VS1.1 - Example: An Artificial 2D Dataset
First we load the pcds package:
We start our exposition of pcds functions for
testing/detecting spatial interaction between classes (or species) in
\(\mathbb R^2\) (i.e. 2D space) using
two data sets, an artificial data and a real-life forestry data (see
file “VS1_2_SwampTrees” for the latter).
Illustration of PCDs on an Artificial 2D Dataset
This data set consists of simulated points from two classes, \(\mathcal{X}\) and \(\mathcal{Y}\), where \(\mathcal{X}\) points are uniformly distributed on the unit square \([0,1]^2\), while \(\mathcal{Y}\) points are chosen closer to the vertices of the unit square for better illustration. Here \(n_x\) is the size of class \(\mathcal{X}\) points, \(n_y\) is the size of class \(\mathcal{Y}\) points, and for better visualization of certain structures and graph constructs.
nx<-10; ny<-5; #try also nx<-40; ny<-10 or nx<-1000; ny<-20;
set.seed(123)
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))We take \(n_x=\) 10 and \(n_y=\) 5 (however, one is encouraged to try
the specifications that follow in the comments after “#try also” in the
commented script here and henceforth.) \(\mathcal{X}\) points are denoted as
Xp and \(\mathcal{Y}\)
points are denoted as Yp in the following scripts.
The scatterplot of \(\mathcal{X}\) points (black circles) and \(\mathcal{Y}\) points (red triangles) can be obtained with the below code.
XYpts = rbind(Xp,Yp) #combined Xp and Yp
lab=c(rep(1,nx),rep(2,ny))
lab.fac=as.factor(lab)
plot(XYpts,col=lab,pch=lab,xlab="x",ylab="y",main="Scatterplot of 2D Points from Two Classes")The PCDs are constructed with vertices from \(\mathcal{X}\) points and the binary relation that determines the arcs are based on proximity regions which depend on the Delaunay triangulation of \(\mathcal{Y}\) points. More specifically, the proximity regions are defined with respect to the Delaunay triangles based on \(\mathcal{Y}\) points and vertex regions in each triangle are based on the center \(M\) or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of each Delaunay triangle.
Convex hull of \(\mathcal{Y}\) points is partitioned by the Delaunay triangles constructed with the same \(\mathcal{Y}\) points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of \(\mathcal{Y}\) points).
See Ceyhan (2005), Ceyhan (2010), and Ceyhan (2012) for more on AS-PCDs. Also see Okabe et al. (2000), Ceyhan (2010), and Sinclair (2016) for more on Delaunay triangulation and the corresponding algorithm (to compute the triangulation).
Below we plot the \(\mathcal{X}\) points together with the Delaunay triangulation of \(\mathcal{Y}\) points.
Xlim<-range(Xp[,1],Yp[,1])
Ylim<-range(Xp[,2],Yp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
plot(Xp,xlab="x", ylab="y",xlim=Xlim+xd*c(-.05,.05),
ylim=Ylim+yd*c(-.05,.05),pch=".",cex=3,main="X points and Delaunay Triangulation of Y Points")
#now, we add the Delaunay triangulation based on $Y$ points
DT<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove")
interp::plot.triSht(DT, add=TRUE, do.points = TRUE)
The scatterplot of the X points in the artificial data set together with the Delaunay triangulation of \(Y\) points (dashed lines).
Or, alternatively, we can use the plotDelaunay.tri
function in pcds to obtain the same plot by executing
plotDelaunay.tri(Xp,Yp,xlab="x",ylab="y",main="X points and Delaunay Triangulation of Y Points")
command.
The number of Delaunay triangles based on \(\mathcal{Y}\) points can be obtained by the
function num.delaunay.tri.
Summary and Visualization with Arc-Slice PCDs
For AS-PCDs, the default center used to construct the vertex regions
is M="CC" i.e., circumcenter in each triangle.
