inci.mat.undPE {pcds.ugraph}

Incidence matrix for the underlying or reflexivity graph of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - multiple triangle case

Description

Returns the incidence matrix for the underlying or reflexivity graph of the PE-PCD whose vertices are the data points in Xp in the multiple triangle case.

PE proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter r \ge 1 and vertex regions in each triangle are based on the center M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of each Delaunay triangle or based on circumcenter 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 incidence matrix loops are allowed, so the diagonal entries are all equal to 1.

See (Ceyhan (2005, 2016)) for more on the PE-PCDs. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

inci.mat.undPE(
  Xp,
  Yp,
  r,
  M = c(1, 1, 1),
  ugraph = c("underlying", "reflexivity")
)

Arguments

Xp

A set of 2D points which constitute the vertices of the underlying or reflexivity graph of the PE-PCD.

Yp

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

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \ge 1.

M

A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as M="CC"), default for M=(1,1,1) which is the center of mass of each triangle.

ugraph

The type of the graph based on PE-PCDs, "underlying" is for the underlying graph, and "reflexivity" is for the reflexivity graph (default is "underlying").

Value

Incidence matrix for the underlying or reflexivity graph of the PE-PCD whose vertices are the 2D data set, Xp. PE proximity regions are constructed with respect to the Delaunay triangles and M-vertex regions.

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 (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2016). “Edge Density of New Graph Types Based on a Random Digraph Family.” Statistical Methodology, 33, 31-54.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

See Also

inci.mat.undPEtri, inci.mat.undAS, inci.mat.undCS, and inci.matPE

Examples

#\donttest{
nx<-20; ny<-5;

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))

M<-c(1,1,1)
r<-1.5

IM<-inci.mat.undPE(Xp,Yp,r,M)
IM
pcds::dom.num.greedy(IM)
#}