Package 'pcds.ugraph'

Description

.onAttach start message

Usage

.onAttach(libname, pkgname)

Arguments

Value

invisible()

Description

.onLoad getOption package settings

Usage

.onLoad(libname, pkgname)

Arguments

Value

invisible()

Examples

getOption("pcds.ugraph.name")

Description

Returns the edge density of the underlying or reflexivity graph of Arc Slice Proximity Catch Digraphs (AS-PCDs) whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

AS proximity regions are 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 edges, loops are not allowed so edges are only possible for points inside tri for this function.

in.tri.only is a logical argument (default is FALSE) for considering only the points inside the triangle or all the points as the vertices of the digraph. if in.tri.only=TRUE, edge density is computed only for the points inside the triangle (i.e., edge density of the subgraph of the underlying or reflexivity graph induced by the vertices in the triangle is computed), otherwise edge density of the entire graph (i.e., graph with all the vertices) is computed.

See also (Ceyhan (2005, 2016)).

Usage

ASedge.dens.tri(
  Xp,
  tri,
  M = "CC",
  ugraph = c("underlying", "reflexivity"),
  in.tri.only = FALSE
)

Arguments

Value

Edge density of the underlying or reflexivity graphs based on the AS-PCD whose vertices are the 2D numerical data set, Xp; AS proximity regions are defined with respect to the triangle tri 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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

PEedge.dens.tri, CSedge.dens.tri, and ASarc.dens.tri

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

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

#For the underlying graph
(num.edgesAStri(Xp,Tr,M)$num.edges)/(n*(n-1)/2)
ASedge.dens.tri(Xp,Tr,M)
ASedge.dens.tri(Xp,Tr,M,in.tri.only = TRUE)

#For the reflexivity graph
(num.edgesAStri(Xp,Tr,M,ugraph="r")$num.edges)/(n*(n-1)/2)
ASedge.dens.tri(Xp,Tr,M,ugraph="r")
ASedge.dens.tri(Xp,Tr,M,in.tri.only = TRUE,ugraph="r")

## End(Not run)

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 edge density of the underlying or reflexivity graph of CS-PCD for uniform 2D data.

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 edge 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, edge density of underlying or reflexivity graph 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 have a substantial overlap. Currently, the Xp points outside the convex hull of Yp points are handled with a correction factor which is derived under the assumption of uniformity of Xp and Yp points in the study window, (see the description below for the argument ch.cor and the function code.) However, in the special case of no Xp points in the convex hull of Yp points, edge 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, 2016)) for more on the test based on the edge density of underlying or reflexivity graphs of CS-PCDs.

Usage

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

Arguments

Value

A list with the elements

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

PEedge.dens.test and CSarc.dens.test

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))

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

CSedge.dens.test(Xp,Yp,t=1.5)
CSedge.dens.test(Xp,Yp,t=1.5,ch=TRUE)

CSedge.dens.test(Xp,Yp,t=1.5,ugraph="r")
CSedge.dens.test(Xp,Yp,t=1.5,ugraph="r",ch=TRUE)
#since Y points are not uniform, convex hull correction is invalid here


## End(Not run)

Description

Returns the edge density of the underlying or reflexivity graphs of Central Similarity Proximity Catch Digraphs (CS-PCDs) 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 edge density standardized by the mean and asymptotic variance of the edge density of the underlying or reflexivity graphs of CS-PCD for uniform data in the triangle tri only when M is the center of mass. For the number of edges, loops are not allowed.

in.tri.only is a logical argument (default is FALSE) for considering only the points inside the triangle or all the points as the vertices of the digraph. if in.tri.only=TRUE, edge density is computed only for the points inside the triangle (i.e., edge density of the subgraph of the underlying or reflexivity graph induced by the vertices in the triangle is computed), otherwise edge density of the entire graph (i.e., graph with all the vertices) is computed.

See also (Ceyhan (2005, 2016)).

