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 i.e., cluster around the centers of the Delaunay
triangles) and association (where Xp points cluster around Yp points) based on the (asymptotic) binomial
d distribution of the domination number of 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 \(Pr(\)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., \(Pr(\)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).
PE proximity region is constructed with the expansion parameter \(r \ge 1\) and \(M\)-vertex regions where M is a center that yields non-degenerate asymptotic distribution of the domination number.
The test statistic 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).
That is, the test statistic is based on the domination number for Xp points inside convex hull of Yp points
for the PE-PCD and default convex hull correction, ch.cor, is FALSE
where M is the center that yields nondegenerate asymptotic distribution for the domination number.
For this approximation to work, number of Xp points must be at least 7 times more than number of Yp points.
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 7 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 convex hull correction factor
(see the description below and the function code.)
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.
See also (ceyhan:dom-num-NPE-Spat2011;textualpcds).
TSDomPEBin(
Xp,
Yp,
r,
ch.cor = FALSE,
ndt = NULL,
alternative = c("two.sided", "less", "greater"),
conf.level = 0.95
)A list with the elements
Test statistic
The \(p\)-value for the hypothesis test for the corresponding alternative
Confidence interval for \(Pr(\)Domination Number\(\le 2)\) at the given level conf.level and
depends on the type of alternative.
A vector with two entries: first is is the estimate of the parameter, i.e.,
\(Pr(\)Domination Number\(=3)\) and second is the domination number
Hypothesized value for the parameter, i.e., the null value for \(Pr(\)Domination Number\(\le 2)\)
Type of the alternative hypothesis in the test, one of "two.sided", "less", "greater"
Description of the hypothesis test
Name of the data set
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 in \((1,1.5]\).
A logical argument for convex hull correction, default ch.cor=FALSE,
recommended when both Xp and Yp have the same rectangular support.
Number of Delaunay triangles based on Yp points, default is NULL.
Type of the alternative hypothesis in the test, one of "two.sided", "less", "greater".
Level of the confidence interval, default is 0.95, for the probability of success
(i.e., \(Pr(\)domination number\(=3)\) for PE-PCD whose vertices are the 2D data set Xp.
Elvan Ceyhan
TSDomPENorm
if (FALSE) {
nx<-100; ny<-5 #try also nx<-1000; ny<-10
r<-1.4 #try also r<-1.5
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))
plotDeltri(Xp,Yp,xlab="",ylab="")
TSDomPEBin(Xp,Yp,r) #try also #TSDomPEBin(Xp,Yp,r,alt="l") and # TSDomPEBin(Xp,Yp,r,alt="g")
TSDomPEBin(Xp,Yp,r,ch=TRUE)
#or try
ndt<-num.del.tri(Yp)
TSDomPEBin(Xp,Yp,r,ndt=ndt)
#values might differ due to the random of choice of the three centers M1,M2,M3
#for the non-degenerate asymptotic distribution of the domination number
}
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