vertices $M_1$ , $M_2$ , and $M_3$ ; with default 1 (see the examples for an illustration).

See also (Ceyhan (2005, 2011)).

Usage

rel.verts.tri.nondegPE(Xp, tri, r, cent = 1)
rel.verts.tri.nondegPE(Xp, tri, r, cent = 1)

Arguments

`Xp`	A set of 2D points representing the set of data points for which indices of the vertex regions containing them are to be determined.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`r`	A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,1.5]$ for this function.
`cent`	Index of the center (as $1,2,3$ corresponding to $M_1,\,M_2,\,M_3$ ) which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in `tri` for expansion parameter `r` in $(1,1.5]$ ; default `cent=1`.

Value

A list with two elements

`rv`	Indices (i.e., a `vector` of indices) of the vertices whose region contains points in `Xp` in the triangle `tri`
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
r<-1.35
cent<-2

P<-c(1.4,1.0)
rel.verts.tri.nondegPE(P,Tr,r,cent)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

rel.verts.tri.nondegPE(Xp,Tr,r,cent)
rel.verts.tri.nondegPE(rbind(Xp,c(2,2)),Tr,r,cent)

rv<-rel.verts.tri.nondegPE(Xp,Tr,r,cent)

M<-center.nondegPE(Tr,r)[cent,];
Ds<-prj.nondegPEcent2edges(Tr,r,cent)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.03,.05,.05)
yc<-Tr[,2]+c(-.06,.02,.05)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.02,.04,-.03,0)
yc<-txt[,2]+c(.07,.03,.05,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
r<-1.35
cent<-2

P<-c(1.4,1.0)
rel.verts.tri.nondegPE(P,Tr,r,cent)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

rel.verts.tri.nondegPE(Xp,Tr,r,cent)
rel.verts.tri.nondegPE(rbind(Xp,c(2,2)),Tr,r,cent)

rv<-rel.verts.tri.nondegPE(Xp,Tr,r,cent)

M<-center.nondegPE(Tr,r)[cent,];
Ds<-prj.nondegPEcent2edges(Tr,r,cent)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.03,.05,.05)
yc<-Tr[,2]+c(-.06,.02,.05)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.02,.04,-.03,0)
yc<-txt[,2]+c(.07,.03,.05,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

The indices of the $CC$ -vertex regions in a triangle that contains the points in a give data set.

Description

Returns the indices of the vertices whose regions contain the points in data set Xp in a triangle tri $=(A,B,C)$ and vertex regions are based on the circumcenter $CC$ of tri. (see the plots in the example for illustrations).

The vertices of the triangle tri are labeled as $1=A$ , $2=B$ , and $3=C$ also according to the row number the vertex is recorded in tri. If a point in Xp is not inside tri, then the function yields NA as output. The corresponding vertex region is the polygon whose interior points are closest to that vertex. If tri is equilateral triangle, then $CC$ and $CM$ (center of mass) coincide.

See also (Ceyhan (2005, 2010)).

Usage

rel.verts.triCC(Xp, tri)
rel.verts.triCC(Xp, tri)

Arguments

`Xp`	A set of 2D points representing the set of data points for which indices of the vertex regions containing them are to be determined.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with two elements

`rv`	Indices (i.e., a `vector` of indices) of the vertices whose region contains points in `Xp` in the triangle `tri`
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(.4,.2)
rel.verts.triCC(P,Tr)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

rel.verts.triCC(Xp,Tr)
rel.verts.triCC(rbind(Xp,c(2,2)),Tr)

(rv<-rel.verts.triCC(Xp,Tr))

CC<-circumcenter.tri(Tr)
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1],CC[1])
Ylim<-range(Tr[,2],Xp[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",asp=1,xlab="",ylab="",
main="Scatterplot of data points \n and the CC-vertex regions",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]
yc<-Tr[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(CC,Ds)
xc<-txt[,1]+c(.04,.04,-.03,0)
yc<-txt[,2]+c(-.07,.04,.06,-.08)
txt.str<-c("CC","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(.4,.2)
rel.verts.triCC(P,Tr)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

rel.verts.triCC(Xp,Tr)
rel.verts.triCC(rbind(Xp,c(2,2)),Tr)

(rv<-rel.verts.triCC(Xp,Tr))

CC<-circumcenter.tri(Tr)
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1],CC[1])
Ylim<-range(Tr[,2],Xp[,2],CC[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",asp=1,xlab="",ylab="",
main="Scatterplot of data points \n and the CC-vertex regions",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-matrix(rep(CC,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]
yc<-Tr[,2]
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(CC,Ds)
xc<-txt[,1]+c(.04,.04,-.03,0)
yc<-txt[,2]+c(-.07,.04,.06,-.08)
txt.str<-c("CC","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

The indices of the $CM$ -vertex regions in a triangle that contains the points in a give data set

Description

Returns the indices of the vertices whose regions contain the points in data set Xp in a triangle tri $=(A,B,C)$ and vertex regions are based on the center of mass $CM$ of tri. (see the plots in the example for illustrations).

The vertices of the triangle tri are labeled as $1=A$ , $2=B$ , and $3=C$ also according to the row number the vertex is recorded in tri. If a point in Xp is not inside tri, then the function yields NA as output for that entry. The corresponding vertex region is the polygon with the vertex, $CM$ , and midpoints the edges crossing the vertex.

See also (Ceyhan (2005, 2010)).

Usage

rel.verts.triCM(Xp, tri)
rel.verts.triCM(Xp, tri)

Arguments

`Xp`	A set of 2D points representing the set of data points for which indices of the vertex regions containing them are to be determined.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with two elements

`rv`	Indices (i.e., a `vector` of indices) of the vertices whose region contains points in `Xp` in the triangle `tri`
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv`.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(.4,.2)
rel.verts.triCM(P,Tr)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

rv<-rel.verts.triCM(Xp,Tr)
rv

CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-matrix(rep(CM,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.04,.05,.05)
yc<-Tr[,2]+c(-.05,.05,.03)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(CM,Ds)
xc<-txt[,1]+c(.04,.04,-.03,0)
yc<-txt[,2]+c(-.07,.04,.06,-.08)
txt.str<-c("CM","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

P<-c(.4,.2)
rel.verts.triCM(P,Tr)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

rv<-rel.verts.triCM(Xp,Tr)
rv

CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-matrix(rep(CM,3),ncol=2,byrow=TRUE); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.04,.05,.05)
yc<-Tr[,2]+c(-.05,.05,.03)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(CM,Ds)
xc<-txt[,1]+c(.04,.04,-.03,0)
yc<-txt[,2]+c(-.07,.04,.06,-.08)
txt.str<-c("CM","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

The alternative function for the indices of the M-vertex regions in a triangle that contains the points in a give data set

Description

An alternative function to the function rel.verts.tri when the center M is not the circumcenter falling outside the triangle. This function only works for a center $M$ in the interior of the triangle, with the projections of $M$ to the edges along the lines joining $M$ to the vertices.

Usage

rel.verts.triM(Xp, tri, M)
rel.verts.triM(Xp, tri, M)

Arguments

`Xp`	A set of 2D points representing the set of data points for which indices of the vertex regions containing them are to be determined.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`M`	A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri`.

Value

A list with two elements

`rv`	Indices of the vertices whose regions contains points in `Xp`.
`tri`	The vertices of the triangle, where row number corresponds to the vertex index in `rv`.

Author(s)

Elvan Ceyhan

References

There are no references for Rd macro ⁠\insertAllCites⁠ on this help page.

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
M<-c(1.6,1.0)

P<-c(.4,.2)
rel.verts.triM(P,Tr,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-c(1.6,1.0)  #try also M<-c(1.3,1.3)

(rv<-rel.verts.tri(Xp,Tr,M))
rel.verts.triM(rbind(Xp,c(2,2)),Tr,M)

Ds<-prj.cent2edges(Tr,M)

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.03,.05,.05)
yc<-Tr[,2]+c(-.06,.02,.05)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.02,.04,-.03,0)
yc<-txt[,2]+c(.07,.03,.05,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
M<-c(1.6,1.0)

P<-c(.4,.2)
rel.verts.triM(P,Tr,M)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-c(1.6,1.0)  #try also M<-c(1.3,1.3)

(rv<-rel.verts.tri(Xp,Tr,M))
rel.verts.triM(rbind(Xp,c(2,2)),Tr,M)

Ds<-prj.cent2edges(Tr,M)

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.03,.05,.05)
yc<-Tr[,2]+c(-.06,.02,.05)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.02,.04,-.03,0)
yc<-txt[,2]+c(.07,.03,.05,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))

## End(Not run)

Generation of points segregated (in a radial or circular fashion) from a given set of points

Description

An object of class "Patterns". Generates n 2D points uniformly in $(a_1-e,a_1+e) \times (a_1-e,a_1+e) \setminus B(y_i,e)$ ( $a_1$ and $b1$ are denoted as a1 and b1 as arguments) where $Y_p=(y_1,y_2,\ldots,y_{n_y})$ with $n_y$ being number of Yp points for various values of e under the segregation pattern and $B(y_i,e)$ is the ball centered at $y_i$ with radius e.

