| CSarc.dens.test {pcds} | R Documentation |
A test of segregation/association based on arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D data
Description
An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial
randomness (CSR) or uniformity of Xp points in the convex hull of Yp points against the alternatives
of segregation (where Xp points cluster away from Yp points) and association (where Xp points cluster around
Yp points) based on the normal approximation of the arc density of the CS-PCD for uniform 2D data
in the convex hull of Yp points.
The function yields the test statistic, p-value for the corresponding alternative,
the confidence interval, estimate and null value for the parameter of interest (which is the arc density),
and method and name of the data set used.
Under the null hypothesis of uniformity of Xp points in the convex hull of Yp points, arc density
of CS-PCD whose vertices are Xp points equals to its expected value under the uniform distribution and
alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association)
or right-sided (i.e., data is accumulated around the centers of the triangles, or segregation).
CS proximity region is constructed with the expansion parameter t>0 and CM-edge regions
(i.e., the test is not available for a general center M at this version of the function).
**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are
assumed to be fixed (i.e., the test is conditional on Yp points).
Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points,
say at least 5 times more.
This test is more appropriate when supports of Xp and Yp has a substantial overlap.
Currently, the Xp points
outside the convex hull of Yp points
are handled with a convex hull correction factor, ch.cor,
which is derived under the assumption of
uniformity of Xp and Yp points in the study window,
(see the description below and the function code.)
However, in the special case of no Xp points in the convex hull of Yp points, arc density is taken to be 1,
as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is
an ongoing line of research of the author of the package.
ch.cor is for convex hull correction (default is "no convex hull correction", i.e., ch.cor=FALSE)
which is recommended when both Xp and Yp have the same rectangular support.
See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).
Usage
CSarc.dens.test(
Xp,
Yp,
t,
ch.cor = FALSE,
alternative = c("two.sided", "less", "greater"),
conf.level = 0.95
)
Arguments
Xp |
A set of 2D points which constitute the vertices of the CS-PCD. |
Yp |
A set of 2D points which constitute the vertices of the Delaunay triangles. |
t |
A positive real number which serves as the expansion parameter in CS proximity region. |
ch.cor |
A logical argument for convex hull correction, default |
alternative |
Type of the alternative hypothesis in the test, one of |
conf.level |
Level of the confidence interval, default is |
Value
A list with the elements
statistic |
Test statistic |
p.value |
The |
conf.int |
Confidence interval for the arc density at the given confidence level |
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 |
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 (2014).
“Comparison of Relative Density of Two Random Geometric Digraph Families in Testing Spatial Clustering.”
TEST, 23(1), 100-134.
Ceyhan E, Priebe CE, Marchette DJ (2007).
“A new family of random graphs for testing spatial segregation.”
Canadian Journal of Statistics, 35(1), 27-50.
See Also
PEarc.dens.test and CSarc.dens.test1D
Examples
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-5; #try also nx<-40; ny<-10 or nx<-1000; ny<-10;
set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))
plotDelaunay.tri(Xp,Yp,xlab="",ylab = "")
CSarc.dens.test(Xp,Yp,t=.5)
CSarc.dens.test(Xp,Yp,t=.5,ch=TRUE)
#try also t=1.0 and 1.5 above