num.edgesAS {pcds.ugraph}

Number of edges of the underlying or reflexivity graph of Arc Slice Proximity Catch Digraphs (AS-PCDs) - multiple triangle case

Description

An object of class "NumEdges". Returns the number of edges of the underlying or reflexivity graph of Arc Slice Proximity Catch Digraph (AS-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.

AS proximity regions are defined with respect to the Delaunay triangles based on Yp points and vertex regions in each triangle are based on the center M="CC" for circumcenter of each Delaunay triangle or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of each Delaunay triangle; default is M="CC", i.e., circumcenter of each triangle. Each Delaunay triangle is first converted to a (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 AS-PCDs. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

num.edgesAS(Xp, Yp, M = "CC", ugraph = c("underlying", "reflexivity"))

Arguments

Xp

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

Yp

A set of 2D points which constitute the vertices of the Delaunay triangles.

M

The center of the triangle. "CC" stands for circumcenter of each Delaunay triangle or 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle; default is M="CC", i.e., the circumcenter of each triangle.

ugraph

The type of the graph based on AS-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 AS-PCD) in the Delaunay triangles

und.graph

Type of the graph as "Underlying" or "Reflexivity" for the AS-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 AS-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 AS-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

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.edgesAStri, num.edgesPE, num.edgesCS, and num.arcsAS

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.edgesAS(Xp,Yp,M)
Nedges
summary(Nedges)
plot(Nedges)
#}