num.arcsPE: Number of arcs of Proportional Edge Proximity Catch Digraphs...

In pcds: Proximity Catch Digraphs and Their Applications

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-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 Proportional Edge Proximity Catch Digraph (PE-PCD) whose vertices 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 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 (\insertCiteceyhan:Phd-thesis,ceyhan:arc-density-PE;textualpcds) for more on PE-PCDs. Also, see (\insertCiteokabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds) for more on Delaunay triangulation and the corresponding algorithm.

Usage

num.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

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 PE-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 PE-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

\insertAllCited

See Also

num.arcsPEtri, num.arcsPEstd.tri, num.arcsCS, and num.arcsAS

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),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.arcsPE(Xp,Yp,r=1.25,M)
Narcs
summary(Narcs)
plot(Narcs)

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