num.edgesPEtri {pcds.ugraph}

Number of edges in the underlying or reflexivity graph of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

An object of class "NumEdges". Returns the number of edges of the underlying or reflexivity graph of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertices are the given 2D numerical data set, Xp in a given triangle. It also provides number of vertices (i.e., number of data points inside the triangle) and indices of the data points that reside in the triangle.

PE proximity region N_{PE}(x,r) is defined with respect to the 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 or based on circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri. For the number of edges, loops are not allowed, so edges are only possible for points inside the triangle tri for this function.

See also (Ceyhan (2005, 2016)).

Usage

num.edgesPEtri(
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

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.

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

A list with the elements

desc

A short description of the output: number of edges and quantities related to the triangle

und.graph

Type of the graph as "Underlying" or "Reflexivity" for the PE-PCD

num.edges

Number of edges of the underlying or reflexivity graphs based on the PE-PCD with vertices in the given triangle tri

num.in.tri

Number of Xp points in the triangle, tri

ind.in.tri

The vector of indices of the Xp points that reside in the triangle

tess.points

Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation is the support triangle.

vertices

Vertices of the underlying or reflexivity graph, Xp.

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

num.edgesPE, num.edgesAStri, num.edgesCStri, and num.arcsPEtri

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)

Nedges = num.edgesPEtri(Xp,Tr,r=1.25,M)
Nedges
summary(Nedges)
plot(Nedges)
#}