edgesPE {pcds.ugraph}

The edges of the underlying or reflexivity graph of the Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - multiple triangle case

Description

An object of class "UndPCDs". Returns edges of the underlying or reflexivity graph of PE-PCD as left and right end points and related parameters and the quantities of these graphs. The vertices of these graphs are the data points in Xp in the multiple triangle case.

PE proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter r \ge 1 and vertex regions in each triangle are based on the center M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of each Delaunay triangle or based on circumcenter of each Delaunay triangle (default for M=(1,1,1) which is the center of mass of the triangle). The different consideration of circumcenter vs any other interior center of the triangle is because the projections from circumcenter are orthogonal to the edges, while projections of M on the edges are on the extensions of the lines connecting M and the vertices. 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 the PE-PCDs. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

edgesPE(Xp, Yp, r, 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 graph of the PE-PCD.

Yp

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

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \ge 1.

M

A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as M="CC"), default for M=(1,1,1) which is the center of mass of each triangle.

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

type

A description of the underlying or reflexivity graph of the digraph

parameters

Parameters of the underlying or reflexivity graph of the digraph, the center M used to construct the vertex regions and the expansion parameter r.

tess.points

Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation is Delaunay triangulation based on Yp points.

tess.name

Name of the tessellation points tess.points

vertices

Vertices of the underlying or reflexivity graph of the digraph, Xp points

vert.name

Name of the data set which constitute the vertices of the underlying or reflexivity graph of the digraph

LE, RE

Left and right end points of the edges of the underlying or reflexivity graph of PE-PCD for 2D data set Xp as vertices of the underlying or reflexivity graph of the digraph

mtitle

Text for "main" title in the plot of the underlying or reflexivity graph of the digraph

quant

Various quantities for the underlying or reflexivity graph of the digraph: number of vertices, number of partition points, number of intervals, number of edges, and edge density.

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

edgesPEtri, edgesAS, edgesCS, and arcsPE

Examples

#\donttest{
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;

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

M<-c(1,1,1)
r<-1.5

Edges<-edgesPE(Xp,Yp,r,M)
Edges
summary(Edges)
plot(Edges)
#}