inci.matPEtri {pcds}

Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns the incidence matrix for the PE-PCD whose vertices are the given 2D numerical data set, Xp, in the triangle tri=T(v=1,v=2,v=3).

PE proximity regions are constructed with respect to triangle tri with expansion parameter r \ge 1 and vertex 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 the triangle tri; default is M=(1,1,1), i.e., the center of mass of tri. Loops are allowed, so the diagonal entries are all equal to 1.

See also (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)).

Usage

inci.matPEtri(Xp, tri, r, M = c(1, 1, 1))

Arguments

Xp

A set of 2D points which constitute the vertices of 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.

Value

Incidence matrix for the PE-PCD with vertices being 2D data set, Xp, in the triangle tri with vertex regions based on center M

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, Priebe CE, Wierman JC (2006). “Relative density of the random r-factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

See Also

inci.matPE, inci.matCStri, and inci.matAStri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

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)
IM<-inci.matPEtri(Xp,Tr,r=1.25,M)

IM
dom.num.greedy(IM) #try also dom.num.exact(IM)
Idom.num.up.bnd(IM,3)