num.arcsCS {pcds}

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and related quantities of the induced subdigraphs for points in the Delaunay triangles - multiple triangle case

Description

An object of class "NumArcs". Returns the number of arcs and various other quantities related to the Delaunay triangles for Central Similarity Proximity Catch Digraph (CS-PCD) whose vertices are the data points in Xp in the multiple triangle case.

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 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. (2007); Ceyhan (2014)) 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.arcsCS(Xp, Yp, t, M = c(1, 1, 1))

Arguments

Xp

A set of 2D points which constitute the vertices 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.

Value

A list with the elements

desc

A short description of the output: number of arcs and related quantities for the induced subdigraphs in the Delaunay triangles

num.arcs

Total number of arcs in all triangles, i.e., the number of arcs for the entire 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.arcs

The vector of the number of arcs 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 digraph, 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 (2014). “Comparison of Relative Density of Two Random Geometric Digraph Families in Testing Spatial Clustering.” TEST, 23(1), 100-134.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

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.arcsCStri, num.arcsCSstd.tri, num.arcsPE, and num.arcsAS

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

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

Narcs = num.arcsCS(Xp,Yp,t=1,M)
Narcs
summary(Narcs)
plot(Narcs)