Pdom.num2PEtri {pcds}

Asymptotic probability that domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs) equals 2 where vertices of the digraph are uniform points in a triangle

Description

Returns P(domination number=2) for PE-PCD for uniform data in a triangle, when the sample size n goes to infinity (i.e., asymptotic probability of domination number = 2).

PE proximity regions are constructed with respect to the triangle with the expansion parameter r \ge 1 and M-vertex regions where M is the vertex that renders the asymptotic distribution of the domination number non-degenerate for the given value of r in (1,1.5].

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011)).

Usage

Pdom.num2PEtri(r)

Arguments

r

A positive real number which serves as the expansion parameter in PE proximity region; must be in (1,1.5] to attain non-degenerate asymptotic distribution for the domination number.

Value

P(domination number=2) for PE-PCD for uniform data on an triangle as the sample size n goes to infinity

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE (2007). “On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs.” Model Assisted Statistics and Applications, 1(4), 231-255.

See Also

Pdom.num2PE1D

Examples

Pdom.num2PEtri(r=1.5)
Pdom.num2PEtri(r=1.4999999999)

Pdom.num2PEtri(r=1.5) / Pdom.num2PEtri(r=1.4999999999)

rseq<-seq(1.01,1.49999999999,l=20)  #try also l=100
lrseq<-length(rseq)

pg2<-vector()
for (i in 1:lrseq)
{
  pg2<-c(pg2,Pdom.num2PEtri(rseq[i]))
}

plot(rseq, pg2,type="l",xlab="r",
ylab=expression(paste("P(", gamma, "=2)")),
     lty=1,xlim=range(rseq)+c(0,.01),ylim=c(0,1))
points(rbind(c(1.50,Pdom.num2PEtri(1.50))),pch=".",cex=3)