Number of arcs of the AS-PCD can be computed by the function
num.arcsAS, which is an object of class
“NumArcs” and takes the arguments
Xp, a set of 2D points which constitute the vertices of the AS-PCD (i.e., class \(\mathcal{X}\) points),Yp, a set of 2D points which constitute the vertices of the Delaunay triangles (i.e., class \(\mathcal{Y}\) points),M, the center of the triangle used to construct the vertex regions."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 isM="CC"i.e., the circumcenter of each triangle.
Its call (with Narcs in the below script)
just returns the description of the digraph. Its summary
returns a description of the digraph, number of arcs of the AS-PCD,
number of data (Xp) points in the convex hull of
Yp (nontarget) points, number of data points in the
Delaunay triangles based on Yp points, numbers of arcs in
the induced subdigraphs in the Delaunay triangles, areas of the Delaunay
triangles, indices of the vertices of the Delaunay triangles, indices of
the Delaunay triangles data points resides. The plot
function (i.e., plot.NumArcs) returns the plot of the
Delaunay triangulation of Yp points, scatter plot of the
Xp points and the number of arcs of the induced subdigraphs
for each Delaunay triangle in the centroid of the triangle.
This function returns the following list as output:
desc: A description of the PCD and the outputnum.arcs: Total number of arcs in all triangles, i.e., the number of arcs for the entire AS-PCDtri.num.arcs: The vector of the number of arcs of the component of the AS-PCD in the Delaunay triangles based on \(\mathcal{Y}\) pointsnum.in.conv.hull: Number of \(\mathcal{X}\) points in the convex hull of \(\mathcal{Y}\) pointsnum.in.tris: The vector of number of \(\mathcal{X}\) points in the Delaunay triangles based on \(\mathcal{Y}\) pointsweight.vec: The vector of the areas of Delaunay triangles based on \(\mathcal{Y}\) pointsdel.tri.ind: Indices of Delaunay triangles based on \(\mathcal{Y}\) points, each column is the vector of indices of the vertices of one triangle.data.tri.ind: A vector of indices of Delaunay triangles in which data points reside, i.e., column number ofdel.tri.indfor each \(\mathcal{X}\) point.tess.points: Triangulation points (i.e.Yppoints)vertices: vertices of the PCD (i.e.Xppoints)
Narcs = num.arcsAS(Xp,Yp,M)
Narcs
#> Call:
#> num.arcsAS(Xp = Xp, Yp = Yp, M = M)
#>
#> Description:
#> Number of Arcs of the AS-PCD with vertices Xp and Related Quantities for the Induced Subdigraphs for the Points in the Delaunay Triangles
summary(Narcs)
#> Call:
#> num.arcsAS(Xp = Xp, Yp = Yp, M = M)
#>
#> Description of the output:
#> Number of Arcs of the AS-PCD with vertices Xp and Related Quantities for the Induced Subdigraphs for the Points in the Delaunay Triangles
#>
#> Number of data (Xp) points in the convex hull of Yp (nontarget) points = 7
#> Number of data points in the Delaunay triangles based on Yp points = 2 1 1 3
#> Number of arcs in the entire digraph = 3
#> Numbers of arcs in the induced subdigraphs in the Delaunay triangles = 0 0 0 3
#> Areas of the Delaunay triangles (used as weights in the arc density of multi-triangle case):
#> 0.2214646 0.2173192 0.2593852 0.2648197
#>
#> Indices of the vertices of the Delaunay triangles (each column refers to a triangle):
#> [,1] [,2] [,3] [,4]
#> [1,] 1 5 3 3
#> [2,] 3 2 4 1
#> [3,] 2 3 5 4
#>
#> Indices of the Delaunay triangles data points resides:
#> 1 4 1 3 NA NA 4 NA 4 2
plot(Narcs)The incidence matrix of the AS-PCD can be found by
inci.matAS. Below, we only print the top 6 rows and columns
of the incidence matrix.