Positive values of e yield realizations from the segregation pattern and nonpositive values of e provide a type of complete spatial randomness (CSR), e should not be too large to make the support of generated points empty, a1 is defaulted to the minimum of the $x$ -coordinates of the Yp points, a2 is defaulted to the maximum of the $x$ -coordinates of the Yp points, b1 is defaulted to the minimum of the $y$ -coordinates of the Yp points, b2 is defaulted to the maximum of the $y$ -coordinates of the Yp points.

Usage

rseg.circular(
  n,
  Yp,
  e,
  a1 = min(Yp[, 1]),
  a2 = max(Yp[, 1]),
  b1 = min(Yp[, 2]),
  b2 = max(Yp[, 2])
)
rseg.circular(
  n,
  Yp,
  e,
  a1 = min(Yp[, 1]),
  a2 = max(Yp[, 1]),
  b1 = min(Yp[, 2]),
  b2 = max(Yp[, 2])
)

Arguments

`n`	A positive integer representing the number of points to be generated.
`Yp`	A set of 2D points representing the reference points. The generated points are segregated (in a circular or radial fashion) from these points.
`e`	A positive real number representing the radius of the balls centered at `Yp` points. These balls are forbidden for the generated points (i.e., generated points would be in the complement of union of these balls).
`a1`, `a2`	Real numbers representing the range of $x$ -coordinates in the region (default is the range of $x$ -coordinates of the `Yp` points).
`b1`, `b2`	Real numbers representing the range of $y$ -coordinates in the region (default is the range of $y$ -coordinates of the `Yp` points).

Value

A list with the elements

`type`	The type of the point pattern
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	Radial (i.e., circular) exclusion parameter of the segregation pattern
`ref.points`	The input set of reference points `Yp`, i.e., points from which generated points are segregated.
`gen.points`	The output set of generated points segregated from `Yp` points
`tri.Yp`	Logical output for triangulation based on `Yp` points should be implemented or not. if `TRUE` triangulation based on `Yp` points is to be implemented (default is set to `FALSE`).
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of reference (i.e., `Yp`) points.
`xlimit`, `ylimit`	The possible ranges of the $x$ - and $y$ -coordinates of the generated points

Author(s)

Elvan Ceyhan

Examples

## Not run: 
nx<-100; ny<-4;  #try also nx<-1000; ny<-10
e<-.15;  #try also e<- -.1; #a negative e provides a CSR realization
#with default bounding box (i.e., unit square)
Y<-cbind(runif(ny),runif(ny))

Xdt<-rseg.circular(nx,Y,e)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

#with default bounding box (i.e., unit square)
Y<-cbind(runif(ny),runif(ny))
Xdt<-Xdt$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,pch=16,col=2,lwd=2, xlab="x",ylab="y",
     main="Circular Segregation of X points from Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
points(Xdt)

#with a rectangular bounding box
a1<-0; a2<-10;
b1<-0; b2<-5;
e<-1.5;
Y<-cbind(runif(ny,a1,a2),runif(ny,b1,b2))
#try also Y<-cbind(runif(ny,a1,a2/2),runif(ny,b1,b2/2))

Xdt<-rseg.circular(nx,Y,e,a1,a2,b1,b2)$gen.points
Xlim<-range(Xdt[,1],Y[,1]); Ylim<-range(Xdt[,2],Y[,2])

plot(Y,pch=16,asp=1,col=2,lwd=2, xlab="x",ylab="y",
     main="Circular Segregation of X points from Y Points",
     xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xdt)

## End(Not run)

## Not run: 
nx<-100; ny<-4;  #try also nx<-1000; ny<-10
e<-.15;  #try also e<- -.1; #a negative e provides a CSR realization
#with default bounding box (i.e., unit square)
Y<-cbind(runif(ny),runif(ny))

Xdt<-rseg.circular(nx,Y,e)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

#with default bounding box (i.e., unit square)
Y<-cbind(runif(ny),runif(ny))
Xdt<-Xdt$gen.points
Xlim<-range(Xdt[,1],Y[,1]);
Ylim<-range(Xdt[,2],Y[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Y,asp=1,pch=16,col=2,lwd=2, xlab="x",ylab="y",
     main="Circular Segregation of X points from Y Points",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
points(Xdt)

#with a rectangular bounding box
a1<-0; a2<-10;
b1<-0; b2<-5;
e<-1.5;
Y<-cbind(runif(ny,a1,a2),runif(ny,b1,b2))
#try also Y<-cbind(runif(ny,a1,a2/2),runif(ny,b1,b2/2))

Xdt<-rseg.circular(nx,Y,e,a1,a2,b1,b2)$gen.points
Xlim<-range(Xdt[,1],Y[,1]); Ylim<-range(Xdt[,2],Y[,2])

plot(Y,pch=16,asp=1,col=2,lwd=2, xlab="x",ylab="y",
     main="Circular Segregation of X points from Y Points",
     xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
points(Xdt)

## End(Not run)

Generation of points segregated (in a Type I fashion) from a given set of points

Description

An object of class "Patterns". Generates n points uniformly in the support for Type I segregation in the convex hull of set of points, Yp.

delta is the parameter of segregation (that is, $\delta 100$ % of the area around each vertex in each Delaunay triangle is forbidden for point generation). delta corresponds to eps in the standard equilateral triangle $T_e$ as $delta=4 eps ^2/3$ (see rseg.std.tri function).

If Yp consists only of 3 points, then the function behaves like the function rseg.tri.

See (Ceyhan et al. (2006); Ceyhan et al. (2007); Ceyhan (2011)) for more on the segregation pattern. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

rseg.multi.tri(n, Yp, delta, DTmesh = NULL, DTr = NULL)
rseg.multi.tri(n, Yp, delta, DTmesh = NULL, DTr = NULL)

Arguments

`n`	A positive integer representing the number of points to be generated.
`Yp`	A set of 2D points from which Delaunay triangulation is constructed.
`delta`	A positive real number in $(0,4/9)$ . `delta` is the parameter of segregation (that is, $\delta 100$ area around each vertex in each Delaunay triangle is forbidden for point generation).
`DTmesh`	Delaunay triangulation of `Yp`, default is `NULL`, which is computed via `tri.mesh` function in `interp` package. `tri.mesh` function yields the triangulation nodes with their neighbours, and creates a triangulation object.
`DTr`	Delaunay triangles based on `Yp`, default is `NULL`, which is computed via `tri.mesh` function in `interp` package. `triangles` function yields a triangulation data structure from the triangulation object created by `tri.mesh`.

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	Exclusion parameter, `delta`, of the Type I segregation pattern. `delta` is in $(0,4/9)$ $\delta 100$ % area around each vertex in each Delaunay triangle is forbidden for point generation.
`ref.points`	The input set of points `Yp`; reference points, i.e., points from which generated points are segregated.
`gen.points`	The output set of generated points segregated from `Yp` points.
`tri.Y`	Logical output, `TRUE`, if triangulation based on `Yp` points should be implemented.
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of reference (i.e., `Yp`) points.
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the reference points, which are the `Yp` points

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random $r$ -factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-1000; ny<-10;

set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
del<-.4

Xdt<-rseg.multi.tri(nx,Yp,del)
Xdt
summary(Xdt)
plot(Xdt)

#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points
TRY<-interp::triangles(DTY)[,1:3];
Xp<-rseg.multi.tri(nx,Yp,del,DTY,TRY)$gen.points
#data under CSR in the convex hull of Ypoints

Xlim<-range(Yp[,1])
Ylim<-range(Yp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot of the data in the convex hull of Y points together with the Delaunay triangulation
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points

par(pty="s")
plot(Xp,main="Points from Type I Segregation \n in Multipe Triangles",
xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05),type="n")
interp::plot.triSht(DTY, add=TRUE,
do.points=TRUE,col="blue")
points(Xp,pch=".",cex=3)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-1000; ny<-10;

set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
del<-.4

Xdt<-rseg.multi.tri(nx,Yp,del)
Xdt
summary(Xdt)
plot(Xdt)

#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points
TRY<-interp::triangles(DTY)[,1:3];
Xp<-rseg.multi.tri(nx,Yp,del,DTY,TRY)$gen.points
#data under CSR in the convex hull of Ypoints

Xlim<-range(Yp[,1])
Ylim<-range(Yp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot of the data in the convex hull of Y points together with the Delaunay triangulation
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points

par(pty="s")
plot(Xp,main="Points from Type I Segregation \n in Multipe Triangles",
xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05),type="n")
interp::plot.triSht(DTY, add=TRUE,
do.points=TRUE,col="blue")
points(Xp,pch=".",cex=3)

## End(Not run)

Generation of points segregated (in a Type I fashion) from the vertices of $T_e$

Description

An object of class "Patterns". Generates n points uniformly in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ under the type I segregation alternative for eps in $(0,\sqrt{3}/3=0.5773503]$ .

