pcds (version 0.1.6)

NumArcsPE: Number of arcs and related quantities of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - multiple triangle case

Description

Returns the number of arcs and various other quantities, vectors, and lists 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 (ceyhan:Phd-thesis,ceyhan:arc-density-PE;textualpcds) for more on PE-PCDs. Also see (okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds) for more on Delaunay triangulation and the corresponding algorithm.

Usage

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

Value

A list with the elements

num.arcs

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

num.in.conhull

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 component 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.

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.

Author

Elvan Ceyhan

References

See Also

NumArcsPEtri, NumArcsPETe, NumArcsCS, and NumArcsAS

Examples

if (FALSE) {
#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))

plotDeltri(Xp,Yp,xlab="",ylab="")

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

NumArcsPE(Xp,Yp,r=1.25,M)
NumArcsPE(Xp,Yp,r=1.5,M)
}