pcds: A package for Proximity Catch Digraphs and Their Applications

Description

pcds is a package for construction and visualization of proximity catch digraphs (PCDs) and computation of two graph invariants of the PCDs and testing spatial patterns using these invariants.

Details

The PCD families considered are Arc-Slice (AS) PCDs, Proportional-Edge (PE) PCDs and Central Similarity (CS) PCDs.

The graph invariants used in testing spatial point data are the domination number (Ceyhan (2011)) and arc density (Ceyhan et al. (2006); Ceyhan et al. (2007)) of for two-dimensional data.

The pcds package also contains the functions for generating patterns of segregation, association, CSR (complete spatial randomness) and Uniform data in one, two and three dimensional cases, for testing these patterns based on two invariants of various families of the proximity catch digraphs (PCDs), (see (Ceyhan (2005)).

Moreover, the package has visualization tools for these digraphs for 1D-3D vertices. The AS-PCD and CS-PCD tools are provided for 1D and 2D data and PE-PCD related tools are provided for 1D-3D data.

The pcds functions

The pcds functions can be grouped as Auxiliary Functions, AS-PCD Functions, PE-PCD Functions, and CS-PCD Functions.

Auxiliary Functions

Contains the auxiliary (or utility) functions for constructing and visualizing Delaunay tessellations in 1D and 2D settings, computing the domination number, constructing the geometrical tools, such as equation of lines for two points, distances between lines and points, checking points inside the triangle etc., finding the (local) extrema (restricted to Delaunay cells or vertex or edge regions in them).

Arc-Slice PCD Functions

Contains the functions used in AS-PCD construction, estimation of domination number, arc density, etc in the 2D setting.

Proportional-Edge PCD Functions

Contains the functions used in PE-PCD construction, estimation of domination number, arc density, etc in the 1D-3D settings.

Central-Similarity PCD Functions

Contains the functions used in CS-PCD construction, estimation of domination number, arc density, etc in the 1D and 2D setting.

Point Generation Functions

Contains functions for generation of points from uniform (or CSR), segregation and association patterns.

In all these functions points are vectors, and data sets are either matrices or data frames.

Author(s)

Maintainer: Elvan Ceyhan elvanceyhan@gmail.com

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

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.

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.

.onAttach start message

Description

.onAttach start message

Usage

.onAttach(libname, pkgname)

Arguments

Value

invisible()

.onLoad getOption package settings

Description

.onLoad getOption package settings

Usage

.onLoad(libname, pkgname)

Arguments

Value

invisible()

Examples

getOption("pcds.name")

Arc density of Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns the arc density of AS-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

AS proximity regions are defined with respect to tri and vertex regions are defined with the center M="CC" for circumcenter of tri; or 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="CC" i.e., circumcenter of tri. For the number of arcs, loops are not allowed so arcs are only possible for points inside tri for this function.

in.tri.only is a logical argument (default is FALSE) for considering only the points inside the triangle or all the points as the vertices of the digraph. if in.tri.only=TRUE, arc density is computed only for the points inside the triangle (i.e., arc density of the subdigraph induced by the vertices in the triangle is computed), otherwise arc density of the entire digraph (i.e., digraph with all the vertices) is computed.

See also (Ceyhan (2005, 2010)).

Usage

ASarc.dens.tri(Xp, tri, M = "CC", in.tri.only = FALSE)

Arguments

Value

Arc density of AS-PCD whose vertices are the 2D numerical data set, Xp; AS proximity regions are defined with respect to the triangle tri and CC-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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

ASarc.dens.tri, CSarc.dens.tri, and num.arcsAStri

Examples

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

set.seed(1)
n<-10  #try also n<-20

Xp<-runif.tri(n,Tr)$g

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

narcs = num.arcsAStri(Xp,Tr,M)$num.arcs; narcs/(n*(n-1))
ASarc.dens.tri(Xp,Tr,M)
ASarc.dens.tri(Xp,Tr,M,in.tri.only = FALSE)

ASarc.dens.tri(Xp,Tr,M)

A test of segregation/association based on arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 2D data

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of Xp points in the convex hull of Yp points against the alternatives of segregation (where Xp points cluster away from Yp points) and association (where Xp points cluster around Yp points) based on the normal approximation of the arc density of the CS-PCD for uniform 2D data in the convex hull of Yp points.

