| PEdom.num.binom.test {pcds} | R Documentation |
A test of segregation/association based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - Binomial Approximation
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
i.e., cluster around the centers of the Delaunay triangles)
and association (where Xp points cluster around Yp points)
based on the (asymptotic) binomial 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 topic of research of the author of the package.
See also (Ceyhan (2011)).
Usage
PEdom.num.binom.test(
Xp,
Yp,
r,
ch.cor = FALSE,
ndt = NULL,
alternative = c("two.sided", "less", "greater"),
conf.level = 0.95
)
Arguments
Xp |
A set of 2D points which constitute the vertices 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 in |
ch.cor |
A logical argument for convex hull correction,
default |
ndt |
Number of Delaunay triangles based on |
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 |
estimate |
A |
null.value |
Hypothesized value for the parameter,
i.e., the null value for |
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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.
See Also
Examples
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))
plotDelaunay.tri(Xp,Yp,xlab="",ylab="")
PEdom.num.binom.test(Xp,Yp,r) #try also #PEdom.num.binom.test(Xp,Yp,r,alt="l") and
# PEdom.num.binom.test(Xp,Yp,r,alt="g")
PEdom.num.binom.test(Xp,Yp,r,ch=TRUE)
#or try
ndt<-num.delaunay.tri(Yp)
PEdom.num.binom.test(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