Usage

CSedge.dens.tri(
  Xp,
  tri,
  t,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity"),
  in.tri.only = FALSE
)

Arguments

Value

A list with the elements

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

ASedge.dens.tri, PEedge.dens.tri, and CSarc.dens.tri

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

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

#For the underlying graph
num.edgesCStri(Xp,Tr,t=1.5,M)$num.edges
CSedge.dens.tri(Xp,Tr,t=1.5,M)
CSedge.dens.tri(Xp,Tr,t=1.5,M,in.tri.only = TRUE)

#For the reflexivity graph
num.edgesCStri(Xp,Tr,t=1.5,M,ugraph="r")$num.edges
CSedge.dens.tri(Xp,Tr,t=1.5,M,ugraph="r")
CSedge.dens.tri(Xp,Tr,t=1.5,M,in.tri.only = TRUE,ugraph="r")

## End(Not run)

Description

An object of class "UndPCDs". Returns edges of the underlying or reflexivity graph of AS-PCD as left and right end points and related parameters and the quantities of these graphs. The vertices of these graphs 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, i.e., AS proximity regions are defined only for Xp points inside the convex hull of Yp points. That is, edges may exist for points only inside the convex hull of Yp points. It also provides various descriptions and quantities about the edges of the AS-PCD such as number of edges, edge 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. 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. 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 edges, loops are not allowed so edges are only possible for points inside the convex hull of Yp points.

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

Usage

edgesAS(Xp, Yp, M = "CC", ugraph = c("underlying", "reflexivity"))

Arguments

Value

A list with the elements

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

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.

See Also

edgesAStri, edgesPE, edgesCS, and arcsAS

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)

Edges<-edgesAS(Xp,Yp,M) #or try Edges<-edgesAS(Xp,Yp,M,ugraph="r")
#or try with the default center Edges<-edgesAS(Xp,Yp)
Edges
summary(Edges)
plot(Edges)

## End(Not run)

Description

An object of class "UndPCDs". Returns edges of the underlying or reflexivity graph of AS-PCD as left and right end points and related parameters and the quantities of these graphs. The vertices of these graphs are the data points in Xp in the multiple triangle case.

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

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="CC", i.e., 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, 2016)).

Usage

edgesAStri(Xp, tri, M = "CC", ugraph = c("underlying", "reflexivity"))

Arguments

Value

A list with the elements

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

edgesAS, edgesPEtri, edgesCStri, and arcsAStri

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

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

#for underlying graph
Edges<-edgesAStri(Xp,Tr,M)
#for reflexivity graph, try Edges<-edgesAStri(Xp,Tr,M,ugraph="r")
#or try with the default center Edges<-edgesAStri(Xp,Tr); M= (Edges$param)$cent
Edges
summary(Edges)
plot(Edges)

#for reflexivity graph
Edges<-edgesAStri(Xp,Tr,M,ugraph="r")
#or try with the default center Edges<-edgesAStri(Xp,Tr); M= (Edges$param)$cent
Edges
summary(Edges)
plot(Edges)

#can add vertex regions
#but we first need to determine center is the circumcenter or not,
#see the description for more detail.
CC<-pcds::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<-pcds::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)

Description

An object of class "UndPCDs". Returns edges of the underlying or reflexivity graph of CS-PCD as left and right end points and related parameters and the quantities of these graphs. The vertices of these graphs 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 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 edges, loops are not allowed so edges are only possible for points inside the convex hull of Yp points.

See (Ceyhan (2005, 2016)) for 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

edgesCS(Xp, Yp, t, M = c(1, 1, 1), ugraph = c("underlying", "reflexivity"))

Arguments

Value

A list with the elements

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

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.

