PEedge.dens.test {pcds.ugraph}

A test of segregation/association based on edge density of underlying or reflexivity graph 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 edge density of the underlying or reflexivity graph of PE-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 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 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 are topics of ongoing 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 graph of PE-PCDs.

Usage

PEedge.dens.test(
  Xp,
  Yp,
  r,
  ugraph = c("underlying", "reflexivity"),
  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 underlying or reflexivity graphs 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.

ugraph

The type of the graph based on PE-PCDs, "underlying" is for the underlying graph, and "reflexivity" is for the reflexivity graph (default is "underlying").

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 edge density of underlying or reflexivity graph 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 edge density at the given confidence level conf.level and depends on the type of alternative.

estimate

Estimate of the parameter, i.e., edge density

null.value

Hypothesized value for the parameter, i.e., the null edge density, which is usually the mean edge 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 (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

CSedge.dens.test and PEarc.dens.test

Examples

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-5;

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))

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

PEedge.dens.test(Xp,Yp,r=1.25)
PEedge.dens.test(Xp,Yp,r=1.25,ch=TRUE)

PEedge.dens.test(Xp,Yp,r=1.25,ugraph="r")
PEedge.dens.test(Xp,Yp,r=1.25,ugraph="r",ch=TRUE)
#since Y points are not uniform, convex hull correction is invalid here