Idom.num2PEtri {pcds}

The indicator for two points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, and returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL) and point, p2, is in the region of vertex rv2 (default is NULL); vertices (and hence rv1 and rv2) are labeled as 1,2,3 in the order they are stacked row-wise in tri.

PE proximity regions are defined with respect to the triangle tri 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 circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri.

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would be a dominating set if they actually were in the data set.

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

Usage

Idom.num2PEtri(
  p1,
  p2,
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

p1, p2

Two 2D points to be tested for constituting a dominating set 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.

rv1, rv2

The indices of the vertices whose regions contains p1 and p2, respectively. They take the vertex labels as 1,2,3 as in the row order of the vertices in tri (default is NULL for both).

ch.data.pnts

A logical argument for checking whether points p1 and p2 are data points in Xp or not (default is FALSE).

Value

I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, 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.num2PEbasic.tri, Idom.num2AStri, and Idom.num2PEtetra

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.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2PEtri(Xp[i,],Xp[j,],Xp,Tr,r,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.tri(Xp[1,],Tr,M)$rv;
rv2<-rel.vert.tri(Xp[2,],Tr,M)$rv
Idom.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M,rv1,rv2)

Idom.num2PEtri(Xp[1,],c(1,2),Xp,Tr,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp