pcds (version 0.1.6)

IncMatCSTe: Incidence matrix for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns the incidence matrix for the CS-PCD whose vertices are the given 2D numerical data set, Xp, in the standard equilateral triangle \(T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))\).

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 \(T_e\). Loops are allowed, so the diagonal entries are all equal to 1.

See also (ceyhan:Phd-thesis,ceyhan:arc-density-CS,ceyhan:test2014;textualpcds).

Usage

IncMatCSTe(Xp, t, M = c(1, 1, 1))

Value

Incidence matrix for the CS-PCD with vertices being 2D data set, Xp and CS proximity regions are defined in the standard equilateral triangle \(T_e\) with M-edge regions.

Arguments

Xp

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

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 \(T_e\).

Author

Elvan Ceyhan

References

See Also

IncMatCStri, IncMatCS and IncMatPETe

Examples

if (FALSE) {
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

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

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

inc.mat<-IncMatCSTe(Xp,t=1.25,M)
inc.mat
sum(inc.mat)-n
NumArcsCSTe(Xp,t=1.25)

dom.greedy(inc.mat) #try also dom.exact(inc.mat)  #might take a long time for large n
IndUBdom(inc.mat,1)
}