See Also

edgesCStri, edgesAS, edgesPE, and arcsCS

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

Edges<-edgesCS(Xp,Yp,t,M)
#or try with the default center Edges<-edgesCS(Xp,Yp,t)
Edges
summary(Edges)
plot(Edges)

Edges<-edgesCS(Xp,Yp,t,M,ugraph="r")
#or try with the default center Edges<-edgesCS(Xp,Yp,t)
Edges
summary(Edges)
plot(Edges)

## End(Not run)

Description

An object of class "UndPCDs". Returns edges of the underlying or reflexivity graph of CS-PCD as left and right end points and related parameters and the quantities of these graphs. The vertices of these graphs are the data points in Xp in the multiple triangle case.

CS proximity regions are constructed with respect to the triangle tri with expansion parameter $t > 0$, i.e., edges may exist only for points inside tri. It also provides various descriptions and quantities about the edges of the underlying or reflexivity graphs of the CS-PCD such as number of edges, edge 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. With any interior center M, the edge regions are constructed using the extensions of the lines combining vertices with M.

See also (Ceyhan (2005, 2016)).

Usage

edgesCStri(Xp, tri, t, M = c(1, 1, 1), ugraph = c("underlying", "reflexivity"))

Arguments

Value

A list with the elements

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

edgesCS, edgesAStri, edgesPEtri, and arcsCStri

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

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

t<-1.5  #try also t<-2

#for underlying graph
Edges<-edgesCStri(Xp,Tr,t,M)
#or try with the default center Edges<-edgesCStri(Xp,Tr,t); M= (Edges$param)$cent
Edges
summary(Edges)
plot(Edges)

#for reflexivity graph
Edges<-edgesCStri(Xp,Tr,t,M,ugraph="r")
#or try with the default center Edges<-edgesCStri(Xp,Tr,t); M= (Edges$param)$cent
Edges
summary(Edges)
plot(Edges)

#can add edge regions
cent<-M
cent.name<-"M"
Ds<-pcds::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
txr<-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)

Description

An object of class "UndPCDs". Returns edges of the underlying or reflexivity graph of PE-PCD as left and right end points and related parameters and the quantities of these graphs. The vertices of these graphs 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 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). 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. 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 edges, loops are not allowed so edges are only possible for points inside the convex hull of Yp points.

See (Ceyhan (2005, 2016)) 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

edgesPE(Xp, Yp, r, M = c(1, 1, 1), ugraph = c("underlying", "reflexivity"))

Arguments

Value

A list with the elements

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

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.

See Also

edgesPEtri, edgesAS, edgesCS, and arcsPE

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

Edges<-edgesPE(Xp,Yp,r,M) #try also Edges<-edgesPE(Xp,Yp,r,M,ugraph="r")
#or try with the default center Edges<-edgesPE(Xp,Yp,r)
Edges
summary(Edges)
plot(Edges)

## End(Not run)

Description

An object of class "UndPCDs". Returns edges of the underlying or reflexivity graph of PE-PCD as left and right end points and related parameters and the quantities of these graphs. The vertices of these graphs are the data points in Xp in the multiple triangle case.

PE proximity regions are constructed with respect to the triangle tri with expansion parameter $r \ge 1$, i.e., edges may exist only for points inside tri. It also provides various descriptions and quantities about the edges of the underlying or reflexivity graph of the PE-PCD such as number of edges, edge 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. 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. M-vertex regions are recommended spatial inference, due to geometry invariance property of the edge density and domination number the PE-PCDs based on uniform data.

See also (Ceyhan (2005, 2016)).

Usage

edgesPEtri(Xp, tri, r, M = c(1, 1, 1), ugraph = c("underlying", "reflexivity"))

Arguments

Value

A list with the elements

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

edgesPE, edgesAStri, edgesCStri, and arcsPEtri

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

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

r<-1.5  #try also r<-2

#for underlying graph
Edges<-edgesPEtri(Xp,Tr,r,M)
#or try with the default center Edges<-edgesPEtri(Xp,Tr,r); M= (Edges$param)$cent
Edges
summary(Edges)
plot(Edges)

