PEedge.dens.tri {pcds.ugraph}

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

Description

Returns the edge density of the underlying or reflexivity graph of Proportional Edge Proximity Catch Digraphs (PE-PCDs) whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

PE proximity regions is defined with respect to tri with expansion parameter r \ge 1 and vertex regions are based on 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. The function also provides edge density standardized by the mean and asymptotic variance of the edge density of the underlying or reflexivity graph of PE-PCD for uniform data in the triangle tri only when M is the center of mass. For the number of edges, loops are not allowed.

in.tri.only is a logical argument (default is FALSE) for considering only the points inside the triangle or all the points as the vertices of the digraph. if in.tri.only=TRUE, edge density is computed only for the points inside the triangle (i.e., edge density of the subgraph of the underlying or reflexivity graph induced by the vertices in the triangle is computed), otherwise edge density of the entire graph (i.e., graph with all the vertices) is computed.

See also (Ceyhan (2005, 2016)).

Usage

PEedge.dens.tri(
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity"),
  in.tri.only = FALSE
)

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").

in.tri.only

A logical argument (default is in.tri.only=FALSE) for computing the edge density for only the points inside the triangle, tri. That is, if in.tri.only=TRUE edge density of the induced subgraph with the vertices inside tri is computed, otherwise otherwise edge density of the entire graph (i.e., graph with all the vertices) is computed.

Value

A list with the elements

edge.dens

Edge density of the underlying or reflexivity graphs of the PE-PCD whose vertices are the 2D numerical data set, Xp; PE proximity regions are defined with respect to the triangle tri and M-vertex regions

std.edge.dens

Edge density standardized by the mean and asymptotic variance of the edge density of the underlying or reflexivity graph of the PE-PCD for uniform data in the triangle tri. This will only be returned, if M is the center of mass.

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

ASedge.dens.tri, CSedge.dens.tri, and PEarc.dens.tri

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)

#For the underlying graph
num.edgesPEtri(Xp,Tr,r=1.5,M)$num.edges
PEedge.dens.tri(Xp,Tr,r=1.5,M)
PEedge.dens.tri(Xp,Tr,r=1.5,M,in.tri.only = TRUE)

#For the reflexivity graph
num.edgesPEtri(Xp,Tr,r=1.5,M,ugraph="r")$num.edges
PEedge.dens.tri(Xp,Tr,r=1.5,M,ugraph="r")
PEedge.dens.tri(Xp,Tr,r=1.5,M,in.tri.only = TRUE,ugraph="r")
#}