center.nondegPE {pcds}

Centers for non-degenerate asymptotic distribution of domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs)

Description

Returns the centers which yield nondegenerate asymptotic distribution for the domination number of PE-PCD for uniform data in a triangle, tri=T(v_1,v_2,v_3).

PE proximity region is defined with respect to the triangle tri with expansion parameter r in (1,1.5].

Vertex regions are defined with the centers that are output of this function. Centers are stacked row-wise with row number is corresponding to the vertex row number in tri (see the examples for an illustration). The center labels 1,2,3 correspond to the vertices M_1, M_2, and M_3 (which are the three centers for r in (1,1.5) which becomes center of mass for r=1.5.).

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

center.nondegPE(tri, r)

Arguments

tri

A 3 \times 2 matrix with each row representing a vertex of the triangle.

r

A positive real number which serves as the expansion parameter in PE proximity region; must be in (1,1.5] for this function.

Value

The centers (stacked row-wise) which give nondegenerate asymptotic distribution for the domination number of PE-PCD for uniform data in a triangle, tri.

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

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.

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
r<-1.35

Ms<-center.nondegPE(Tr,r)
Ms

Xlim<-range(Tr[,1])
Ylim<-range(Tr[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",
main="Centers of nondegeneracy\n for the PE-PCD in a triangle",
axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Ms,pch=".",col=1)
polygon(Ms,lty = 2)

xc<-Tr[,1]+c(-.02,.02,.02)
yc<-Tr[,2]+c(.02,.02,.03)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

xc<-Ms[,1]+c(-.04,.04,.03)
yc<-Ms[,2]+c(.02,.02,.05)
txt.str<-c("M1","M2","M3")
text(xc,yc,txt.str)