#for reflexivity graph
Edges<-edgesPEtri(Xp,Tr,r,M,ugraph="r")
#or try with the default center Edges<-edgesPEtri(Xp,Tr,r); M= (Edges$param)$cent
Edges
summary(Edges)
plot(Edges)

#can add vertex regions
#but we first need to determine center is the circumcenter or not,
#see the description for more detail.
CC<-pcds::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<-pcds::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)

Description

The mean and (asymptotic) variance functions for the underlying or reflexivity graphs of Central Similarity Proximity Catch Digraphs (CS-PCDs): muOrCS2D and asyvarOrCS2D for the underlying graph and muAndCS2D and asyvarAndCS2D for the reflexivity graph.

muOrCS2D and muAndCS2D return the mean of the (edge) density of the underlying or reflexivity graphs of CS-PCDs, respectively, for 2D uniform data in a triangle. Similarly, asyvarOrCS2D and asyvarAndCS2D return the asymptotic variance of the edge density of the underlying or reflexivity graphs of CS-PCDs, respectively, for 2D uniform data in a triangle.

CS proximity regions are defined with expansion parameter $t > 0$ with respect to the triangle in which the points reside and edge regions are based on center of mass, $CM$ of the triangle.

See also (Ceyhan (2016)).

Usage

muOrCS2D(t)

muAndCS2D(t)

mu.undCS2D(t, ugraph = c("underlying", "reflexivity"))

asyvarOrCS2D(t)

asyvarAndCS2D(t)

asy.var.undCS2D(t, ugraph = c("underlying", "reflexivity"))

Arguments

Value

mu.undCS2D returns the mean and asyvarUndOrCS2D returns the (asymptotic) variance of the edge density of the underlying graph of the CS-PCD for uniform data in any triangle if ugraph="underlying", and those of the reflexivity graph if ugraph="reflexivity". The functions muOrCS2D, muAndCS2D, asyvarOrCS2D, and asyvarAndCS2D are the corresponding mean and asymptotic variance functions for the edge density of the reflexivity graph of the CS-PCD, respectively, for uniform data in any triangle.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

mu.undCS2D, asy.var.undCS2D muCS2D, and asyvarCS2D,

Examples

## Not run: 
mu.undCS2D(1.2)
mu.undCS2D(1.2,ugraph="r")

tseq<-seq(0.01,10,by=.05)
ltseq<-length(tseq)

muOR = muAND <- vector()
for (i in 1:ltseq)
{
  muOR<-c(muOR,mu.undCS2D(tseq[i]))
  muAND<-c(muAND,mu.undCS2D(tseq[i],ugraph="r"))
}

plot(tseq, muOR,type="l",xlab="t",ylab=expression(mu(t)),lty=1,
     xlim=range(tseq),ylim=c(0,1))
lines(tseq,muAND,type="l",lty=2,col=2)
legend("bottomright", inset=.02,
       legend=c(expression(mu[or](t)),expression(mu[and](t))),
       lty=1:2,col=1:2)

## End(Not run)

## Not run: 
asy.var.undCS2D(1.2)
asy.var.undCS2D(1.2,ugraph="r")

asyvarOrCS2D(.2)

tseq<-seq(.05,25,by=.05)
ltseq<-length(tseq)

avarOR<-avarAND<-vector()
for (i in 1:ltseq)
{
  avarOR<-c(avarOR,asy.var.undCS2D(tseq[i]))
  avarAND<-c(avarAND,asy.var.undCS2D(tseq[i],ugraph="r"))
}

par(mar=c(5,5,4,2))
plot(tseq, 4*avarAND,type="l",lty=2,col=2,xlab="t",
     ylab=expression(paste(sigma^2,"(t)")),xlim=range(tseq))
lines(tseq,4*avarOR,type="l")
legend(18,.1,
       legend=c(expression(paste(sigma["underlying"]^"2","(t)")),
                 expression(paste(sigma["reflexivity"]^"2","(t)")) ),
       lty=1:2,col=1:2)

