arcsPE {pcds}

The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - multiple triangle case

Description

An object of class "PCDs". Returns arcs of PE-PCD as tails (or sources) and heads (or arrow ends) and related parameters and the quantities of the digraph. The vertices of the PE-PCD 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). 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 arcs, loops are not allowed so arcs are only possible for points inside the convex hull of Yp points.

See (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)) 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

arcsPE(Xp, Yp, r, M = c(1, 1, 1))

Arguments

Xp

A set of 2D points which constitute the vertices 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.

Value

A list with the elements

type

A description of the type of the digraph

parameters

Parameters of the digraph, the center used to construct the vertex regions and the expansion parameter.

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.

tess.name

Name of the tessellation points tess.points

vertices

Vertices of the digraph, Xp points

vert.name

Name of the data set which constitute the vertices of the digraph

S

Tails (or sources) of the arcs of PE-PCD for 2D data set Xp as vertices of the digraph

E

Heads (or arrow ends) of the arcs of PE-PCD for 2D data set Xp as vertices of the digraph

mtitle

Text for "main" title in the plot of the digraph

quant

Various quantities for the digraph: number of vertices, number of partition points, number of triangles, number of arcs, and arc 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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E, Priebe CE, Wierman JC (2006). “Relative density of the random r-factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

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

arcsPEtri, arcsAS, and arcsCS

Examples

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

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))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

M<-c(1,1,1)  #try also M<-c(1,2,3)

r<-1.5  #try also r<-2

Arcs<-arcsPE(Xp,Yp,r,M)
#or try with the default center Arcs<-arcsPE(Xp,Yp,r)
Arcs
summary(Arcs)
plot(Arcs)