num.arcsPEtri {pcds}

Number of arcs of Proportional Edge Proximity Catch Digraphs (PE-PCDs) and quantities related to the triangle - one triangle case

Description

An object of class "NumArcs". Returns the number of arcs of Proportional Edge Proximity Catch Digraphs (PE-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 triangle) and indices of the data points that reside in the triangle.

PE proximity region N_{PE}(x,r) is defined with respect to the triangle, tri 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 the triangle tri or based on circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri. For the number of arcs, loops are not allowed so arcs are only possible for points inside the triangle tri for this function.

See also (Ceyhan (2005, 2016)).

Usage

num.arcsPEtri(Xp, tri, r, M = c(1, 1, 1))

Arguments

Xp

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

tri

A 3 \times 2 matrix with each row representing a vertex of the triangle.

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 triangle tri or the circumcenter of tri which may be entered as "CC" as well; default is M=(1,1,1), i.e., the center of mass of tri.

Value

A list with the elements

desc

A short description of the output: number of arcs and quantities related to the triangle

num.arcs

Number of arcs of the PE-PCD

tri.num.arcs

Number of arcs of the induced subdigraph of the PE-PCD for vertices in the triangle tri

num.in.tri

Number of Xp points in the triangle, tri

ind.in.tri

The vector of indices of the Xp points that reside in the triangle

tess.points

Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation points are the vertices of the support triangle tri.

vertices

Vertices of the digraph, Xp.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

See Also

num.arcsPEstd.tri, num.arcsPE, num.arcsCStri, and num.arcsAStri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);

n<-10  #try also n<-20
set.seed(1)
Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

Narcs = num.arcsPEtri(Xp,Tr,r=1.25,M)
Narcs
summary(Narcs)
plot(Narcs)