In the type I segregation, the triangular forbidden regions around the vertices are determined by the parameter eps which serves as the height of these triangles (see examples for a sample plot.)

See also (Ceyhan et al. (2006); Ceyhan et al. (2007); Ceyhan (2011)).

Usage

rseg.std.tri(n, eps)
rseg.std.tri(n, eps)

Arguments

`n`	A positive integer representing the number of points to be generated.
`eps`	A positive real number representing the parameter of type I segregation (which is the height of the triangular forbidden regions around the vertices).

Value

A list with the elements

`type`	The type of the point pattern
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	The exclusion parameter, `eps`, of the segregation pattern, which is the height of the triangular forbidden regions around the vertices
`ref.points`	The input set of points `Y`; reference points, i.e., points from which generated points are segregated (i.e., vertices of $T_e$ ).
`gen.points`	The output set of generated points segregated from `Y` points (i.e., vertices of $T_e$ ).
`tri.Y`	Logical output for triangulation based on `Y` points should be implemented or not. if `TRUE` triangulation based on `Y` points is to be implemented (default is set to `FALSE`).
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of reference (i.e., `Y`) points, which is 3 here.
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the reference points, which are the vertices of $T_e$ here.

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-100
eps<-.3  #try also .15, .5, .75

set.seed(1)
Xdt<-rseg.std.tri(n,eps)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-Xdt$gen.points

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Type I segregation in the \n standard equilateral triangle",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

#The support for the Type I segregation alternative
sr<-eps/(sqrt(3)/2)
C1<-C+sr*(A-C); C2<-C+sr*(B-C)
A1<-A+sr*(B-A); A2<-A+sr*(C-A)
B1<-B+sr*(A-B); B2<-B+sr*(C-B)
supp<-rbind(A1,B1,B2,C2,C1,A2)

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Support of the Type I Segregation",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
if (sr<=.5)
{
  polygon(Te)
  polygon(supp,col=5)
} else
{
  polygon(Te,col=5,lwd=2.5)
  polygon(rbind(A,A1,A2),col=0,border=NA)
  polygon(rbind(B,B1,B2),col=0,border=NA)
  polygon(rbind(C,C1,C2),col=0,border=NA)
}
points(Xp)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-100
eps<-.3  #try also .15, .5, .75

set.seed(1)
Xdt<-rseg.std.tri(n,eps)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-Xdt$gen.points

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Type I segregation in the \n standard equilateral triangle",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

#The support for the Type I segregation alternative
sr<-eps/(sqrt(3)/2)
C1<-C+sr*(A-C); C2<-C+sr*(B-C)
A1<-A+sr*(B-A); A2<-A+sr*(C-A)
B1<-B+sr*(A-B); B2<-B+sr*(C-B)
supp<-rbind(A1,B1,B2,C2,C1,A2)

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Support of the Type I Segregation",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
if (sr<=.5)
{
  polygon(Te)
  polygon(supp,col=5)
} else
{
  polygon(Te,col=5,lwd=2.5)
  polygon(rbind(A,A1,A2),col=0,border=NA)
  polygon(rbind(B,B1,B2),col=0,border=NA)
  polygon(rbind(C,C1,C2),col=0,border=NA)
}
points(Xp)

## End(Not run)

Generation of points segregated (in a Type I fashion) from the vertices of a triangle

Description

An object of class "Patterns". Generates n points uniformly in the support for Type I segregation in a given triangle, tri.

delta is the parameter of segregation (that is, $\delta 100$ % of the area around each vertex in the triangle is forbidden for point generation). delta corresponds to eps in the standard equilateral triangle $T_e$ as $delta=4 eps^2/3$ (see rseg.std.tri function).

See (Ceyhan et al. (2006); Ceyhan et al. (2007); Ceyhan (2011)) for more on the segregation pattern.

Usage

rseg.tri(n, tri, delta)
rseg.tri(n, tri, delta)

Arguments

`n`	A positive integer representing the number of points to be generated from the segregation pattern in the triangle, `tri`.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.
`delta`	A positive real number in $(0,4/9)$ . `delta` is the parameter of segregation (that is, $\delta 100$ % area around each vertex in each Delaunay triangle is forbidden for point generation).

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	Exclusion parameter, `delta`, of the Type I segregation pattern. `delta` is in $(0,4/9)$ $\delta 100$ % area around each vertex in the triangle `tri` is forbidden for point generation.
`ref.points`	The input set of points, i.e., vertices of `tri`; reference points, i.e., points from which generated points are segregated.
`gen.points`	The output set of generated points segregated from the vertices of `tri`.
`tri.Y`	Logical output, if `TRUE` the triangle `tri` is also plotted when the corresponding plot function from the `Patterns` object is called.
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of reference (i.e., vertex of `tri`, which is 3 here).
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the reference points, which are the vertices of the triangle `tri`

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
n<-100
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)
del<-.4

Xdt<-rseg.tri(n,Tr,del)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g
Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Points from Type I Segregation \n in one Triangle",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.03)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
n<-100
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)
del<-.4

Xdt<-rseg.tri(n,Tr,del)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g
Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Points from Type I Segregation \n in one Triangle",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.03)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

## End(Not run)

Generation of points segregated (in a Type II fashion) from the vertices of $T_e$

Description

An object of class "Patterns". Generates n points uniformly in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ under the type II segregation alternative for eps in $(0,\sqrt{3}/6=0.2886751]$ .

In the type II segregation, the annular forbidden regions around the edges are determined by the parameter eps which is the distance from the interior triangle (i.e., support for the segregation) to $T_e$ (see examples for a sample plot.)

Usage

rsegII.std.tri(n, eps)
rsegII.std.tri(n, eps)

Arguments

`n`	A positive integer representing the number of points to be generated.
`eps`	A positive real number representing the parameter of type II segregation (which is the distance from the interior triangle points to the boundary of $T_e$ ).

Value

A list with the elements

`type`	The type of the point pattern
`mtitle`	The `"main"` title for the plot of the point pattern
`parameters`	The exclusion parameter, `eps`, of the segregation pattern, which is the distance from the interior triangle to $T_e$
`ref.points`	The input set of points `Y`; reference points, i.e., points from which generated points are segregated (i.e., vertices of $T_e$ ).
`gen.points`	The output set of generated points segregated from `Y` points (i.e., vertices of $T_e$ ).
`tri.Y`	Logical output for triangulation based on `Y` points should be implemented or not. if `TRUE` triangulation based on `Y` points is to be implemented (default is set to `FALSE`).
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of reference (i.e., `Y`) points, which is 3 here.
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the reference points, which are the vertices of $T_e$ here

Author(s)

Elvan Ceyhan

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10  #try also n<-20 or n<-100 or 1000
eps<-.15  #try also .2

set.seed(1)
Xdt<-rsegII.std.tri(n,eps)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-Xdt$gen.points

plot(Te,pch=".",xlab="",ylab="",
main="Type II segregation in the \n standard equilateral triangle",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

#The support for the Type II segregation alternative
C1<-c(1/2,sqrt(3)/2-2*eps);
A1<-c(eps*sqrt(3),eps); B1<-c(1-eps*sqrt(3),eps);
supp<-rbind(A1,B1,C1)

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Support of the Type II Segregation",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
  polygon(Te)
  polygon(supp,col=5)
points(Xp)

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10  #try also n<-20 or n<-100 or 1000
eps<-.15  #try also .2

set.seed(1)
Xdt<-rsegII.std.tri(n,eps)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-Xdt$gen.points

plot(Te,pch=".",xlab="",ylab="",
main="Type II segregation in the \n standard equilateral triangle",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

#The support for the Type II segregation alternative
C1<-c(1/2,sqrt(3)/2-2*eps);
A1<-c(eps*sqrt(3),eps); B1<-c(1-eps*sqrt(3),eps);
supp<-rbind(A1,B1,C1)

plot(Te,asp=1,pch=".",xlab="",ylab="",
main="Support of the Type II Segregation",
     xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01))
  polygon(Te)
  polygon(supp,col=5)
points(Xp)

Generation of Uniform Points in the standard basic triangle

Description

An object of class "Uniform". Generates n points uniformly in the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Any given triangle can be mapped to the basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan et al. (2006)). Hence, standard basic triangle is useful for simulation studies under the uniformity hypothesis.