IM<-inci.matAS(Xp,Yp,M)
IM[1:6,1:6]
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 0 0 0 0 0
#> [2,] 0 1 0 0 0 0
#> [3,] 0 0 1 0 0 0
#> [4,] 0 0 0 1 0 0
#> [5,] 0 0 0 0 1 0
#> [6,] 0 0 0 0 0 1Technical aside: Once we have the incidence matrix
of a digraph (or a graph), we can find an exact or an approximate
dominating set and hence the exact or approximate domination number of
the digraph (or graph). The function dom.num.greedy finds
the approximate domination number (an upper bound) and also provides the
indices of the points in the approximate dominating set. On the other
hand, the function dom.num.exact finds the exact domination
number and the indices of the points in an exact dominating set.
dom.num.exact might take a long time for large \(n_x\) (e.g. \(n_x
\ge 19\)), as it checks all possible subsets of the dataset to
find a minimum dominating set.
dom.num.greedy(IM) #try also dom.num.exact(IM) #this might take a longer time for large nx (i.e. nx >= 19)
#> $approx.dom.num
#> [1] 8
#>
#> $ind.approx.mds
#> [1] 9 1 10 5 8 4 3 6Plot of the arcs in the digraph AS-PCD is provided by the function
plotASarcs, which take the arguments
Xp,YpandMare same as in the functionnum.arcsAS,asp, a numeric value, giving the aspect ratio for \(y\) axis to \(x\)-axis \(y/x\) (default isNA); see the official help page foraspby typing “? asp”,mainan overall title for the plot (default=NULL),xlab,ylabtitles for the \(x\) and \(y\) axes, respectively (default=NULLfor both),xlim,ylim, two numeric vectors of length 2, giving the \(x\)- and \(y\)-coordinate ranges (default=NULLfor both), and..., additionalplotparameters.
For all the plots for AS-PCD, we use the option asp=1 so
that the circles actually do look like circles (i.e., the arc-slices
which are the boundary of the circles restricted to triangles look
circular as they should).
The arcs of the AS-PCD for the 2D artificial data set using the CC-vertex regions together with the Delaunay triangles based on the \(Y\) points (dashed lines).
Plot of the AS proximity regions is provided by the function
plotASregs, which has the same arguments as the function
plotASarcs.
The AS proximity regions for all \(X\) points in the 2D artificial data set using the CC-vertex regions together with the Delaunay triangles based on the \(Y\) points (dashed lines).
The function arcsAS is an object of class
“PCDs” and has the same arguments as in
num.arcsAS. Its call (with Arcs
in the below script) just returns the description of the digraph. Its
summary returns a description of the digraph, selected tail
(or source) points of the arcs in the digraph (first 6 or fewer are
printed), selected head (or end) points of the arcs in the digraph
(first 6 or fewer are printed), the parameters of the digraph (here it
is only the center, “CC”), and various quantities of the digraph
(namely, the number of vertices, number of partition points, number of
triangles, number of arcs, and arc density. The plot
function (i.e., plot.PCDs) returns the same plot as in
plotASarcs, i.e., the plot of the arcs in the digraph
together with the Delaunay triangles based on the \(\mathcal{Y}\) points, hence we comment it
out below.
