arcsPEtri {pcds}

The arcs of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - one triangle case

Description

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

PE proximity regions are constructed with respect to the triangle tri with expansion parameter r \ge 1, i.e., arcs may exist only for points inside tri. It also provides various descriptions and quantities about the arcs of the PE-PCD such as number of arcs, arc density, etc.

Vertex 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 or based on the circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri. When the center is the circumcenter, CC, the vertex regions are constructed based on the orthogonal projections to the edges, while with any interior center M, the vertex regions are constructed using the extensions of the lines combining vertices with M. M-vertex regions are recommended spatial inference, due to geometry invariance property of the arc density and domination number the PE-PCDs based on uniform data.

See also (Ceyhan (2005); Ceyhan et al. (2006)).

Usage

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

Arguments

Xp

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

type

A description of the type of the digraph

parameters

Parameters of the digraph, the center M used to construct the vertex regions and the expansion parameter r.

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 constitutes the vertices of the digraph

S

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

E

Heads (or arrow ends) of the arcs of PE-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, Priebe CE, Wierman JC (2006). “Relative density of the random r-factor proximity catch digraphs for testing spatial patterns of segregation and association.” Computational Statistics & Data Analysis, 50(8), 1925-1964.

See Also

arcsPE, arcsAStri, and arcsCStri

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)

r<-1.5  #try also r<-2

Arcs<-arcsPEtri(Xp,Tr,r,M)
#or try with the default center Arcs<-arcsPEtri(Xp,Tr,r); M= (Arcs$param)$cent
Arcs
summary(Arcs)
plot(Arcs)

#can add vertex regions
#but we first need to determine center is the circumcenter or not,
#see the description for more detail.
CC<-circumcenter.tri(Tr)
if (isTRUE(all.equal(M,CC)))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
}
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)

#now we can add the vertex names and annotation
txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]+c(-.02,.02,.02,.02,.03,-.03,-.01)
yc<-txt[,2]+c(.02,.02,.03,.06,.04,.05,-.07)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)