prj.nondegPEcent2edges {pcds}

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

Description

Returns the projections from center cent to the edges on the extension of the lines joining cent to the vertices in the triangle, tri. Here M is one of the three centers which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in tri for a given expansion parameter r in (1,1.5]. The center label cent values 1,2,3 correspond to the vertices M_1, M_2, and M_3 (i.e., row numbers in the output of center.nondegPE(tri,r)); default for cent is 1. cent becomes center of mass CM for r=1.5.

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

Usage

prj.nondegPEcent2edges(tri, r, cent = 1)

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.

cent

Index of the center (as 1,2,3 corresponding to M_1,\,M_2,\,M_3) which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in tri for expansion parameter r in (1,1.5]; default cent=1.

Value

Three projection points (stacked row-wise) from one of the centers (as 1,2,3 corresponding to M_1,\,M_2,\,M_3) which gives nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in tri for expansion parameter r in (1,1.5].

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

prj.cent2edges.basic.tri and prj.cent2edges

Examples

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

prj.nondegPEcent2edges(Tr,r,cent=2)

Ms<-center.nondegPE(Tr,r)
M1=Ms[1,]

Ds<-prj.nondegPEcent2edges(Tr,r,cent=1)

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

plot(Tr,pch=".",xlab="",ylab="",
main="Projections from a non-degeneracy center\n to the edges of the 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,.03,.02)
yc<-Tr[,2]+c(-.02,.04,.04)
txt.str<-c("A","B","C")
text(xc,yc,txt.str)

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

points(Ds,pch=4,col=2)
L<-rbind(M1,M1,M1); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2,lwd=2,col=4)
txt<-Ds
xc<-txt[,1]+c(-.02,.04,-.04)
yc<-txt[,2]+c(-.02,.04,.04)
txt.str<-c("D1","D2","D3")
text(xc,yc,txt.str)

prj.nondegPEcent2edges(Tr,r,cent=3)
#gives an error message if center index, cent, is different from 1, 2 or 3
prj.nondegPEcent2edges(Tr,r=1.49,cent=2)
#gives an error message if r>1.5