Arcs<-arcsAS(Xp,Yp,M)
Arcs
#> Call:
#> arcsAS(Xp = Xp, Yp = Yp, M = M)
#>
#> Type:
#> [1] "Arc Slice Proximity Catch Digraph (AS-PCD) for 2D Points in Multiple Triangles with CC-Vertex Regions"
summary(Arcs)
#> Call:
#> arcsAS(Xp = Xp, Yp = Yp, M = M)
#>
#> Type of the digraph:
#> [1] "Arc Slice Proximity Catch Digraph (AS-PCD) for 2D Points in Multiple Triangles with CC-Vertex Regions"
#>
#> Vertices of the digraph = Xp
#> Partition points of the region = Yp
#>
#> Selected tail (or source) points of the arcs in the digraph
#> (first 6 or fewer are printed)
#> [,1] [,2]
#> [1,] 0.5281055 0.2460877
#> [2,] 0.5514350 0.3279207
#> [3,] 0.5514350 0.3279207
#>
#> Selected head (or end) points of the arcs in the digraph
#> (first 6 or fewer are printed)
#> [,1] [,2]
#> [1,] 0.5514350 0.3279207
#> [2,] 0.7883051 0.4533342
#> [3,] 0.5281055 0.2460877
#>
#> Parameters of the digraph
#> $center
#> [1] "CC"
#>
#> Various quantities of the digraph
#> number of vertices number of partition points
#> 7.00000000 5.00000000
#> number of triangles number of arcs
#> 4.00000000 3.00000000
#> arc density
#> 0.07142857
plot(Arcs, asp=1)Summary and Visualization with Proportional Edge PCDs
The functions for PE-PCD have similar arguments as the AS-PCDs except
(i) PE-PCDs have the additional argument for the expansion
parameter \(r \ge 1\) and (ii) for
PE-PCDs, the default center used to construct the vertex regions is
M="CM" i.e., center of mass of each triangle.
Number of arcs of the PE-PCD can be computed by the function
num.arcsPE which is an object of class
“NumArcs”. The function takes arguments
Xp,Yp,r,M where r is the expansion parameter
and returns the same type of output as the function
num.arcsAS.
Narcs = num.arcsPE(Xp,Yp,r,M)
summary(Narcs)
#> Call:
#> num.arcsPE(Xp = Xp, Yp = Yp, r = r, M = M)
#>
#> Description of the output:
#> Number of Arcs of the PE-PCD with vertices Xp and Related Quantities for the Induced Subdigraphs for the Points in the Delaunay Triangles
#>
#> Number of data (Xp) points in the convex hull of Yp (nontarget) points = 7
#> Number of data points in the Delaunay triangles based on Yp points = 2 1 1 3
#> Number of arcs in the entire digraph = 3
#> Numbers of arcs in the induced subdigraphs in the Delaunay triangles = 1 0 0 2
#> Areas of the Delaunay triangles (used as weights in the arc density of multi-triangle case):
#> 0.2214646 0.2173192 0.2593852 0.2648197
#>
#> Indices of the vertices of the Delaunay triangles (each column refers to a triangle):
#> [,1] [,2] [,3] [,4]
#> [1,] 1 5 3 3
#> [2,] 3 2 4 1
#> [3,] 2 3 5 4
#>
#> Indices of the Delaunay triangles data points resides:
#> 1 4 1 3 NA NA 4 NA 4 2
plot(Narcs)The incidence matrix of the PE-PCD can be found by
inci.matPE.
Once the incidence matrix is found, approximate and exact dominating
sets and hence domination numbers can be found by the functions
dom.num.greedy and dom.num.exact,
respectively.
Plot of the arcs in the digraph can be obtained by
plotPEarcs.
The arcs of the PE-PCD for the 2D artificial data set using the CM-vertex regions and expansion parameter \(r=1.5\) together with the Delaunay triangles based on the \(Y\) points (dashed lines).
Plot of the PE proximity regions can be obtained by
plotPEregs.
The PE proximity regions for all the points the 2D artificial data set using the CM-vertex regions and expansion parameter \(r=1.5\) together with the Delaunay triangles based on the \(Y\) points (dashed lines).
The function arcsPE is an object of class
“PCDs”. Its call, summary, and
plot are as in arcsAS with the addition of the
expansion parameter (see the Parameters of the digraph
part) in the summary.PCDs. The plot function
returns the same plot as in plotPEarcs, hence we comment it
out below.
