rel.verts.tri.nondegPE {pcds}

The indices of the vertex regions in a triangle that contains the points in a give data set

Description

Returns the indices of the vertices whose regions contain the points in data set Xp in a triangle tri=(A,B,C) and vertex regions are based on the center cent which yields nondegenerate asymptotic distribution of the domination number of PE-PCD for uniform data in tri for expansion parameter r in (1,1.5].

Vertices of triangle tri are labeled as 1,2,3 according to the row number the vertex is recorded if a point in Xp is not inside tri, then the function yields NA as output for that entry. The corresponding vertex region is the polygon with the vertex, cent, and projection points on the edges. The center label cent values 1,2,3 correspond to the vertices M_1, M_2, and M_3; with default 1 (see the examples for an illustration).

See also (Ceyhan (2005, 2011)).

Usage

rel.verts.tri.nondegPE(Xp, tri, r, cent = 1)

Arguments

Xp

A set of 2D points representing the set of data points for which indices of the vertex regions containing them are to be determined.

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

A list with two elements

rv

Indices (i.e., a vector of indices) of the vertices whose region contains points in Xp in the triangle tri

tri

The vertices of the triangle, where row number corresponds to the vertex index in rv.

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 (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

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.

See Also

rel.verts.triCM, rel.verts.triCC, and rel.verts.tri

Examples

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

P<-c(1.4,1.0)
rel.verts.tri.nondegPE(P,Tr,r,cent)

n<-20  #try also n<-40
set.seed(1)
Xp<-runif.tri(n,Tr)$g

rel.verts.tri.nondegPE(Xp,Tr,r,cent)
rel.verts.tri.nondegPE(rbind(Xp,c(2,2)),Tr,r,cent)

rv<-rel.verts.tri.nondegPE(Xp,Tr,r,cent)

M<-center.nondegPE(Tr,r)[cent,];
Ds<-prj.nondegPEcent2edges(Tr,r,cent)

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

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=".",col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty = 2)

xc<-Tr[,1]+c(-.03,.05,.05)
yc<-Tr[,2]+c(-.06,.02,.05)
txt.str<-c("rv=1","rv=2","rv=3")
text(xc,yc,txt.str)

txt<-rbind(M,Ds)
xc<-txt[,1]+c(.02,.04,-.03,0)
yc<-txt[,2]+c(.07,.03,.05,-.07)
txt.str<-c("M","D1","D2","D3")
text(xc,yc,txt.str)
text(Xp,labels=factor(rv$rv))