inci.mat.undPEtri {pcds.ugraph}

Incidence matrix for the underlying or reflexivity graph of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns the incidence matrix for the underlying or reflexivity graph of 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, 2016)).

Usage

inci.mat.undPEtri(
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

Arguments

Xp

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

ugraph

The type of the graph based on PE-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 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 (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

inci.mat.undPE, inci.mat.undAStri, inci.mat.undCStri, and inci.matPEtri

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.undPEtri(Xp,Tr,r=1.25,M))
pcds::dom.num.greedy(IM)
pcds::Idom.num.up.bnd(IM,3)

(IM<-inci.mat.undPEtri(Xp,Tr,r=1.25,M,ugraph="r"))
pcds::dom.num.greedy(IM)
pcds::Idom.num.up.bnd(IM,3)
#}