Arcs<-arcsPE(Xp,Yp,r,M)
Arcs
#> Call:
#> arcsPE(Xp = Xp, Yp = Yp, r = r, M = M)
#>
#> Type:
#> [1] "Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D points in Multiple Triangles with Expansion parameter r = 1.5 and Center M = (1,1,1)"
summary(Arcs)
#> Call:
#> arcsPE(Xp = Xp, Yp = Yp, r = r, M = M)
#>
#> Type of the digraph:
#> [1] "Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D points in Multiple Triangles with Expansion parameter r = 1.5 and Center M = (1,1,1)"
#>
#> Vertices of the digraph = Xp
#> Partition points of the region = Yp
#>
#> Selected tail (or source) points of the arcs in the digraph
#> (first 6 or fewer are printed)
#> [,1] [,2]
#> [1,] 0.4089769 0.6775706
#> [2,] 0.5281055 0.2460877
#> [3,] 0.5514350 0.3279207
#>
#> Selected head (or end) points of the arcs in the digraph
#> (first 6 or fewer are printed)
#> [,1] [,2]
#> [1,] 0.2875775 0.9568333
#> [2,] 0.5514350 0.3279207
#> [3,] 0.5281055 0.2460877
#>
#> Parameters of the digraph
#> $center
#> [1] 1 1 1
#>
#> $`expansion parameter`
#> [1] 1.5
#>
#> Various quantities of the digraph
#> number of vertices number of partition points
#> 7.00000000 5.00000000
#> number of triangles number of arcs
#> 4.00000000 3.00000000
#> arc density
#> 0.07142857
plot(Arcs)Testing Spatial Interaction with the PE-PCDs
We can test the spatial pattern or interaction of segregation/association in the 2D setting based on arc density or domination number of PE-PCDs.
The Use of Arc Density of PE-PCDs for Testing Spatial Interaction
We can test the spatial interaction between two classes or species
based on the arc density of PE-PCDs using the function
PEarc.dens.test which takes the arguments
Xp, a set of 2D points which constitute the vertices of the AS-PCD (i.e., class \(\mathcal{X}\) points),Yp, a set of 2D points which constitute the vertices of the Delaunay triangles (i.e., class \(\mathcal{Y}\) points),ra 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, defaultch.cor=FALSE, recommended when bothXpandYphave the same rectangular support.alternative, type of the alternative hypothesis in the test, one of"two.sided","less","greater".conf.level, the level of the confidence interval, default is0.95, for the arc density of PE-PCD based on the 2D data setXp.
This function is an object of class “htest” (i.e.,
hypothesis test) and 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 PE-PCD for uniform 2D data utilizing the
asymptotic normality of the \(U\)-statistics.
The function returns 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 here), 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 PE-PCD whose
vertices are Xp points equals to its expected value under
the uniform distribution and alternative could be
“two-sided" (i.e.,”two.sided”), or "left-sided" (i.e.,“less”for the case in which $\X$ points are accumulated around the $\Y$ points, or association) or "right-sided" (i.e.,“greater”`
for the case in which \(\mathcal{X}\)
points are accumulated around the centers of the triangles whose
vertices are from the \(\mathcal{Y}\)
points, or segregation).
See Ceyhan et al. (2006) and Ceyhan (2014) for more detail on the arc density of PE-PCD and its use for testing 2D spatial interactions. We only provide the two-sided test below, although both one-sided versions are also available.
PEarc.dens.test(Xp,Yp,r) #try also PEarc.dens.test(Xp,Yp,r,alt="l") or with alt="g"
#>
#> Large Sample z-Test Based on Arc Density of PE-PCD for Testing
#> Uniformity of 2D Data ---
#> without Convex Hull Correction
#>
#> data: Xp
#> standardized arc density (i.e., Z) = -0.21983, p-value = 0.826
#> alternative hypothesis: true (expected) arc density is not equal to 0.09712203
#> 95 percent confidence interval:
#> 0.04234726 0.14084889
#> sample estimates:
#> arc density
#> 0.09159807The Use of Domination Number of PE-PCDs for Testing Spatial Interaction
We first provide two functions to compute the domination number of
PE-PCDs: PEdom.num and PEdom.num.nondeg.