## End(Not run)

Description

The mean and (asymptotic) variance functions for the underlying or reflexivity graph of Proportional Edge Proximity Catch Digraphs (PE-PCDs): muOrPE2D and asyvarOrPE2D for the underlying graph and muAndPE2D and asyvarAndPE2D for the reflexivity graph.

muOrPE2D and muAndPE2D return the mean of the (edge) density of the underlying or reflexivity graph of PE-PCDs, respectively, for 2D uniform data in a triangle. Similarly, asyvarOrPE2D and asyvarAndPE2D return the asymptotic variance of the edge density of the underlying or reflexivity graph of PE-PCDs, respectively, 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.

See also (Ceyhan (2016)).

Usage

muOrPE2D(r)

muAndPE2D(r)

mu.undPE2D(r, ugraph = c("underlying", "reflexivity"))

asyvarOrPE2D(r)

asyvarAndPE2D(r)

asy.var.undPE2D(r, ugraph = c("underlying", "reflexivity"))

Arguments

Value

mu.undPE2D returns the mean and asyvarUndOrPE2D returns the (asymptotic) variance of the edge density of the underlying graph of the PE-PCD for uniform data in any triangle if ugraph="underlying", and those of the reflexivity graph if ugraph="reflexivity". The functions muOrPE2D, muAndPE2D, asyvarOrPE2D, and asyvarAndPE2D are the corresponding mean and asymptotic variance functions for the edge density of the reflexivity graph of the PE-PCD, respectively, for uniform data in any triangle.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

#' mu.undCS2D, asy.var.undCS2D, muPE2D, asyvarPE2D, muAndCS2D, and asyvarAndCS2D

Examples

## Not run: 
mu.undPE2D(1.2)
mu.undPE2D(1.2,ugraph="r")

rseq<-seq(1.01,5,by=.05)
lrseq<-length(rseq)

muOR = muAND <- vector()
for (i in 1:lrseq)
{
  muOR<-c(muOR,mu.undPE2D(rseq[i]))
  muAND<-c(muAND,mu.undPE2D(rseq[i],ugraph="r"))
}

plot(rseq, muOR,type="l",xlab="r",ylab=expression(mu(r)),lty=1,
     xlim=range(rseq),ylim=c(0,1))
lines(rseq,muAND,type="l",lty=2,col=2)
legend("bottomright", inset=.02,
       legend=c(expression(mu[or](r)),expression(mu[and](r))),
       lty=1:2,col=1:2)

## End(Not run)

## Not run: 
asy.var.undPE2D(1.2)
asy.var.undPE2D(1.2,ugraph="r")

rseq<-seq(1.01,5,by=.05)
lrseq<-length(rseq)

avarOR<-avarAND<-vector()
for (i in 1:lrseq)
{
  avarOR<-c(avarOR,asy.var.undPE2D(rseq[i]))
  avarAND<-c(avarAND,asy.var.undPE2D(rseq[i],ugraph="r"))
}

par(mar=c(5,5,4,2))
plot(rseq, avarAND,type="l",lty=2,col=2,xlab="r",
     ylab=expression(paste(sigma^2,"(r)")),xlim=range(rseq))
lines(rseq,avarOR,type="l")
legend(3.75,.02,
       legend=c(expression(paste(sigma["underlying"]^"2","(r)")),
                 expression(paste(sigma["reflexivity"]^"2","(r)")) ),
       lty=1:2,col=1:2)

## End(Not run)

Description

Returns $I($p1p2 is an edge in the underlying or reflexivity graph of AS-PCDs $)$ for points p1 and p2 in the standard basic triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator for the AS-PCD underlying graph, that is, returns 1 if p2 is in $N_{AS}(p1)$ **or** p1 is in $N_{AS}(p2)$, returns 0 otherwise. On the other hand, when ugraph="reflexivity", it returns the edge indicator for the AS-PCD reflexivity graph, that is, returns 1 if p2 is in $N_{AS}(p1)$ **and** p1 is in $N_{AS}(p2)$, returns 0 otherwise.