Usage

runif.basic.tri(n, c1, c2)
runif.basic.tri(n, c1, c2)

Arguments

`n`	A positive integer representing the number of uniform points to be generated in the standard basic triangle.
`c1`, `c2`	Positive real numbers representing the top vertex in standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$ , $c_1$ must be in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`tess.points`	The vertices of the support of the uniformly generated points, it is the standard basic triangle $T_b$ for this function
`gen.points`	The output set of generated points uniformly in the standard basic triangle
`out.region`	The outer region which contains the support region, `NULL` for this function.
`desc.pat`	Description of the point pattern from which points are to be generated
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of vertices of the support points (here it is 3).
`txt4pnts`	Description of the two numbers in `num.points`.
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the support, Tb

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);
n<-100

set.seed(1)
runif.basic.tri(1,c1,c2)
Xdt<-runif.basic.tri(n,c1,c2)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-runif.basic.tri(n,c1,c2)$g

Xlim<-range(Tb[,1])
Ylim<-range(Tb[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,xlab="",ylab="",xlim=Xlim+xd*c(-.01,.01),
ylim=Ylim+yd*c(-.01,.01),type="n")
polygon(Tb)
points(Xp)

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);
n<-100

set.seed(1)
runif.basic.tri(1,c1,c2)
Xdt<-runif.basic.tri(n,c1,c2)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-runif.basic.tri(n,c1,c2)$g

Xlim<-range(Tb[,1])
Ylim<-range(Tb[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tb,xlab="",ylab="",xlim=Xlim+xd*c(-.01,.01),
ylim=Ylim+yd*c(-.01,.01),type="n")
polygon(Tb)
points(Xp)

## End(Not run)

Generation of Uniform Points in the Convex Hull of Points

Description

An object of class "Uniform". Generates n points uniformly in the Convex Hull of set of points, Yp. That is, generates uniformly in each of the triangles in the Delaunay triangulation of Yp, i.e., in the multiple triangles partitioning the convex hull of Yp.

If Yp consists only of 3 points, then the function behaves like the function runif.tri.

DTmesh is the Delaunay triangulation of Yp, default is DTmesh=NULL. DTmesh yields triangulation nodes with neighbours (result of tri.mesh function from interp package).

See (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

runif.multi.tri(n, Yp, DTmesh = NULL)
runif.multi.tri(n, Yp, DTmesh = NULL)

Arguments

`n`	A positive integer representing the number of uniform points to be generated in the convex hull of the point set `Yp`.
`Yp`	A set of 2D points whose convex hull is the support of the uniform points to be generated.
`DTmesh`	Triangulation nodes with neighbours (result of `tri.mesh` function from `interp` package).

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`tess.points`	The points which constitute the vertices of the triangulation and whose convex hull determines the support of the generated points.
`gen.points`	The output set of generated points uniformly in the convex hull of `Yp`
`out.region`	The outer region which contains the support region, `NULL` for this function.
`desc.pat`	Description of the point pattern from which points are to be generated
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of vertices in the triangulation (i.e., size of `Yp`) points.
`txt4pnts`	Description of the two numbers in `num.points`
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the points in Yp

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-1000; ny<-10;
set.seed(1)
Yp<-cbind(runif(ny,0,10),runif(ny,0,10))

Xdt<-runif.multi.tri(nx,Yp)
#data under CSR in the convex hull of Ypoints
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g
#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points
Xp<-runif.multi.tri(nx,Yp,DTY)$g
#data under CSR in the convex hull of Ypoints

Xlim<-range(Yp[,1])
Ylim<-range(Yp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot of the data in the convex hull of Y points together with the Delaunay triangulation
plot(Xp, xlab=" ", ylab=" ",
main="Uniform Points in Convex Hull of Y Points",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),type="n")
interp::plot.triSht(DTY, add=TRUE,
do.points = TRUE,pch=16,col="blue")
points(Xp,pch=".",cex=3)

Yp<-rbind(c(.3,.2),c(.4,.5),c(.14,.15))
runif.multi.tri(nx,Yp)

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-1000; ny<-10;
set.seed(1)
Yp<-cbind(runif(ny,0,10),runif(ny,0,10))

Xdt<-runif.multi.tri(nx,Yp)
#data under CSR in the convex hull of Ypoints
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g
#or use
DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
#Delaunay triangulation based on Y points
Xp<-runif.multi.tri(nx,Yp,DTY)$g
#data under CSR in the convex hull of Ypoints

Xlim<-range(Yp[,1])
Ylim<-range(Yp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot of the data in the convex hull of Y points together with the Delaunay triangulation
plot(Xp, xlab=" ", ylab=" ",
main="Uniform Points in Convex Hull of Y Points",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),type="n")
interp::plot.triSht(DTY, add=TRUE,
do.points = TRUE,pch=16,col="blue")
points(Xp,pch=".",cex=3)

Yp<-rbind(c(.3,.2),c(.4,.5),c(.14,.15))
runif.multi.tri(nx,Yp)

## End(Not run)

Generation of Uniform Points in the Standard Regular Tetrahedron $T_h$

Description

An object of class "Uniform". Generates n points uniformly in the standard regular tetrahedron $T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))$ .

Usage

runif.std.tetra(n)
runif.std.tetra(n)

Arguments

`n`	A positive integer representing the number of uniform points to be generated in the standard regular tetrahedron $T_h$ .

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`tess.points`	The vertices of the support region of the uniformly generated points, it is the standard regular tetrahedron $T_h$ for this function
`gen.points`	The output set of generated points uniformly in the standard regular tetrahedron $T_h$ .
`out.region`	The outer region which contains the support region, `NULL` for this function.
`desc.pat`	Description of the point pattern from which points are to be generated
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of vertices of the support points (here it is 4).
`txt4pnts`	Description of the two numbers in `num.points`
`xlimit`, `ylimit`, `zlimit`	The ranges of the $x$ -, $y$ -, and $z$ -coordinates of the support, $T_h$

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-100

set.seed(1)
Xdt<-runif.std.tetra(n)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-runif.std.tetra(n)$g

Xlim<-range(tetra[,1])
Ylim<-range(tetra[,2])
Zlim<-range(tetra[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3],
phi =20,theta=15, bty = "g", pch = 20, cex = 1,
ticktype = "detailed",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.05,.05))
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],
add=TRUE,lwd=2)

plot3D::text3D(tetra[,1]+c(.05,0,0,0),tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

## End(Not run)

## Not run: 
#need to install scatterplot3d package and call "library(scatterplot3d)"
s3d<-scatterplot3d(Xp, highlight.3d=TRUE,xlab="x",
ylab="y",zlab="z",
              col.axis="blue", col.grid="lightblue",
               main="3D Scatterplot of the data", pch=20)
s3d$points3d(tetra,pch=20,col="blue")

## End(Not run)

## Not run: 
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-100

set.seed(1)
Xdt<-runif.std.tetra(n)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-runif.std.tetra(n)$g

Xlim<-range(tetra[,1])
Ylim<-range(tetra[,2])
Zlim<-range(tetra[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3],
phi =20,theta=15, bty = "g", pch = 20, cex = 1,
ticktype = "detailed",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.05,.05))
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],
add=TRUE,lwd=2)

plot3D::text3D(tetra[,1]+c(.05,0,0,0),tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

## End(Not run)

## Not run: 
#need to install scatterplot3d package and call "library(scatterplot3d)"
s3d<-scatterplot3d(Xp, highlight.3d=TRUE,xlab="x",
ylab="y",zlab="z",
              col.axis="blue", col.grid="lightblue",
               main="3D Scatterplot of the data", pch=20)
s3d$points3d(tetra,pch=20,col="blue")

## End(Not run)

Generation of Uniform Points in the Standard Equilateral Triangle

Description

An object of class "Uniform". Generates n points uniformly in the standard equilateral triangle $T_e=T(A,B,C)$ with vertices $A=(0,0)$ , $B=(1,0)$ , and $C=(1/2,\sqrt{3}/2)$ .