The function PEdom.num takes the same arguments as
num.arcsPE and returns a list with three
elements as output:
dom.num, the overall domination number of the PE-PCD whose vertices areXppoints,ind.mds, the data indices of the minimum dominating set of the PE-PCD whose vertices areXppoints,tri.dom.nums, the vector of domination numbers of the PE-PCD components for the Delaunay triangles.
This function takes any center in the interior of the triangles as its argument or circumcenter (“CC”). The 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).
On the other hand, PEdom.num.nondeg takes only the
arguments Xp,Yp,r and returns the same output as in
PEdom.num function, but uses one of the non-degeneracy
centers in the multiple triangle case (hence M is not an
argument for this function). That is, the center M is one
of the three 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 \(r=1.5\).
These two functions are different from the function
dom.num.greedy since they give an exact minimum dominating
set and the exact domination number and from dom.num.exact,
since they give a minimum dominating set and number in polynomial time
(in the number of vertices of the digraph, i.e., number of
Xp points).
PEdom.num(Xp,Yp,r,M) #try also PEdom.num(Xp,Yp,r=2,M)
#> $dom.num
#> [1] 5
#>
#> $ind.mds
#> [1] 3 10 4 9 2
#>
#> $tri.dom.nums
#> [1] 1 1 1 2
PEdom.num.nondeg(Xp,Yp,r) #try also PEdom.num.nondeg(Xp,Yp,r=1.25)
#> $dom.num
#> [1] 5
#>
#> $ind.mds
#> [1] 3 10 4 2 9
#>
#> $tri.dom.nums
#> [1] 1 1 1 2We can test the spatial patterns of segregation or association based
on domination number of PE-PCD using the functions
PEdom.num.binom.test or PEdom.num.norm.test.
Each of these functions is an object of class “htest”
(i.e., hypothesis test) and performs the same hypothesis test as in
PEarc.dens.test. Both functions take the arguments
Xp,Yp,r,ch.cor,ndt,alternative,conf.level where
ndt is the number of Delaunay triangles based on
Yp points with default NULL and other
variables are as in PEarc.dens.test. They return the test
statistic, \(p\)-value for the
corresponding alternative, the confidence interval,
estimate and null value for the parameter of interest (which is \(P(\mbox{domination number}\le 2)\)), 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.,
\(P(\mbox{domination number}\le 2)\))
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).
In this case, the PE proximity region is constructed with the
expansion parameter \(r \in (1,1.5]\)
and \(M\)-vertex regions where
M is a center that yields non-degenerate asymptotic
distribution of the domination number.
The test statistic in PEdom.num.binom.test 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). For this approximation to work, number of
Xp points must be at least 7 times more than number of
Yp points.
We only provide the two-sided tests below, although both one-sided versions are also available.
PEdom.num.binom.test(Xp,Yp,r) #try also PEdom.num.binom.test(Xp,Yp,r,alt="g") or with alt="l"
#>
#> Large Sample Binomial Test based on the Domination Number of PE-PCD for
#> Testing Uniformity of 2D Data ---
#> without Convex Hull Correction
#>
#> data: Xp
#> # of times domination number is <= 2 = 4, p-value = 0.5785
#> alternative hypothesis: true Pr(Domination Number <=2) is not equal to 0.7413
#> 95 percent confidence interval:
#> 0.3976354 1.0000000
#> sample estimates:
#> domination number || Pr(domination number <= 2)
#> 5 1The test statistic in PEdom.num.norm.test is based on
the normal approximation to the binomial distribution. 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 number of
Yp points.
We only provide the two-sided tests below, although both one-sided versions are also available.