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 $(1-c_1)^2+c_2^2 \leq 1$.

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="CC", i.e., circumcenter of $T_b$.

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)).

Usage

IedgeASbasic.tri(
  p1,
  p2,
  c1,
  c2,
  M = "CC",
  ugraph = c("underlying", "reflexivity")
)

Arguments

Value

Returns 1 if there is an edge between points p1 and p2 in the underlying or reflexivity graph of AS-PCDs in the standard basic triangle, and 0 otherwise.

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

IedgeAStri, IedgeCSbasic.tri, IedgePEbasic.tri and IarcASbasic.tri

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(pcds::runif.basic.tri(1,c1,c2)$g)
set.seed(4)
P1<-as.numeric(pcds::runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(pcds::runif.basic.tri(1,c1,c2)$g)
IedgeASbasic.tri(P1,P2,c1,c2,M)
IedgeASbasic.tri(P1,P2,c1,c2,M,ugraph = "reflexivity")

P1<-c(.4,.2)
P2<-c(.5,.26)
IedgeASbasic.tri(P1,P2,c1,c2,M)
IedgeASbasic.tri(P1,P2,c1,c2,M,ugraph="r")

## End(Not run)

Description

Returns $I($p1p2 is an edge in the underlying or reflexivity graph of AS-PCDs $)$ for points p1 and p2 in a given triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator for the AS-PCD underlying graph, that is, returns 1 if p2 is in $N_{AS}(p1)$ **or** p1 is in $N_{AS}(p2)$, returns 0 otherwise. On the other hand, when ugraph="reflexivity", it returns the edge indicator for the AS-PCD reflexivity graph, that is, returns 1 if p2 is in $N_{AS}(p1)$ **and** p1 is in $N_{AS}(p2)$, returns 0 otherwise.

In both cases AS 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 the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri.

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, 2016)).

Usage

IedgeAStri(p1, p2, tri, M = "CC", ugraph = c("underlying", "reflexivity"))

Arguments

Value

Returns 1 if there is an edge between points p1 and p2 in the underlying or reflexivity graph of AS-PCDs in a given triangle tri, and 0 otherwise.

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

IedgeASbasic.tri, IedgePEtri, IedgeCStri and IarcAStri

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
M<-as.numeric(pcds::runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0);

n<-3
set.seed(1)
Xp<-pcds::runif.tri(n,Tr)$g

IedgeAStri(Xp[1,],Xp[3,],Tr,M)
IedgeAStri(Xp[1,],Xp[3,],Tr,M,ugraph = "reflexivity")

set.seed(1)
P1<-as.numeric(pcds::runif.tri(1,Tr)$g)
P2<-as.numeric(pcds::runif.tri(1,Tr)$g)
IedgeAStri(P1,P2,Tr,M)
IedgeAStri(P1,P2,Tr,M,ugraph="r")

## End(Not run)

Description

Returns $I($p1p2 is an edge in the underlying or reflexivity graph of CS-PCDs $)$ for points p1 and p2 in the standard basic triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator for the CS-PCD underlying graph, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ or p1 is in $N_{CS}(p2,t)$, returns 0 otherwise. On the other hand, when ugraph="reflexivity", it returns the edge indicator for the CS-PCD reflexivity graph, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ and p1 is in $N_{CS}(p2,t)$, returns 0 otherwise.

In both cases $N_{CS}(x,t)$ is the CS proximity region for point $x$ with expansion parameter $t > 0$. 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 $T_b$.

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)).

Usage

IedgeCSbasic.tri(
  p1,
  p2,
  t,
  c1,
  c2,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

Arguments

Value

Returns 1 if there is an edge between points p1 and p2 in the underlying or reflexivity graph of CS-PCDs in the standard basic triangle, and 0 otherwise.

