arcsCS: The arcs of Central Similarity Proximity Catch Digraph...

In pcds: Proximity Catch Digraphs and Their Applications

The arcs of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D data - multiple triangle case

Description

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

See (\insertCiteceyhan:Phd-thesis,ceyhan:arc-density-CS,ceyhan:test2014;textualpcds) for more on CS-PCDs. Also see (\insertCiteokabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds) for more on Delaunay triangulation and the corresponding algorithm.

Usage

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

type	A description of the type of the digraph
parameters	Parameters of the digraph, here, it is the center used to construct the edge regions.
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 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 CS-PCD for 2D data set Xp as vertices of the digraph
E	Heads (or arrow ends) of the arcs of CS-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

\insertAllCited

See Also

arcsCStri, arcsAS and arcsPE

Examples

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; 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)

tau<-1.5  #try also tau<-2

Arcs<-arcsCS(Xp,Yp,tau,M)
#or use the default center Arcs<-arcsCS(Xp,Yp,tau)
Arcs
summary(Arcs)
plot(Arcs)

pcds documentation built on May 29, 2024, 9:36 a.m.