pcds (version 0.1.2)

IncMatPEMT: Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - multiple triangle case

Description

Returns the incidence matrix of 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 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 (unscaled) 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 incidence matrix loops are allowed, so the diagonal entries are all equal to 1.

See (ceyhan:Phd-thesis,ceyhan:arc-density-PE,ceyhan:dom-num-NPE-Spat2011;textualpcds) for more on the PE-PCDs. Also see (okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds) for more on Delaunay triangulation and the corresponding algorithm.

Usage

IncMatPEMT(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

Incidence matrix for the PE-PCD with vertices being 2D data set, Xp. PE proximity regions are constructed with respect to the Delaunay triangles and M-vertex regions.

References

See Also

IncMatPEtri, IncMatPETe, IncMatASMT, and IncMatCSMT

Examples

# NOT RUN {
nx<-20; ny<-4;  #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,0,1),runif(ny,0,1))

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

r<-1.5  #try also r<-2

IM<-IncMatPEMT(Xp,Yp,r,M)
IM
dom.greedy(IM)
# }
# NOT RUN {
dom.exact(IM)  #might take a long time in this brute-force fashion ignoring the
#disconnected nature of the digraph inherent by the geometric construction of it
# }
# NOT RUN {
PEdomMTnd(Xp,Yp,r)

Arcs<-ArcsPEMT(Xp,Yp,r,M)
Arcs
summary(Arcs)
plot(Arcs)

IncMatPEMT(Xp,Yp,r,M)
IncMatPEMT(Xp,Yp[1:3,],r,M)

IncMatPEMT(Xp,rbind(Yp,Yp),r,M)

dat.fr<-data.frame(a=Xp)
IncMatPEMT(dat.fr,Yp,r,M)

dat.fr<-data.frame(a=Yp)
IncMatPEMT(Xp,dat.fr,r,M)

# }