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

IedgeCStri, IedgeASbasic.tri, IedgePEbasic.tri and IarcCSbasic.tri

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(pcds::runif.basic.tri(1,c1,c2)$g)

t<-1.5

P1<-as.numeric(pcds::runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(pcds::runif.basic.tri(1,c1,c2)$g)
IedgeCSbasic.tri(P1,P2,t,c1,c2,M)
IedgeCSbasic.tri(P1,P2,t,c1,c2,M,ugraph = "reflexivity")

P1<-c(.4,.2)
P2<-c(.5,.26)
IedgeCSbasic.tri(P1,P2,t=2,c1,c2,M)
IedgeCSbasic.tri(P1,P2,t=2,c1,c2,M,ugraph="a")

## End(Not run)

Description

Returns $I($p1p2 is an edge in the underlying or reflexivity graph of CS-PCDs $)$ for points p1 and p2 in the standard equilateral triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator for points p1 and p2 in the standard equilateral triangle, for the CS-PCD underlying graph, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ or p1 is in $N_{CS}(p2,t)$, returns 0 otherwise. On the other hand, when ugraph="reflexivity", it returns the edge indicator for points p1 and p2 in the standard equilateral triangle, for the CS-PCD reflexivity graph, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ and p1 is in $N_{CS}(p2,t)$, returns 0 otherwise.

In both cases $N_{CS}(x,t)$ is the CS proximity region for point $x$ with expansion parameter $t > 0$. 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 $T_e$.

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)).

Usage

IedgeCSstd.tri(
  p1,
  p2,
  t,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

Arguments

Value

Returns 1 if there is an edge between points p1 and p2 in the underlying or reflexivity graph of CS-PCDs in the standard equilateral triangle, and 0 otherwise.

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

IedgeCSbasic.tri, IedgeCStri, and IarcCSstd.tri

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<-pcds::runif.std.tri(n)$gen.points

M<-as.numeric(pcds::runif.std.tri(1)$g)  #try also M<-c(.6,.2)

IedgeCSstd.tri(Xp[1,],Xp[3,],t=1.5,M)
IedgeCSstd.tri(Xp[1,],Xp[3,],t=1.5,M,ugraph="reflexivity")

P1<-c(.4,.2)
P2<-c(.5,.26)
t<-2
IedgeCSstd.tri(P1,P2,t,M)
IedgeCSstd.tri(P1,P2,t,M,ugraph = "reflexivity")

## End(Not run)

Description

Returns $I($p1p2 is an edge in the underlying or reflexivity graph of CS-PCDs $)$ for points p1 and p2 in a given triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator for the CS-PCD underlying graph, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ or p1 is in $N_{CS}(p2,t)$, returns 0 otherwise. On the other hand, when ugraph="reflexivity", it returns the edge indicator for the CS-PCD reflexivity graph, that is, returns 1 if p2 is in $N_{CS}(p1,t)$ and p1 is in $N_{CS}(p2,t)$, returns 0 otherwise.

In both cases 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; default is $M=(1,1,1)$, i.e., the center of mass of tri.

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, 2016)).

Usage

IedgeCStri(
  p1,
  p2,
  tri,
  t,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

Arguments

Value

Returns 1 if there is an edge between points p1 and p2 in the underlying or reflexivity graph of CS-PCDs in a given triangle tri, and 0 otherwise.

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

IedgeCSbasic.tri, IedgeAStri, IedgePEtri and IarcCStri

Examples

## Not run: 
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
M<-as.numeric(pcds::runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0);

t<-1.5
n<-3
set.seed(1)
Xp<-pcds::runif.tri(n,Tr)$g

IedgeCStri(Xp[1,],Xp[2,],Tr,t,M)
IedgeCStri(Xp[1,],Xp[2,],Tr,t,M,ugraph = "reflexivity")

P1<-as.numeric(pcds::runif.tri(1,Tr)$g)
P2<-as.numeric(pcds::runif.tri(1,Tr)$g)
IedgeCStri(P1,P2,Tr,t,M)
IedgeCStri(P1,P2,Tr,t,M,ugraph="r")

## End(Not run)

Description

Returns $I($p1p2 is an edge in the underlying or reflexivity graph of PE-PCDs $)$ for points p1 and p2 in the standard basic triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator for the PE-PCD underlying graph, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ **or** p1 is in $N_{PE}(p2,r)$, returns 0 otherwise. On the other hand, when ugraph="reflexivity", it returns the edge indicator for the PE-PCD reflexivity graph, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ **and** p1 is in $N_{PE}(p2,r)$, returns 0 otherwise.

