pcds (version 0.1.4)

IncMatPETe: Incidence matrix for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns the incidence matrix for the PE-PCD whose vertices are the given 2D numerical data set, dat, in the standard equilateral triangle \(T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2))\).

PE proximity region is constructed with respect to the standard equilateral triangle \(T_e\) with expansion parameter \(r \ge 1\) and vertex regions are based on the center \(M=(m_1,m_2)\) in Cartesian coordinates or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of \(T_e\); default is \(M=(1,1,1)\) i.e., the center of mass of \(T_e\). Loops are allowed, so the diagonal entries are all equal to 1.

See also (ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:dom-num-NPE-Spat2011;textualpcds).

Usage

IncMatPETe(dat, r, M = c(1, 1, 1))

Value

Incidence matrix for the PE-PCD with vertices being 2D data set, dat and PE proximity regions are defined in the standard equilateral triangle \(T_e\) with M-vertex regions.

Arguments

dat

A set of 2D points which constitute the vertices of the PE-PCD.

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \(\ge 1\).

M

A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle \(T_e\); default is \(M=(1,1,1)\) i.e. the center of mass of \(T_e\).

Author

Elvan Ceyhan

References

See Also

IncMatPEtri, IncMatPEMT and IncMatCSTe

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-10

set.seed(1)
dat<-runifTe(n)$gen.points

M<-as.numeric(runifTe(1)$g)  #try also M<-c(.6,.2)

NumArcsPETe(dat,r=1.25)

inc.mat<-IncMatPETe(dat,r=1.25,M)
inc.mat
sum(inc.mat)-n

dom.greedy(inc.mat)
IndUBdom(inc.mat,2)
dom.exact(inc.mat)

inc.mat<-IncMatPETe(rbind(dat,c(0,1)),r=1.25,M)
inc.mat
sum(inc.mat)-(n+1)

IncMatPETe(dat,r=1.5,M);

IncMatPETe(rbind(dat,dat),r=1.5,M)

dat.fr<-data.frame(a=dat)
IncMatPETe(dat.fr,r=1.5,M);