Idom.num1CSstd.tri {pcds}

The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns I(p is a dominating point of the CS-PCD) where the vertices of the CS-PCD are the 2D data set Xp in the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)), that is, returns 1 if p is a dominating point of CS-PCD, returns 0 otherwise.

CS proximity region is constructed with respect to T_e with expansion parameter t>0 and edge regions are based on center of mass CM=(1/2,\sqrt{3}/6).

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, 2010)).

Usage

Idom.num1CSstd.tri(p, Xp, t, ch.data.pnt = FALSE)

Arguments

p

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

Xp

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

t

A positive real number which serves as the expansion parameter in CS proximity region.

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 CS-PCD) where the vertices of the CS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, 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 (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

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

Idom.num1CSt1std.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
Te<-rbind(A,B,C);
t<-1.5
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num1CSstd.tri(Xp[3,],Xp,t)
Idom.num1CSstd.tri(c(1,2),c(1,2),t)
Idom.num1CSstd.tri(c(1,2),c(1,2),t,ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSstd.tri(Xp[i,],Xp,t))}

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

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

plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE);
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(Te,CM)
xc<-txt[,1]+c(-.02,.02,.01,.05)
yc<-txt[,2]+c(.02,.02,.03,.02)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

Idom.num1CSstd.tri(c(1,2),Xp,t,ch.data.pnt = FALSE)
#gives an error if ch.data.pnt = TRUE message since p is not a data point