num.arcsCStri {pcds}

Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the triangle - one 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 triangle) and indices of the data points that reside in the triangle.

CS proximity region N_{CS}(x,t) is defined with respect to the triangle, tri 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 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 tri for this function.

See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).

Usage

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

Arguments

Xp

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

tri

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

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 triangle tri; 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 CS-PCD

tri.num.arcs

Number of arcs of the induced subdigraph of the CS-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 (2014). “Comparison of Relative Density of Two Random Geometric Digraph Families in Testing Spatial Clustering.” TEST, 23(1), 100-134.

Ceyhan E, Priebe CE, Marchette DJ (2007). “A new family of random graphs for testing spatial segregation.” Canadian Journal of Statistics, 35(1), 27-50.

See Also

num.arcsCSstd.tri, num.arcsCS, num.arcsPEtri, 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.arcsCStri(Xp,Tr,t=.5,M)
Narcs
summary(Narcs)
plot(Narcs)