Usage

runif.std.tri(n)
runif.std.tri(n)

Arguments

`n`	A positive integer representing the number of uniform points to be generated in the standard equilateral triangle $T_e$ .

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`tess.points`	The vertices of the support region of the uniformly generated points, it is the standard equilateral triangle $T_e$ for this function
`gen.points`	The output set of generated points uniformly in the standard equilateral triangle $T_e$ .
`out.region`	The outer region which contains the support region, `NULL` for this function.
`desc.pat`	Description of the point pattern from which points are to be generated
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of vertices of the support points (here it is 3).
`txt4pnts`	Description of the two numbers in `num.points`
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the support, $T_e$

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-100

set.seed(1)
Xdt<-runif.std.tri(n)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-runif.std.tri(n)$gen.points
plot(Te,asp=1,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.01,.01),
ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-100

set.seed(1)
Xdt<-runif.std.tri(n)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xp<-runif.std.tri(n)$gen.points
plot(Te,asp=1,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.01,.01),
ylim=Ylim+yd*c(-.01,.01))
polygon(Te)
points(Xp)

## End(Not run)

Generation of Uniform Points in the first one-sixth of standard equilateral triangle

Description

An object of class "Uniform". Generates n points uniformly in the first 1/6th of the standard equilateral triangle $T_e=(A,B,C)$ with vertices with $A=(0,0)$ ; $B=(1,0)$ , $C=(1/2,\sqrt{3}/2)$ (see the examples below). The first 1/6th of the standard equilateral triangle is the triangle with vertices $A=(0,0)$ , $(1/2,0)$ , $C=(1/2,\sqrt{3}/6)$ .

Usage

runif.std.tri.onesixth(n)
runif.std.tri.onesixth(n)

Arguments

`n`	a positive integer representing number of uniform points to be generated in the first one-sixth of $T_e$ .

Value

A list with the elements

`type`	The type of the point pattern
`mtitle`	The `"main"` title for the plot of the point pattern
`support`	The vertices of the support of the uniformly generated points
`gen.points`	The output set of uniformly generated points in the first 1/6th of the standard equilateral triangle.
`out.region`	The outer region for the one-sixth of $T_e$ , which is just $T_e$ here.
`desc.pat`	Description of the point pattern
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of vertices of the support (i.e., `Y`) points.
`txt4pnts`	Description of the two numbers in `num.points`.
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the generated, support and outer region points

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
CM<-(A+B+C)/3;
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
nx<-100  #try also nx<-1000

#data generation step
set.seed(1)
Xdt<-runif.std.tri.onesixth(nx)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xd<-Xdt$gen.points

#plot of the data with the regions in the equilateral triangle
Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,asp=1,pch=".",xlim=Xlim+xd*c(-.01,.01),
ylim=Ylim+yd*c(-.01,.01),xlab=" ",ylab=" ",
     main="first 1/6th of the \n standard equilateral triangle")
polygon(Te)
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
polygon(rbind(A,D3,CM),col=5)
points(Xd)

#letter annotation of the plot
txt<-rbind(A,B,C,CM,D1,D2,D3)
xc<-txt[,1]+c(-.02,.02,.02,.04,.05,-.03,0)
yc<-txt[,2]+c(.02,.02,.02,.03,0,.03,-.03)
txt.str<-c("A","B","C","CM","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
CM<-(A+B+C)/3;
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
nx<-100  #try also nx<-1000

#data generation step
set.seed(1)
Xdt<-runif.std.tri.onesixth(nx)
Xdt
summary(Xdt)
plot(Xdt,asp=1)

Xd<-Xdt$gen.points

#plot of the data with the regions in the equilateral triangle
Xlim<-range(Te[,1])
Ylim<-range(Te[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,asp=1,pch=".",xlim=Xlim+xd*c(-.01,.01),
ylim=Ylim+yd*c(-.01,.01),xlab=" ",ylab=" ",
     main="first 1/6th of the \n standard equilateral triangle")
polygon(Te)
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
polygon(rbind(A,D3,CM),col=5)
points(Xd)

#letter annotation of the plot
txt<-rbind(A,B,C,CM,D1,D2,D3)
xc<-txt[,1]+c(-.02,.02,.02,.04,.05,-.03,0)
yc<-txt[,2]+c(.02,.02,.02,.03,0,.03,-.03)
txt.str<-c("A","B","C","CM","D1","D2","D3")
text(xc,yc,txt.str)

## End(Not run)

Generation of Uniform Points in a tetrahedron

Description

An object of class "Uniform". Generates n points uniformly in the general tetrahedron th whose vertices are stacked row-wise.

Usage

runif.tetra(n, th)
runif.tetra(n, th)

Arguments

`n`	A positive integer representing the number of uniform points to be generated in the tetrahedron.
`th`	A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron.

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`tess.points`	The vertices of the support of the uniformly generated points, it is the tetrahedron' `th` for this function
`gen.points`	The output set of generated points uniformly in the tetrahedron, `th`.
`out.region`	The outer region which contains the support region, `NULL` for this function.
`desc.pat`	Description of the point pattern from which points are to be generated
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of vertices of the support points (here it is 4).
`txt4pnts`	Description of the two numbers in `num.points`
`xlimit`, `ylimit`, `zlimit`	The ranges of the $x$ -, $y$ -, and $z$ -coordinates of the support, `th`

Author(s)

Elvan Ceyhan

Examples

## Not run: 
A<-sample(1:12,3); B<-sample(1:12,3);
C<-sample(1:12,3); D<-sample(1:12,3)
tetra<-rbind(A,B,C,D)

n<-100

set.seed(1)
Xdt<-runif.tetra(n,tetra)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g

Xlim<-range(tetra[,1],Xp[,1])
Ylim<-range(tetra[,2],Xp[,2])
Zlim<-range(tetra[,3],Xp[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3],
theta =225, phi = 30, bty = "g",
main="Uniform Points in a Tetrahedron",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],
add=TRUE,lwd=2)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

#need to install scatterplot3d package and call "library(scatterplot3d)"
s3d<-scatterplot3d(Xp, highlight.3d=TRUE,
xlab="x",ylab="y",zlab="z",
                    col.axis="blue", col.grid="lightblue",
                    main="3D Scatterplot of the data", pch=20)
 s3d$points3d(tetra,pch=20,col="blue")
 
## End(Not run)

## Not run: 
A<-sample(1:12,3); B<-sample(1:12,3);
C<-sample(1:12,3); D<-sample(1:12,3)
tetra<-rbind(A,B,C,D)

n<-100

set.seed(1)
Xdt<-runif.tetra(n,tetra)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g

Xlim<-range(tetra[,1],Xp[,1])
Ylim<-range(tetra[,2],Xp[,2])
Zlim<-range(tetra[,3],Xp[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3],
theta =225, phi = 30, bty = "g",
main="Uniform Points in a Tetrahedron",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.05,.05),
          pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3],
add=TRUE,lwd=2)

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3],
labels=c("A","B","C","D"), add=TRUE)

#need to install scatterplot3d package and call "library(scatterplot3d)"
s3d<-scatterplot3d(Xp, highlight.3d=TRUE,
xlab="x",ylab="y",zlab="z",
                    col.axis="blue", col.grid="lightblue",
                    main="3D Scatterplot of the data", pch=20)
 s3d$points3d(tetra,pch=20,col="blue")
 
## End(Not run)

Generation of Uniform Points in a Triangle

Description

An object of class "Uniform". Generates n points uniformly in a given triangle, tri

Usage

runif.tri(n, tri)
runif.tri(n, tri)

Arguments

`n`	A positive integer representing the number of uniform points to be generated in the triangle.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with the elements

`type`	The type of the pattern from which points are to be generated
`mtitle`	The `"main"` title for the plot of the point pattern
`tess.points`	The vertices of the support of the uniformly generated points, it is the triangle `tri` for this function
`gen.points`	The output set of generated points uniformly in the triangle, `tri`.
`out.region`	The outer region which contains the support region, `NULL` for this function.
`desc.pat`	Description of the point pattern from which points are to be generated
`num.points`	The `vector` of two numbers, which are the number of generated points and the number of vertices of the support points (here it is 3).
`txt4pnts`	Description of the two numbers in `num.points`
`xlimit`, `ylimit`	The ranges of the $x$ - and $y$ -coordinates of the support, `tri`

Author(s)

Elvan Ceyhan

Examples

## Not run: 
n<-100
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)

Xdt<-runif.tri(n,Tr)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g
Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
plot(Tr,pch=".",xlab="",ylab="",main="Uniform Points in One Triangle",
     xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
n<-100
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)

Xdt<-runif.tri(n,Tr)
Xdt
summary(Xdt)
plot(Xdt)

Xp<-Xdt$g
Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
plot(Tr,pch=".",xlab="",ylab="",main="Uniform Points in One Triangle",
     xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

## End(Not run)

The auxiliary triangle to define the support of type I segregation

Description

Returns the triangle whose intersection with a general triangle gives the support for type I segregation given the delta (i.e., $\delta 100$ % area of a triangle around the vertices is chopped off). See the plot in the examples.