In both cases $N_{PE}(x,r)$ 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 $T_b$.

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)).

Usage

IedgePEbasic.tri(
  p1,
  p2,
  r,
  c1,
  c2,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

Arguments

Value

Returns 1 if there is an edge between points p1 and p2 in the underlying or reflexivity graph of PE-PCDs in the standard basic triangle, and 0 otherwise.

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

IedgePEtri, IedgeASbasic.tri, IedgeCSbasic.tri and IarcPEbasic.tri

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(pcds::runif.basic.tri(1,c1,c2)$g)

r<-1.5
set.seed(4)

P1<-as.numeric(pcds::runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(pcds::runif.basic.tri(1,c1,c2)$g)
IedgePEbasic.tri(P1,P2,r,c1,c2,M)
IedgePEbasic.tri(P1,P2,r,c1,c2,M,ugraph = "reflexivity")

P1<-c(.4,.2)
P2<-c(.5,.26)
IedgePEbasic.tri(P1,P2,r,c1,c2,M)
IedgePEbasic.tri(P1,P2,r,c1,c2,M,ugraph = "reflexivity")

## End(Not run)

Description

Returns $I($p1p2 is an edge in the underlying or reflexivity graph of PE-PCDs $)$ for points p1 and p2 in the standard equilateral triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator for points p1 and p2 in the standard equilateral triangle, for the PE-PCD underlying graph, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ **or** p1 is in $N_{PE}(p2,r)$, returns 0 otherwise. On the other hand, when ugraph="reflexivity", it returns the edge indicator for points p1 and p2 in the standard equilateral triangle, for the PE-PCD reflexivity graph, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ **and** p1 is in $N_{PE}(p2,r)$, returns 0 otherwise.

In both cases $N_{PE}(x,r)$ 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 $T_e$.

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)).

Usage

IedgePEstd.tri(
  p1,
  p2,
  r,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

Arguments

Value

Returns 1 if there is an edge between points p1 and p2 in the underlying or reflexivity graph of PE-PCDs in the standard equilateral triangle, and 0 otherwise.

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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

IedgePEbasic.tri, IedgePEtri, and IarcPEstd.tri

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<-pcds::runif.std.tri(n)$gen.points

M<-as.numeric(pcds::runif.std.tri(1)$g) #try also M<-c(.6,.2)

IedgePEstd.tri(Xp[1,],Xp[3,],r=1.5,M)
IedgePEstd.tri(Xp[1,],Xp[3,],r=1.5,M,ugraph="reflexivity")

P1<-c(.4,.2)
P2<-c(.5,.26)
r<-2
IedgePEstd.tri(P1,P2,r,M)
IedgePEstd.tri(P1,P2,r,M,ugraph = "reflexivity")

## End(Not run)

Description

Returns $I($p1p2 is an edge in the underlying or reflexivity graph of PE-PCDs $)$ for points p1 and p2 in a given triangle.

More specifically, when the argument ugraph="underlying", it returns the edge indicator for the PE-PCD underlying graph, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ or p1 is in $N_{PE}(p2,r)$, returns 0 otherwise. On the other hand, when ugraph="reflexivity", it returns the edge indicator for the PE-PCD reflexivity graph, that is, returns 1 if p2 is in $N_{PE}(p1,r)$ and p1 is in $N_{PE}(p2,r)$, returns 0 otherwise.

In both cases 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.