num.arcsCSstd.tri: Number of arcs of Central Similarity Proximity Catch Digraphs...

In elvanceyhan/pcds: Proximity Catch Digraphs and Their Applications

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the triangle - standard equilateral triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) whose vertices are the given 2D numerical data set, Xp. It also provides number of vertices (i.e., number of data points inside the standard equilateral triangle T_e) and indices of the data points that reside in T_e.

CS proximity region N_{CS}(x,t) is defined with respect to the standard equilateral triangle T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) with expansion parameter t>0 and edge 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. For the number of arcs, loops are not allowed so arcs are only possible for points inside T_e for this function.

See also (\insertCiteceyhan:Phd-thesis,ceyhan:arc-density-CS,ceyhan:test2014;textualpcds).

Usage

num.arcsCSstd.tri(Xp, t, M = c(1, 1, 1))

Arguments

Xp	A set of 2D points which constitute the vertices of the digraph.
t	A positive real number which serves as the expansion parameter in CS proximity region.
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.

Value

A list with the elements

desc	A short description of the output: number of arcs and quantities related to the standard equilateral triangle
num.arcs	Number of arcs of the CS-PCD
num.in.tri	Number of Xp points in the standard equilateral triangle, T_e
ind.in.tri	The vector of indices of the Xp points that reside in T_e
tess.points	Points on which the tessellation of the study region is performed, here, tessellation is the support triangle T_e.
vertices	Vertices of the digraph, Xp.

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

num.arcsCStri, num.arcsCS, and num.arcsPEstd.tri,

Examples

## Not run: 
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

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

Narcs = num.arcsCSstd.tri(Xp,t=.5,M)
Narcs
summary(Narcs)
par(pty="s")
plot(Narcs,asp=1)

## End(Not run)

elvanceyhan/pcds documentation built on June 29, 2023, 8:12 a.m.