Caveat: the vertices of this triangle may be outside the triangle, tri, depending on the value of delta (i.e., for small values of delta).

Usage

seg.tri.support(delta, tri)
seg.tri.support(delta, tri)

Arguments

`delta`	A positive real number between 0 and 1 that determines the percentage of area of the triangle around the vertices forbidden for point generation.
`tri`	A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

the vertices of the triangle (stacked row-wise) whose intersection with a general triangle gives the support for type I segregation for the given delta

Author(s)

Elvan Ceyhan

Examples

## Not run: 
#for a general triangle
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
delta<-.3  #try also .5,.75,.85
Tseg<-seg.tri.support(delta,Tr)

Xlim<-range(Tr[,1],Tseg[,1])
Ylim<-range(Tr[,2],Tseg[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

par(pty="s")
plot(Tr,pch=".",xlab="",ylab="",
main="segregation support is the intersection\n of these two triangles",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
polygon(Tseg,lty=2)

txt<-rbind(Tr,Tseg)
xc<-txt[,1]+c(-.03,.03,.03,.06,.04,-.04)
yc<-txt[,2]+c(.02,.02,.04,-.03,0,0)
txt.str<-c("A","B","C","T1","T2","T3")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
#for a general triangle
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
delta<-.3  #try also .5,.75,.85
Tseg<-seg.tri.support(delta,Tr)

Xlim<-range(Tr[,1],Tseg[,1])
Ylim<-range(Tr[,2],Tseg[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

par(pty="s")
plot(Tr,pch=".",xlab="",ylab="",
main="segregation support is the intersection\n of these two triangles",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
polygon(Tseg,lty=2)

txt<-rbind(Tr,Tseg)
xc<-txt[,1]+c(-.03,.03,.03,.06,.04,-.04)
yc<-txt[,2]+c(.02,.02,.04,-.03,0,0)
txt.str<-c("A","B","C","T1","T2","T3")
text(xc,yc,txt.str)

## End(Not run)

The closest points among a data set in the standard equilateral triangle to the median lines in the six half edge regions

Description

An object of class "Extrema". Returns the six closest points among the data set, Xp, in the standard equilateral triangle $T_e=T(A=(0,0),B=(1,0),C=(1/2,\sqrt{3}/2))$ in half edge regions. In particular, in regions $r_1$ and $r_6$ , it finds the closest point in each region to the line segment $[A,CM]$ in regions $r_2$ and $r_3$ , it finds the closest point in each region to the line segment $[B,CM]$ and in regions $r_4$ and $r_5$ , it finds the closest point in each region to the line segment $[C,CM]$ where $CM=(A+B+C)/3$ is the center of mass.

See the example for this function or example for index.six.Te function. If there is no data point in region $r_i$ , then it returns "NA NA" for $i$ -th row in the extrema. ch.all.intri is for checking whether all data points are in $T_e$ (default is FALSE).

Usage

six.extremaTe(Xp, ch.all.intri = FALSE)
six.extremaTe(Xp, ch.all.intri = FALSE)

Arguments

`Xp`	A set of 2D points among which the closest points in the standard equilateral triangle to the median lines in 6 half edge regions.
`ch.all.intri`	A logical argument for checking whether all data points are in $T_e$ (default is `FALSE`).

Value

A list with the elements

`txt1`	Region labels as r1-r6 (correspond to row number in Extremum Points).
`txt2`	A short description of the distances as `"Distances to Line Segments (A,CM), (B,CM), and (C,CM) in the six regions r1-r6"`.
`type`	Type of the extrema points
`mtitle`	The `"main"` title for the plot of the extrema
`ext`	The extrema points, here, closest points in each of regions `r1-r6` to the line segments joining vertices to the center of mass, $CM$ .
`X`	The input data, `Xp`, can be a `matrix` or `data frame`
`num.points`	The number of data points, i.e., size of `Xp`
`supp`	Support of the data points, here, it is $T_e$ .
`cent`	The center point used for construction of edge regions.
`ncent`	Name of the center, `cent`, it is center of mass `"CM"` for this function.
`regions`	The six regions, `r1-r6` and edge regions inside the triangle, $T_e$ , provided as a list.
`region.names`	Names of the regions as `"r1"`-`"r6"` and names of the edge regions as `"er=1"`, `"er=2"`, and `"er=3"`.
`region.centers`	Centers of mass of the regions `r1-r6` and of edge regions inside $T_e$ .
`dist2ref`	Distances from closest points in each of regions `r1-r6` to the line segments joining vertices to the center of mass, $CM$ .

Author(s)

Elvan Ceyhan

Examples

## Not run: 
n<-20  #try also n<-100
Xp<-runif.std.tri(n)$gen.points

Ext<-six.extremaTe(Xp)
Ext
summary(Ext)
plot(Ext)

sixt<-Ext

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

h1<-c(1/2,sqrt(3)/18); h2<-c(2/3, sqrt(3)/9); h3<-c(2/3, 2*sqrt(3)/9);
h4<-c(1/2, 5*sqrt(3)/18); h5<-c(1/3, 2*sqrt(3)/9); h6<-c(1/3, sqrt(3)/9);

r1<-(h1+h6+CM)/3;r2<-(h1+h2+CM)/3;r3<-(h2+h3+CM)/3;
r4<-(h3+h4+CM)/3;r5<-(h4+h5+CM)/3;r6<-(h5+h6+CM)/3;

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
polygon(rbind(h1,h2,h3,h4,h5,h6))
points(Xp)
points(sixt$ext,pty=2,pch=4,col="red")

txt<-rbind(Te,r1,r2,r3,r4,r5,r6)
xc<-txt[,1]+c(-.02,.02,.02,0,0,0,0,0,0)
yc<-txt[,2]+c(.02,.02,.03,0,0,0,0,0,0)
txt.str<-c("A","B","C","1","2","3","4","5","6")
text(xc,yc,txt.str)

## End(Not run)

## Not run: 
n<-20  #try also n<-100
Xp<-runif.std.tri(n)$gen.points

Ext<-six.extremaTe(Xp)
Ext
summary(Ext)
plot(Ext)

sixt<-Ext

A<-c(0,0); B<-c(1,0); C<-c(0.5,sqrt(3)/2);
Te<-rbind(A,B,C)
CM<-(A+B+C)/3
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)

h1<-c(1/2,sqrt(3)/18); h2<-c(2/3, sqrt(3)/9); h3<-c(2/3, 2*sqrt(3)/9);
h4<-c(1/2, 5*sqrt(3)/18); h5<-c(1/3, 2*sqrt(3)/9); h6<-c(1/3, sqrt(3)/9);

r1<-(h1+h6+CM)/3;r2<-(h1+h2+CM)/3;r3<-(h2+h3+CM)/3;
r4<-(h3+h4+CM)/3;r5<-(h4+h5+CM)/3;r6<-(h5+h6+CM)/3;

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
L<-Te; R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
polygon(rbind(h1,h2,h3,h4,h5,h6))
points(Xp)
points(sixt$ext,pty=2,pch=4,col="red")

txt<-rbind(Te,r1,r2,r3,r4,r5,r6)
xc<-txt[,1]+c(-.02,.02,.02,0,0,0,0,0,0)
yc<-txt[,2]+c(.02,.02,.03,0,0,0,0,0,0)
txt.str<-c("A","B","C","1","2","3","4","5","6")
text(xc,yc,txt.str)

## End(Not run)

The slope of a line

Description

Returns the slope of the line joining two distinct 2D points a and b.

Usage

slope(a, b)
slope(a, b)

Arguments

a, b

2D points that determine the straight line (i.e., through which the straight line passes).

