Idom.num1PEbasic.tri {pcds}

The indicator for a point being a dominating point or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic 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 for data in the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)), that is, returns 1 if p is a dominating point of PE-PCD, and returns 0 otherwise.

PE proximity regions are defined with respect to the standard basic triangle T_b. In the standard basic triangle, T_b, c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle is useful for simulation studies under the uniformity hypothesis.

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 a standard basic triangle to the edges on the extension of the lines joining M to the vertices or based on the circumcenter of T_b; default is M=(1,1,1), i.e., the center of mass of T_b. 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.

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, 2011)).

Usage

Idom.num1PEbasic.tri(
  p,
  Xp,
  r,
  c1,
  c2,
  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.

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \ge 1.

c1, c2

Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; c_1 must be in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 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 standard basic triangle T_b or the circumcenter of T_b which may be entered as "CC" as well; default is M=(1,1,1), i.e., the center of mass of T_b.

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 T_b.

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.

See Also

Idom.num1ASbasic.tri and Idom.num1AStri

Examples

c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)
r<-2

P<-c(.4,.2)
Idom.num1PEbasic.tri(P,Xp,r,c1,c2,M)
Idom.num1PEbasic.tri(Xp[1,],Xp,r,c1,c2,M)

Idom.num1PEbasic.tri(c(1,1),Xp,r,c1,c2,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since point p=c(1,1) is not a data point in Xp

#or try
Rv<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv
Idom.num1PEbasic.tri(Xp[1,],Xp,r,c1,c2,M,Rv)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEbasic.tri(Xp[i,],Xp,r,c1,c2,M))}

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

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

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

if (identical(M,circumcenter.tri(Tb)))
{
  plot(Tb,pch=".",asp=1,xlab="",ylab="",axes=TRUE,
  xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
  polygon(Tb)
  points(Xp,pch=1,col=1)
  Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2)
} else
{plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
  polygon(Tb)
  points(Xp,pch=1,col=1)
  Ds<-prj.cent2edges.basic.tri(c1,c2,M)}
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)

txt<-rbind(Tb,M,Ds)
xc<-txt[,1]+c(-.02,.02,.02,-.02,.03,-.03,.01)
yc<-txt[,2]+c(.02,.02,.02,-.02,.02,.02,-.03)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1PEbasic.tri(c(.2,.1),Xp,r,c1,c2,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since point p is not a data point in Xp