Idom.num1PEtri {pcds}

The indicator for a point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp in the triangle tri, that is, returns 1 if p is a dominating point of PE-PCD, and returns 0 otherwise.

Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as 1,2,3 in the order they are stacked row-wise in tri.

PE proximity region is constructed with respect to the triangle tri with expansion parameter r \ge 1 and vertex regions are based on center M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of the triangle tri or based on the circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri.

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

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

Usage

Idom.num1PEtri(p, Xp, tri, r, M = c(1, 1, 1), rv = NULL, ch.data.pnt = FALSE)

Arguments

p

A 2D point that is to be tested for being a dominating point or not of the PE-PCD.

Xp

A set of 2D points which constitutes the vertices of the PE-PCD.

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 \ge 1.

M

A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle tri or the circumcenter of tri which may be entered as "CC" as well; default is M=(1,1,1), i.e., the center of mass of tri.

rv

Index of the vertex whose region contains point p, rv takes the vertex labels as 1,2,3 as in the row order of the vertices in tri.

ch.data.pnt

A logical argument for checking whether point p is a data point in Xp or not (default is FALSE).

Value

I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, and returns 0 otherwise.

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.

See Also

Idom.num1PEbasic.tri and Idom.num1AStri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5  #try also r<-2

Idom.num1PEtri(Xp[1,],Xp,Tr,r,M)
Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M)
Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M,ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEtri(Xp[i,],Xp,Tr,r,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.tri(Xp[1,],Tr,M)$rv
Idom.num1PEtri(Xp[1,],Xp,Tr,r,M,Rv)

Ds<-prj.cent2edges(Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

Xlim<-range(Tr[,1],Xp[,1],M[1])
Ylim<-range(Tr[,2],Xp[,2],M[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=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Tr,M,Ds)
xc<-txt[,1]+c(-.02,.03,.02,-.02,.04,-.03,.0)
yc<-txt[,2]+c(.02,.02,.05,-.03,.04,.06,-.07)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

P<-c(1.4,1)
Idom.num1PEtri(P,P,Tr,r,M)
Idom.num1PEtri(Xp[1,],Xp,Tr,r,M)

Idom.num1PEtri(c(1,2),Xp,Tr,r,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since p is not a data point