Value

Slope of the line joining 2D points a and b

Author(s)

Elvan Ceyhan

Examples

A<-c(-1.22,-2.33); B<-c(2.55,3.75)
slope(A,B)

slope(c(1,2),c(2,3))

A<-c(-1.22,-2.33); B<-c(2.55,3.75)
slope(A,B)

slope(c(1,2),c(2,3))

Return a summary of a `Extrema` `object`

Description

Returns the below information about the object:

call of the function defining the object, the type of the extrema (i.e. the description of the extrema), extrema points, distances from extrema to the reference object (e.g. boundary of a triangle), some of the data points (from which extrema is found).

Usage

## S3 method for class 'Extrema'
summary(object, ...)
## S3 method for class 'Extrema'
summary(object, ...)

Arguments

`object`	An `object` of class `Extrema`.
`...`	Additional parameters for `summary`.

Value

The call of the object of class "Extrema", the type of the extrema (i.e. the description of the extrema), extrema points, distances from extrema to the reference object (e.g. boundary of a triangle), some of the data points (from which extrema is found).

Examples

## Not run: 
n<-10
Xp<-runif.std.tri(n)$gen.points
Ext<-cl2edges.std.tri(Xp)
Ext
summary(Ext)

## End(Not run)

## Not run: 
n<-10
Xp<-runif.std.tri(n)$gen.points
Ext<-cl2edges.std.tri(Xp)
Ext
summary(Ext)

## End(Not run)

Return a summary of a `Lines` `object`

Description

Returns the below information about the object:

call of the function defining the object, the defining points, selected $x$ and $y$ points on the line, equation of the line, and coefficients of the line.

Usage

## S3 method for class 'Lines'
summary(object, ...)
## S3 method for class 'Lines'
summary(object, ...)

Arguments

`object`	An `object` of class `Lines`.
`...`	Additional parameters for `summary`.

Value

The call of the object of class "Lines", the defining points, selected $x$ and $y$ points on the line, equation of the line, and coefficients of the line (in the form: y = slope * x + intercept).

Examples

A<-c(-1.22,-2.33); B<-c(2.55,3.75)
xr<-range(A,B);
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=3) #try also l=10, 20 or 100

lnAB<-Line(A,B,x)
lnAB
summary(lnAB)

A<-c(-1.22,-2.33); B<-c(2.55,3.75)
xr<-range(A,B);
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=3) #try also l=10, 20 or 100

lnAB<-Line(A,B,x)
lnAB
summary(lnAB)

Return a summary of a `Lines3D` `object`

Description

Returns the below information about the object:

call of the function defining the object, the defining vectors (i.e., initial and direction vectors), selected $x$ , $y$ , and $z$ points on the line, equation of the line (in parametric form), and coefficients of the line.

Usage

## S3 method for class 'Lines3D'
summary(object, ...)
## S3 method for class 'Lines3D'
summary(object, ...)

Arguments

`object`	An `object` of class `Lines3D`.
`...`	Additional parameters for `summary`.

Value

Examples

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3);
vecs<-rbind(P,Q)
Line3D(P,Q,.1)
Line3D(P,Q,.1,dir.vec=FALSE)

tr<-range(vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=3) #try also l=10, 20 or 100

lnPQ3D<-Line3D(P,Q,tsq)
lnPQ3D
summary(lnPQ3D)

## End(Not run)

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3);
vecs<-rbind(P,Q)
Line3D(P,Q,.1)
Line3D(P,Q,.1,dir.vec=FALSE)

tr<-range(vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=3) #try also l=10, 20 or 100

lnPQ3D<-Line3D(P,Q,tsq)
lnPQ3D
summary(lnPQ3D)

## End(Not run)

Return a summary of a `NumArcs` `object`

Description

Returns the below information about the object:

call of the function defining the object, the description of the proximity catch digraph (PCD), desc. In the one Delaunay cell case, the function provides the total number of arcs in the digraph, vertices of Delaunay cell, and indices of target points in the Delaunay cell.

In the multiple Delaunay cell case, the function provides total number of arcs in the digraph, number of arcs for the induced digraphs for points in the Delaunay cells, vertices of Delaunay cells or indices of points that form the the Delaunay cells, indices of target points in the convex hull of nontarget points, indices of Delaunay cells in which points reside, and area or length of the the Delaunay cells.

Usage

## S3 method for class 'NumArcs'
summary(object, ...)
## S3 method for class 'NumArcs'
summary(object, ...)

Arguments

`object`	An `object` of class `NumArcs`.
`...`	Additional parameters for `summary`.

Value

The call of the object of class "NumArcs", the desc of the proximity catch digraph (PCD), total number of arcs in the digraph. Moreover, in the one Delaunay cell case, the function also provides vertices of Delaunay cell, and indices of target points in the Delaunay cell; and in the multiple Delaunay cell case, it also provides number of arcs for the induced digraphs for points in the Delaunay cells, vertices of Delaunay cells or indices of points that form the the Delaunay cells, indices of target points in the convex hull of nontarget points, indices of Delaunay cells in which points reside, and area or length of the the Delaunay cells.

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
summary(Arcs)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
summary(Arcs)

## End(Not run)

Return a summary of a `Patterns` `object`

Description

Returns the below information about the object:

call of the function defining the object, the type of the pattern, parameters of the pattern, study window, some sample points from the generated pattern, reference points (if any for the bivariate pattern), and number of points for each class

Usage

## S3 method for class 'Patterns'
summary(object, ...)
## S3 method for class 'Patterns'
summary(object, ...)

Arguments

`object`	An `object` of class `Patterns`.
`...`	Additional parameters for `summary`.

Value

The call of the object of class "Patterns", the type of the pattern, parameters of the pattern, study window, some sample points from the generated pattern, reference points (if any for the bivariate pattern), and number of points for each class

Examples

## Not run: 
nx<-10; #try also 10, 100, and 1000
ny<-5; #try also 1
e<-.15;
Y<-cbind(runif(ny),runif(ny))
#with default bounding box (i.e., unit square)

Xdt<-rseg.circular(nx,Y,e)
Xdt
summary(Xdt)

## End(Not run)

## Not run: 
nx<-10; #try also 10, 100, and 1000
ny<-5; #try also 1
e<-.15;
Y<-cbind(runif(ny),runif(ny))
#with default bounding box (i.e., unit square)

Xdt<-rseg.circular(nx,Y,e)
Xdt
summary(Xdt)

## End(Not run)

Return a summary of a `PCDs` `object`

Description

Returns the below information about the object:

call of the function defining the object, the type of the proximity catch digraph (PCD), (i.e. the description of the PCD), some of the partition (i.e. intervalization in the 1D case and triangulation in the 2D case) points (i.e., vertices of the intervals or the triangles), parameter(s) of the PCD, and various quantities (number of vertices, number of arcs and arc density of the PCDs, number of vertices for the partition and number of partition cells (i.e., intervals or triangles)).

Usage

## S3 method for class 'PCDs'
summary(object, ...)
## S3 method for class 'PCDs'
summary(object, ...)

Arguments

`object`	An `object` of class `PCDs`.
`...`	Additional parameters for `summary`.

Value

The call of the object of class "PCDs", the type of the proximity catch digraph (PCD), (i.e. the description of the PCD), some of the partition (i.e. intervalization in the 1D case and triangulation in the 2D case) points (i.e., vertices of the intervals or the triangles), parameter(s) of the PCD, and various quantities (number of vertices, number of arcs and arc density of the PCDs, number of vertices for the partition and number of partition cells (i.e., intervals or triangles)).

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
summary(Arcs)

## End(Not run)

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10
Xp<-runif.tri(n,Tr)$g
M<-as.numeric(runif.tri(1,Tr)$g)
Arcs<-arcsAStri(Xp,Tr,M)
Arcs
summary(Arcs)

## End(Not run)

Return a summary of a `Planes` `object`

Description

Returns the below information about the object:

call of the function defining the object, the defining 3D points, selected $x$ , $y$ , and $z$ points on the plane, equation of the plane, and coefficients of the plane.

Usage

## S3 method for class 'Planes'
summary(object, ...)
## S3 method for class 'Planes'
summary(object, ...)

Arguments

`object`	An `object` of class `Planes`.
`...`	Additional parameters for `summary`.

Value

The call of the object of class "Planes", the defining 3D points, selected $x$ , $y$ , and $z$ points on the plane, equation of the plane, and coefficients of the plane (in the form: z = A*x + B*y + C).

