CSarc.dens.tri {pcds}

Arc density of Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case

Description

Returns the arc density of CS-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

CS proximity regions is defined with respect to tri with expansion parameter t>0 and edge regions are based on center M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of the triangle tri; default is M=(1,1,1) i.e., the center of mass of tri. The function also provides arc density standardized by the mean and asymptotic variance of the arc density of CS-PCD for uniform data in the triangle tri only when M is the center of mass. For the number of arcs, loops are not allowed.

is a logical argument (default is FALSE) for considering only the points inside the triangle or all the points as the vertices of the digraph. if in.tri.only=TRUE, arc density is computed only for the points inside the triangle (i.e., arc density of the subdigraph induced by the vertices in the triangle is computed), otherwise arc density of the entire digraph (i.e., digraph with all the vertices) is computed.

See (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)) for more on CS-PCDs.

Usage

CSarc.dens.tri(Xp, tri, t, M = c(1, 1, 1), in.tri.only = FALSE)

Arguments

Xp

A set of 2D points which constitute the vertices of the 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.

in.tri.only

A logical argument (default is =FALSE) for computing the arc density for only the points inside the triangle, tri. That is, if =TRUE arc density of the induced subdigraph with the vertices inside tri is computed, otherwise otherwise arc density of the entire digraph (i.e., digraph with all the vertices) is computed.

Value

A list with the elements

arc.dens

Arc density of CS-PCD whose vertices are the 2D numerical data set, Xp; CS proximity regions are defined with respect to the triangle tri and M-edge regions

std.arc.dens

Arc density standardized by the mean and asymptotic variance of the arc density of CS-PCD for uniform data in the triangle tri.This will only be returned if M is the center of mass.

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

ASarc.dens.tri, PEarc.dens.tri, and num.arcsCStri

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)

CSarc.dens.tri(Xp,Tr,t=.5,M)
CSarc.dens.tri(Xp,Tr,t=.5,M, in.tri.only= FALSE)
#try also t=1 and t=1.5 above