TSArcDensPEMT: A test of segregation/association based on arc density of Proportional Edge Proximity Catch Digraph (PE-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 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 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 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). This test is more appropriate when supports of Xp and Yp has a substantial overlap,

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:Phd-thesis,ceyhan:arc-density-PE;textualpcds) for more on the test based on the arc density of PE-PCDs.

Usage

TSArcDensPEMT(
  Xp,
  Yp,
  r,
  ch.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

A set of 2D points which constitute the vertices of the PE-PCD.

A set of 2D points which constitute the vertices of the Delaunay triangles.

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

References

Examples

Run this code

# NOT RUN {
#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<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny),runif(ny))

oldpar <- par(no.readonly = TRUE)
plotDeltri(Xp,Yp,xlab="",ylab="")
par(oldpar)

NumArcsPEMT(Xp,Yp,r=1.25)
TSArcDensPEMT(Xp,Yp,r=1.25)
TSArcDensPEMT(Xp,Yp,r=1.25,ch=TRUE)

NumArcsPEMT(Xp,Yp,r=1.5)
TSArcDensPEMT(Xp,Yp,r=1.5,alt="l")
TSArcDensPEMT(Xp,Yp,r=1.5,ch=TRUE,alt="l")

NumArcsPEMT(Xp,Yp,r=2)
TSArcDensPEMT(Xp,Yp,r=2)
TSArcDensPEMT(Xp,Yp,r=2,ch=TRUE)

r<-2
TSArcDensPEMT(Xp,Yp,r)

Xp<-runif.tri(nx,Yp[1:3,])$g
TSArcDensPEMT(Xp,Yp[1:3,],r)


TSArcDensPEMT(Xp,rbind(Yp,Yp),r)

dat.fr<-data.frame(a=Xp)
TSArcDensPEMT(dat.fr,Yp,r)

dat.fr<-data.frame(a=Yp)
TSArcDensPEMT(Xp,dat.fr,r)

# }