Examples

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3); C<-c(3,9,12)
pts<-rbind(P,Q,C)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5) #try also l=10, 20 or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5) #try also l=10, 20 or 100

plPQC<-Plane(P,Q,C,x,y)
plPQC
summary(plPQC)

## End(Not run)

## Not run: 
P<-c(1,10,3); Q<-c(1,1,3); C<-c(3,9,12)
pts<-rbind(P,Q,C)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.1
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5) #try also l=10, 20 or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5) #try also l=10, 20 or 100

plPQC<-Plane(P,Q,C,x,y)
plPQC
summary(plPQC)

## End(Not run)

Return a summary of a `TriLines` `object`

Description

Returns the below information about the object:

call of the function defining the object, the defining points, selected $x$ and $y$ points on the line, equation of the line, together with the vertices of the triangle, and coefficients of the line.

Usage

## S3 method for class 'TriLines'
summary(object, ...)
## S3 method for class 'TriLines'
summary(object, ...)

Arguments

`object`	An `object` of class `TriLines`.
`...`	Additional parameters for `summary`.

Value

The call of the object of class "TriLines", the defining points, selected $x$ and $y$ points on the line, equation of the line, together with the vertices of the triangle, and coefficients of the line (in the form: y = slope * x + intercept).

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,l=3)

lnACM<-lineA2CMinTe(x)
lnACM
summary(lnACM)

## End(Not run)

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
xfence<-abs(A[1]-B[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(min(A[1],B[1])-xfence,max(A[1],B[1])+xfence,l=3)

lnACM<-lineA2CMinTe(x)
lnACM
summary(lnACM)

## End(Not run)

Return a summary of a `Uniform` `object`

Description

Returns the below information about the object:

call of the function defining the object, the type of the pattern (i.e. the description of the uniform distribution), study window, vertices of the support of the Uniform distribution, some sample points generated from the uniform distribution, and the number of points (i.e., number of generated points and the number of vertices of the support of the uniform distribution.)

Usage

## S3 method for class 'Uniform'
summary(object, ...)
## S3 method for class 'Uniform'
summary(object, ...)

Arguments

`object`	An `object` of class `Uniform`.
`...`	Additional parameters for `summary`.

Value

The call of the object of class "Uniform", the type of the pattern (i.e. the description of the uniform distribution), study window, vertices of the support of the Uniform distribution, some sample points generated from the uniform distribution, and the number of points (i.e., number of generated points and the number of vertices of the support of the uniform distribution.)

Examples

## Not run: 
n<-10 #try also 20, 100, and 1000
A<-c(1,1); B<-c(2,0); R<-c(1.5,2);
Tr<-rbind(A,B,R)

Xdt<-runif.tri(n,Tr)
Xdt
summary(Xdt)

## End(Not run)

## Not run: 
n<-10 #try also 20, 100, and 1000
A<-c(1,1); B<-c(2,0); R<-c(1.5,2);
Tr<-rbind(A,B,R)

Xdt<-runif.tri(n,Tr)
Xdt
summary(Xdt)

## End(Not run)

Tree Species in a Swamp Forest

Description

Locations and species classification of trees in a plot in the Savannah River, SC, USA. Locations are given in meters, rounded to the nearest 0.1 decimal. The data come from a one-hectare (200-by-50m) plot in the Savannah River Site. The 734 mapped stems included 156 Carolina ashes (Fraxinus caroliniana), 215 water tupelos (Nyssa aquatica), 205 swamp tupelos (Nyssa sylvatica), 98 bald cypresses (Taxodium distichum) and 60 stems from 8 additional three species (labeled as Others (OT)). The plots were set up by Bill Good and their spatial patterns described in (Good and Whipple (1982)), the plots have been maintained and resampled by Rebecca Sharitz and her colleagues of the Savannah River Ecology Laboratory. The data and some of its description are borrowed from the swamp data entry in the dixon package in the CRAN repository.

See also (Good and Whipple (1982); Jones et al. (1994); Dixon (2002)).

Usage

data(swamptrees)
data(swamptrees)

Format

A data frame with 734 rows and 4 variables

Details

Text describing the variable (i.e., column) names in the data set.

x,y: x and y (i.e., Cartesian) coordinates of the trees
live: a categorical variable that indicates the tree is alive (labeled as 1) or dead (labeled as 0)
sp: species label of the trees:
- FX: Carolina ash (Fraxinus caroliniana)
- NS: Swamp tupelo (Nyssa sylvatica)
- NX: Water tupelo (Nyssa aquatica)
- TD: Bald cypress (Taxodium distichum)
- OT: Other species

Source

Prof. Philip Dixon's website

References

Dixon PM (2002). “Nearest-neighbor contingency table analysis of spatial segregation for several species.” Ecoscience, 9(2), 142-151.

Good BJ, Whipple SA (1982). “Tree spatial patterns: South Carolina bottomland and swamp forests.” Bulletin of the Torrey Botanical Club, 109(4), 529-536.

Jones RH, Sharitz RR, James SM, Dixon PM (1994). “Tree population dynamics in seven South Carolina mixed-species forests.” Bulletin of the Torrey Botanical Club, 121(4), 360-368.

Examples

data(swamptrees)
plot(swamptrees$x,swamptrees$y, col=as.numeric(swamptrees$sp),pch=19,
xlab='',ylab='',main='Swamp Trees')

data(swamptrees)
plot(swamptrees$x,swamptrees$y, col=as.numeric(swamptrees$sp),pch=19,
xlab='',ylab='',main='Swamp Trees')

Converting a triangle to the standard basic triangle form form

Description

This function transforms any triangle, tri, to the standard basic triangle form.

The standard basic triangle form is $T_b=T((0,0),(1,0),(c_1,c_2))$ where $c_1$ is in $[0,1/2]$ , $c_2>0$ and $(1-c_1)^2+c_2^2 \le 1$ .

Usage

tri2std.basic.tri(tri)
tri2std.basic.tri(tri)

Arguments

tri

A $3 \times 2$ matrix with each row representing a vertex of the triangle.

Value

A list with two elements

`Cvec`	The nontrivial vertex $C=(c_1,c_2)$ in the standard basic triangle form $T_b$ .
`orig.order`	Row order of the input triangle, `tri`, when converted to the standard basic triangle form $T_b$

Author(s)

Elvan Ceyhan

Examples

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);

tri2std.basic.tri(rbind(A,B,C))
tri2std.basic.tri(rbind(B,C,A))

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
tri2std.basic.tri(rbind(A,B,C))
tri2std.basic.tri(rbind(A,C,B))
tri2std.basic.tri(rbind(B,A,C))

## End(Not run)

## Not run: 
c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);

tri2std.basic.tri(rbind(A,B,C))
tri2std.basic.tri(rbind(B,C,A))

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
tri2std.basic.tri(rbind(A,B,C))
tri2std.basic.tri(rbind(A,C,B))
tri2std.basic.tri(rbind(B,A,C))

## End(Not run)

Points from one class inside the convex hull of the points from the other class

Description

Given two 2D data sets, Xp and Yp, it returns the Xp points inside the convex hull of Yp points.

See (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

Xin.convex.hullY(Xp, Yp)
Xin.convex.hullY(Xp, Yp)

Arguments

`Xp`	A set of 2D points which constitute the data set.
`Yp`	A set of 2D points which constitute the vertices of the Delaunay triangles.

Value

Xp points inside the convex hull of Yp points

Author(s)

Elvan Ceyhan

References

Examples

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

DT<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")

Xlim<-range(Xp[,1],Yp[,1])
Ylim<-range(Xp[,2],Yp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xch<-Xin.convex.hullY(Xp,Yp)

plot(Xp,main=" ", xlab=" ", ylab=" ",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),pch=".",cex=3)
interp::convex.hull(DT,plot.it = TRUE, add = TRUE)  # or try polygon(Yp[ch$i,])
points(Xch,pch=4,col="red")

## End(Not run)

## Not run: 
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

DT<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")

Xlim<-range(Xp[,1],Yp[,1])
Ylim<-range(Xp[,2],Yp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

Xch<-Xin.convex.hullY(Xp,Yp)

plot(Xp,main=" ", xlab=" ", ylab=" ",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),pch=".",cex=3)
interp::convex.hull(DT,plot.it = TRUE, add = TRUE)  # or try polygon(Yp[ch$i,])
points(Xch,pch=4,col="red")

## End(Not run)

Package 'pcds'

Help Index

.onAttach start message

Description

Usage

Arguments

Value

.onLoad getOption package settings

Description

Usage

Arguments

Value

Examples

The angles to draw arcs between two line segments

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

The angle between two line segments

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

The arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for a 2D data set - multiple triangle case

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

The arcs of Arc Slice Proximity Catch Digraph (AS-PCD) for 2D data - one triangle case

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D data - multiple triangle case

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - multiple interval case

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - end interval case

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data - one interval case

Description

Usage

Arguments

Value

Author(s)