| num.arcsCSstd.tri {pcds} | R Documentation |
Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and quantities related to the triangle - standard equilateral 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 standard equilateral triangle T_e)
and indices of the data points that reside in T_e.
CS proximity region N_{CS}(x,t) is defined with respect to the standard
equilateral triangle T_e=T(v=1,v=2,v=3)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) 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 T_e;
default is M=(1,1,1) i.e., the center of mass of T_e.
For the number of arcs, loops are not allowed so
arcs are only possible for points inside T_e for this function.
See also (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)).
Usage
num.arcsCSstd.tri(Xp, t, M = c(1, 1, 1))
Arguments
Xp |
A set of 2D points which constitute the vertices of the digraph. |
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 standard equilateral triangle |
Value
A list with the elements
desc |
A short description of the output: number of arcs and quantities related to the standard equilateral 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 standard equilateral triangle |
num.in.tri |
Number of |
ind.in.tri |
The vector of indices of the |
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 |
vertices |
Vertices of the digraph, |
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.arcsCStri, num.arcsCS, and num.arcsPEstd.tri,
Examples
A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
n<-10 #try also n<-20
set.seed(1)
Xp<-runif.std.tri(n)$gen.points
M<-as.numeric(runif.std.tri(1)$g) #try also M<-c(.6,.2)
Narcs = num.arcsCSstd.tri(Xp,t=.5,M)
Narcs
summary(Narcs)
oldpar <- par(pty="s")
plot(Narcs,asp=1)
par(oldpar)