PEdom.num.norm.test(Xp,Yp,r) #try also PEdom.num.norm.test(Xp,Yp,r,alt="g") or with alt="l"
#>
#> Normal Approximation to the Domination Number of PE-PCD for Testing
#> Uniformity of 2D Data ---
#> without Convex Hull Correction
#>
#> data: Xp
#> standardized domination number (i.e., Z) = 1.1815, p-value = 0.2374
#> alternative hypothesis: true expected domination number is not equal to 2.9652
#> 95 percent confidence interval:
#> 3.283383 6.716617
#> sample estimates:
#> domination number || Pr(domination number <= 2)
#> 5 1See Ceyhan and Priebe (2007), Ceyhan (2011), and Ceyhan (2012) for more on the domination number of PE-PCDs and their use in testing spatial interaction.
In all the test functions (based on arc density and domination
number) above, the option ch.cor is for convex hull
correction (default is “no convex hull correction”, i.e.,
ch.cor=FALSE) which is recommended when Xp
points tend to be mostly within the support of the Yp
points. When the symmetric difference of the supports of \(\mathcal{X}\) and \(\mathcal{Y}\) is non-negligible, the tests
can be adjusted to account for the \(\mathcal{X}\) points outside the convex
hull of \(\mathcal{Y}\) points with the
option ch.cor=TRUE. For example,
PEarc.dens.test(Xp,Yp,r,ch=TRUE) would yield the convex
hull corrected version of the arc-based test of spatial interaction.
Summary and Visualization with Central Similarity PCDs
The functions for CS-PCD have similar arguments as the
num.arcsPE with the expansion parameter \(t >0\). For CS-PCDs, the default center
used to construct the edge regions is M="CM" i.e., center
of mass of each triangle.
The functions for CS-PCD have similar arguments as the PE-PCDs except
(i) the expansion parameter is\(t>0\) and (ii) for CS-PCDs, the default
center used to construct the edge regions is M="CM" i.e.,
center of mass of each triangle.
Number of arcs of the CS-PCD can be computed by the function
num.arcsCS, which is an object of class
“NumArcs” and takes similar arguments and returns the
similar output items as in num.arcsPE.
Narcs = num.arcsCS(Xp,Yp,tau,M)
summary(Narcs)
#> Call:
#> num.arcsCS(Xp = Xp, Yp = Yp, t = tau, M = M)
#>
#> Description of the output:
#> Number of Arcs of the CS-PCD with vertices Xp and Related Quantities for the Induced Subdigraphs for the Points in the Delaunay Triangles
#>
#> Number of data (Xp) points in the convex hull of Yp (nontarget) points = 7
#> Number of data points in the Delaunay triangles based on Yp points = 2 1 1 3
#> Number of arcs in the entire digraph = 3
#> Numbers of arcs in the induced subdigraphs in the Delaunay triangles = 1 0 0 2
#> Areas of the Delaunay triangles (used as weights in the arc density of multi-triangle case):
#> 0.2214646 0.2173192 0.2593852 0.2648197
#>
#> Indices of the vertices of the Delaunay triangles (each column refers to a triangle):
#> [,1] [,2] [,3] [,4]
#> [1,] 1 5 3 3
#> [2,] 3 2 4 1
#> [3,] 2 3 5 4
#>
#> Indices of the Delaunay triangles data points resides:
#> 1 4 1 3 NA NA 4 NA 4 2
#>
#plot(Narcs)The incidence matrix of the CS-PCD can be found by
inci.matCS. Below, we only print the top 6 rows of the
incidence matrix. Once the incidence matrix is found, approximate and
exact domination numbers can be found by the functions
dom.num.greedy and dom.num.exact,
respectively.
Plot of the arcs in the digraph can be obtained by the function
plotCSarcs.
The arcs of the CS-PCD for the 2D artificial data set using the CM-edge regions and expansion parameter \(t=1.5\) together with the Delaunay triangles based on the \(Y\) points (dashed lines).
Plot of the CS proximity regions can be obtained by the function
plotCSregs.
