inci.matCSstd.tri {pcds}

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

Usage

inci.matCSstd.tri(Xp, t, M = c(1, 1, 1))

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.

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.

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

inci.matCStri, inci.matCS and inci.matPEstd.tri

Examples

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<-inci.matCSstd.tri(Xp,t=1.25,M)
inc.mat
sum(inc.mat)-n
num.arcsCSstd.tri(Xp,t=1.25)

dom.num.greedy(inc.mat) #try also dom.num.exact(inc.mat)  #might take a long time for large n
Idom.num.up.bnd(inc.mat,1)