num.edgesCS {pcds.ugraph}

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 (Ceyhan (2005, 2016)) for more on CS-PCDs. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) 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

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 (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

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)
#}