An object of class "NumEdges".
Returns the number of edges of
the underlying or reflexivity graph of
Central Similarity Proximity Catch Digraph (CS-PCD)
and various other quantities and vectors such as
the vector of number of vertices (i.e., number of data points)
in the Delaunay triangles,
number of data points in the convex hull of Yp points,
indices of the Delaunay triangles for the data points, etc.
CS proximity regions are defined with respect to the
Delaunay triangles based on Yp points
with expansion parameter \(t > 0\)
and edge regions in each triangle
is based on the center \(M=(\alpha,\beta,\gamma)\)
in barycentric coordinates in the interior of each
Delaunay triangle (default for \(M=(1,1,1)\)
which is the center of mass of the triangle).
Each Delaunay triangle is first converted to
an (nonscaled) basic triangle so that M will be the same
type of center for each Delaunay triangle
(this conversion is not necessary when M is \(CM\)).
Convex hull of Yp is partitioned
by the Delaunay triangles based on Yp points
(i.e., multiple triangles are the set of these Delaunay triangles
whose union constitutes the
convex hull of Yp points).
For the number of edges,
loops are not allowed so edges are only possible
for points inside the convex hull of Yp points.
See (ceyhan:Phd-thesis,ceyhan:stamet2016;textualpcds.ugraph) for more on CS-PCDs. Also, see (okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds.ugraph) for more on Delaunay triangulation and the corresponding algorithm.
num.edgesCS(Xp, Yp, t, M = c(1, 1, 1), ugraph = c("underlying", "reflexivity"))A list with the elements
A short description of the output: number of edges and related quantities for the induced subgraphs of the underlying or reflexivity graphs (of CS-PCD) in the Delaunay triangles
Type of the graph as "Underlying" or "Reflexivity" for the CS-PCD
Total number of edges in all triangles, i.e., the number of edges for the entire underlying or reflexivity graphs of the CS-PCD
Number of Xp points
in the convex hull of Yp points
The vector of number of Xp points
in the Delaunay triangles based on Yp points
The vector of the areas of
Delaunay triangles based on Yp points
The vector of the number of edges
of the components of the CS-PCD in the
Delaunay triangles based on Yp points
A matrix of indices of vertices of
the Delaunay triangles based on Yp points,
each column corresponds to the vector of
indices of the vertices of one triangle.
A vector of indices of vertices of
the Delaunay triangles in which data points reside,
i.e., column number of del.tri.ind for each Xp point.
Tessellation points,
i.e., points on which the tessellation of the study region is performed,
here, tessellation is the Delaunay triangulation based on Yp points.
Vertices of the underlying or reflexivity graph, Xp.
A set of 2D points which constitute the vertices of the underlying or reflexivity graphs of the CS-PCD.
A set of 2D points which constitute the vertices of the Delaunay triangles.
A positive real number which serves as the expansion parameter in CS proximity region.
A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle, default for \(M=(1,1,1)\) which is the center of mass of each triangle.
The type of the graph based on CS-PCDs,
"underlying" is for the underlying graph,
and "reflexivity" is for
the reflexivity graph (default is "underlying").
Elvan Ceyhan
num.edgesCStri, num.edgesAS,
num.edgesPE, and num.arcsCS
#\donttest{
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5;
set.seed(1)
Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
pcds::plotDelaunay.tri(Xp,Yp,xlab="",ylab="")
M<-c(1,1,1)
Nedges = num.edgesCS(Xp,Yp,t=1.5,M)
Nedges
summary(Nedges)
plot(Nedges)
#}
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