inci.mat.undAStri {pcds.ugraph}

Incidence matrix for the underlying or reflexivity graph of Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

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

AS proximity regions are defined with respect to the triangle tri=T(v=1,v=2,v=3) 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 or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri. Loops are allowed, so the diagonal entries are all equal to 1.

See also (Ceyhan (2005, 2016)).

Usage

inci.mat.undAStri(Xp, tri, M = "CC", ugraph = c("underlying", "reflexivity"))

Arguments

Xp

A set of 2D points which constitute the vertices of the underlying or reflexivity graph of the AS-PCD.

tri

A 3 \times 2 matrix with each row representing a vertex of the triangle.

M

The center of the triangle. "CC" stands for circumcenter of the triangle tri or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of tri; default is M="CC", i.e., the circumcenter of tri.

ugraph

The type of the graph based on AS-PCDs, "underlying" is for the underlying graph, and "reflexivity" is for the reflexivity graph (default is "underlying").

Value

Incidence matrix for the underlying or reflexivity graph of the AS-PCD whose vertices are the 2D data set, Xp in the triangle tri with vertex regions based on the 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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

inci.mat.undAS, inci.mat.undPEtri, inci.mat.undCStri, and inci.matAStri

Examples

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

set.seed(1)
Xp<-pcds::runif.tri(n,Tr)$g

M<-as.numeric(pcds::runif.tri(1,Tr)$g)
(IM<-inci.mat.undAStri(Xp,Tr,M))
pcds::dom.num.greedy(IM)
pcds::Idom.num.up.bnd(IM,3)

(IM<-inci.mat.undAStri(Xp,Tr,M,ugraph="r"))
pcds::dom.num.greedy(IM)
pcds::Idom.num.up.bnd(IM,3)
#}