IarcCSedge.reg.std.tri {pcds}

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise, where N_{CS}(x,t) is the CS proximity region for point x with expansion parameter t>0. This function is equivalent to IarcCSstd.tri, except that it computes the indicator using the functions IarcCSstd.triRAB, IarcCSstd.triRBC and IarcCSstd.triRAC which are edge-region specific indicator functions. For example, IarcCSstd.triRAB computes I(p2 is in N_{CS}(p1,t)) for points p1 and p2 when p1 resides in the edge region of edge AB.

CS proximity region is defined with respect to the standard equilateral triangle T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) and edge regions are based on the center M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of T_e; default is M=(1,1,1) i.e., the center of mass of T_e. re is the index of the edge region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_e, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

IarcCSedge.reg.std.tri(p1, p2, t, M = c(1, 1, 1), re = NULL)

Arguments

p1

A 2D point whose CS proximity region is constructed.

p2

A 2D point. The function determines whether p2 is inside the CS proximity region of p1 or not.

t

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

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 equilateral triangle T_e; default is M=(1,1,1) i.e. the center of mass of T_e.

re

The index of the edge region in T_e containing the point, either 1,2,3 or NULL (default is NULL).

Value

I(p2 is in N_{CS}(p1,t)) for p1, that is, returns 1 if p2 is in N_{CS}(p1,t), 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 (2014). “Comparison of Relative Density of Two Random Geometric Digraph Families in Testing Spatial Clustering.” TEST, 23(1), 100-134.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

See Also

IarcCStri and IarcPEstd.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-3

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

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M)
IarcCSstd.tri(Xp[1,],Xp[2,],t,M)

#or try
re<-rel.edge.std.triCM(Xp[1,])$re
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M,re=re)