The CS proximity regions for all the points the 2D artificial data set using the CM-edge regions and expansion parameter \(t=1.5\) together with the Delaunay triangles based on the \(Y\) points (dashed lines).
The function arcsCS is an object of class
“PCDs”. Its arguments and the output for call,
summary, and plot are as in
arcsPE. The plot function returns the same
plot as in plotCSarcs, hence we comment it out below.
Arcs<-arcsCS(Xp,Yp,tau,M)
Arcs
#> Call:
#> arcsCS(Xp = Xp, Yp = Yp, t = tau, M = M)
#>
#> Type:
#> [1] "Central Similarity Proximity Catch Digraph (CS-PCD) for 2D Points in the Multiple Triangles with Expansion Parameter t = 1.5 and Center M = (1,1,1)"
summary(Arcs)
#> Call:
#> arcsCS(Xp = Xp, Yp = Yp, t = tau, M = M)
#>
#> Type of the digraph:
#> [1] "Central Similarity Proximity Catch Digraph (CS-PCD) for 2D Points in the Multiple Triangles with Expansion Parameter t = 1.5 and Center M = (1,1,1)"
#>
#> Vertices of the digraph = Xp
#> Partition points of the region = Yp
#>
#> Selected tail (or source) points of the arcs in the digraph
#> (first 6 or fewer are printed)
#> [,1] [,2]
#> [1,] 0.4089769 0.6775706
#> [2,] 0.5281055 0.2460877
#> [3,] 0.5514350 0.3279207
#>
#> Selected head (or end) points of the arcs in the digraph
#> (first 6 or fewer are printed)
#> [,1] [,2]
#> [1,] 0.2875775 0.9568333
#> [2,] 0.5514350 0.3279207
#> [3,] 0.5281055 0.2460877
#>
#> Parameters of the digraph
#> $center
#> [1] 1 1 1
#>
#> $`expansion parameter`
#> [1] 1.5
#>
#> Various quantities of the digraph
#> number of vertices number of partition points
#> 7.00000000 5.00000000
#> number of triangles number of arcs
#> 4.00000000 3.00000000
#> arc density
#> 0.07142857
plot(Arcs)Testing Spatial Interaction with the CS-PCDs
We can test the spatial pattern or interaction of segregation/association based on arc density of CS-PCDs (as the distribution of the domination number of CS-PCDs is still a topic of ongoing work).
The Use of Arc Density of CS-PCDs for Testing Spatial Interaction
We can test the spatial interaction between two classes or species
based on the arc density of CS-PCDs using the function
CSarc.dens.test. This function is an object of class
“htest” and performs the same hypothesis test as in Section
@ref(sec:testing-arti-data-PE-PCD)
Moreover, it takes similar arguments and returns similar output as in
PEarc.dens.test function, for the same null and alternative
hypotheses. See also Ceyhan et al. (2007)
and Ceyhan (2014) for more detail.
We only provide the two-sided test below, although both one-sided versions are also available.
CSarc.dens.test(Xp,Yp,tau) #try also CSarc.dens.test(Xp,Yp,tau,alt="l") or with alt="g"
#>
#> Large Sample z-Test Based on Arc Density of CS-PCD for Testing
#> Uniformity of 2D Data ---
#> without Convex Hull Correction
#>
#> data: Xp
#> standardized arc density (i.e., Z) = 0.6039, p-value = 0.5459
#> alternative hypothesis: true (expected) arc density is not equal to 0.06749794
#> 95 percent confidence interval:
#> 0.0252619 0.1473522
#> sample estimates:
#> arc density
#> 0.08630702As in the tests based on PE-PCD, it is possible to account for \(\mathcal{X}\) points outside the convex
hull of \(\mathcal{Y}\) points, with
the option ch.cor=TRUE. For example,
CSarc.dens.test(Xp,Yp,tau,ch=TRUE) would yield the convex
hull corrected version of the arc-based test of spatial interaction.
References