arcsCStri {pcds}

The arcs of Central Similarity Proximity Catch Digraphs (CS-PCD) for 2D data - one triangle case

Description

An object of class "PCDs". Returns arcs of CS-PCD as tails (or sources) and heads (or arrow ends) and related parameters and the quantities of the digraph. The vertices of the CS-PCD are the data points in Xp in the one triangle case.

CS proximity regions are constructed with respect to the triangle tri with expansion parameter t>0, i.e., arcs may exist for points only inside tri. It also provides various descriptions and quantities about the arcs of the CS-PCD such as number of arcs, arc density, etc.

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.

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

Usage

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

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 or the circumcenter of tri; default is M=(1,1,1) i.e., the center of mass of tri.

Value

A list with the elements

type

A description of the type of the digraph

parameters

Parameters of the digraph, the center M used to construct the edge regions and the expansion parameter t.

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.

tess.name

Name of the tessellation points tess.points

vertices

Vertices of the digraph, Xp points

vert.name

Name of the data set which constitute the vertices of the digraph

S

Tails (or sources) of the arcs of CS-PCD for 2D data set Xp as vertices of the digraph

E

Heads (or arrow ends) of the arcs of CS-PCD for 2D data set Xp as vertices of the digraph

mtitle

Text for "main" title in the plot of the digraph

quant

Various quantities for the digraph: number of vertices, number of partition points, number of triangles, number of arcs, and arc density.

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

arcsCS, arcsAStri and arcsPEtri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10

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)

t<-1.5  #try also t<-2

Arcs<-arcsCStri(Xp,Tr,t,M)
#or try with the default center Arcs<-arcsCStri(Xp,Tr,t); M= (Arcs$param)$c
Arcs
summary(Arcs)
plot(Arcs)

#can add edge regions
L<-rbind(M,M,M); R<-Tr
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

#now we can add the vertex names and annotation
txt<-rbind(Tr,M)
xc<-txt[,1]+c(-.02,.03,.02,.03)
yc<-txt[,2]+c(.02,.02,.03,.06)
txt.str<-c("A","B","C","M")
text(xc,yc,txt.str)