num.edgesCS: Number of edges of the underlying or reflexivity graphs of...

In pcds.ugraph: Underlying Graphs of Proximity Catch Digraphs and Their Applications

Number of edges of the underlying or reflexivity graphs of Central Similarity Proximity Catch Digraphs (CS-PCDs) - multiple triangle case

Description

An object of class "NumEdges". Returns the number of edges of the underlying or reflexivity graph of Central Similarity Proximity Catch Digraph (CS-PCD) and various other quantities and vectors such as the vector of number of vertices (i.e., number of data points) in the Delaunay triangles, number of data points in the convex hull of Yp points, indices of the Delaunay triangles for the data points, etc.

CS proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter t > 0 and edge regions in each triangle is based on the center M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of each Delaunay triangle (default for M=(1,1,1) which is the center of mass of the triangle). Each Delaunay triangle is first converted to an (nonscaled) basic triangle so that M will be the same type of center for each Delaunay triangle (this conversion is not necessary when M is CM).

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points). For the number of edges, loops are not allowed so edges are only possible for points inside the convex hull of Yp points.

See (\insertCiteceyhan:Phd-thesis,ceyhan:stamet2016;textualpcds.ugraph) for more on CS-PCDs. Also, see (\insertCiteokabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds.ugraph) for more on Delaunay triangulation and the corresponding algorithm.

Usage

num.edgesCS(Xp, Yp, t, 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 graphs of the CS-PCD.
Yp	A set of 2D points which constitute the vertices of the Delaunay triangles.
t	A positive real number which serves as the expansion parameter in CS proximity region.
M	A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle, default for M=(1,1,1) which is the center of mass of each triangle.
ugraph	The type of the graph based on CS-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 related quantities for the induced subgraphs of the underlying or reflexivity graphs (of CS-PCD) in the Delaunay triangles
und.graph	Type of the graph as "Underlying" or "Reflexivity" for the CS-PCD
num.edges	Total number of edges in all triangles, i.e., the number of edges for the entire underlying or reflexivity graphs of the CS-PCD
num.in.conv.hull	Number of Xp points in the convex hull of Yp points
num.in.tris	The vector of number of Xp points in the Delaunay triangles based on Yp points
weight.vec	The vector of the areas of Delaunay triangles based on Yp points
tri.num.edges	The vector of the number of edges of the components of the CS-PCD in the Delaunay triangles based on Yp points
del.tri.ind	A matrix of indices of vertices of the Delaunay triangles based on Yp points, each column corresponds to the vector of indices of the vertices of one triangle.
data.tri.ind	A vector of indices of vertices of the Delaunay triangles in which data points reside, i.e., column number of del.tri.ind for each Xp point.
tess.points	Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation is the Delaunay triangulation based on Yp points.
vertices	Vertices of the underlying or reflexivity graph, Xp.

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

num.edgesCStri, num.edgesAS, num.edgesPE, and num.arcsCS

Examples

#\donttest{
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;

set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))

pcds::plotDelaunay.tri(Xp,Yp,xlab="",ylab="")

M<-c(1,1,1)

Nedges = num.edgesCS(Xp,Yp,t=1.5,M)
Nedges
summary(Nedges)
plot(Nedges)
#}

pcds.ugraph documentation built on May 29, 2024, 10:39 a.m.