The function yields the test statistic, p-value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is the arc density), and method and name of the data set used.

Under the null hypothesis of uniformity of Xp points in the convex hull of Yp points, arc density of CS-PCD whose vertices are Xp points equals to its expected value under the uniform distribution and alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association) or right-sided (i.e., data is accumulated around the centers of the triangles, or segregation).

CS proximity region is constructed with the expansion parameter t>0 and CM-edge regions (i.e., the test is not available for a general center M at this version of the function).

**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are assumed to be fixed (i.e., the test is conditional on Yp points). Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points, say at least 5 times more. This test is more appropriate when supports of Xp and Yp has a substantial overlap. Currently, the Xp points outside the convex hull of Yp points are handled with a convex hull correction factor, ch.cor, which is derived under the assumption of uniformity of Xp and Yp points in the study window, (see the description below and the function code.) However, in the special case of no Xp points in the convex hull of Yp points, arc density is taken to be 1, as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is an ongoing line of research of the author of the package.

ch.cor is for convex hull correction (default is "no convex hull correction", i.e., ch.cor=FALSE) which is recommended when both Xp and Yp have the same rectangular support.

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

Usage

CSarc.dens.test(
  Xp,
  Yp,
  t,
  ch.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

Value

A list with the elements

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

PEarc.dens.test and CSarc.dens.test1D

Examples

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

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))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab = "")

CSarc.dens.test(Xp,Yp,t=.5)
CSarc.dens.test(Xp,Yp,t=.5,ch=TRUE)
#try also t=1.0 and 1.5 above

A test of uniformity of 1D data in a given interval based on Central Similarity Proximity Catch Digraph (CS-PCD)

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of uniformity of 1D data in one interval based on the normal approximation of the arc density of the CS-PCD with expansion parameter t>0 and centrality parameter c \in (0,1).

The function yields the test statistic, p-value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is the arc density), and method and name of the data set used.

The null hypothesis is that data is uniform in a finite interval (i.e., arc density of CS-PCD equals to its expected value under uniform distribution) and alternative could be two-sided, or left-sided (i.e., data is accumulated around the end points) or right-sided (i.e., data is accumulated around the mid point or center M_c).

See also (Ceyhan (2016)).

Usage

CSarc.dens.test.int(
  Xp,
  int,
  t,
  c = 0.5,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

See Also

PEarc.dens.test.int

Examples

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

n<-10
Xp<-runif(n,a,b)

num.arcsCSmid.int(Xp,int,t,c)
CSarc.dens.test.int(Xp,int,t,c)

num.arcsCSmid.int(Xp,int,t,c=.3)
CSarc.dens.test.int(Xp,int,t,c=.3)

num.arcsCSmid.int(Xp,int,t=1.5,c)
CSarc.dens.test.int(Xp,int,t=1.5,c)

Xp<-runif(n,a-1,b+1)
num.arcsCSmid.int(Xp,int,t,c)
CSarc.dens.test.int(Xp,int,t,c)

c<-.4
t<-.5
a<-0; b<-10; int<-c(a,b)
n<-10  #try also n<-20
Xp<-runif(n,a,b)

CSarc.dens.test.int(Xp,int,t,c)

A test of segregation/association based on arc density of Central Similarity Proximity Catch Digraph (CS-PCD) for 1D data

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of Xp points in the range (i.e., range) of Yp points against the alternatives of segregation (where Xp points cluster away from Yp points) and association (where Xp points cluster around Yp points) based on the normal approximation of the arc density of the CS-PCD for uniform 1D data.

The function yields the test statistic, p-value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is the arc density), and method and name of the data set used.

Under the null hypothesis of uniformity of Xp points in the range of Yp points, arc density of CS-PCD whose vertices are Xp points equals to its expected value under the uniform distribution and alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association) or right-sided (i.e., data is accumulated around the centers of the intervals, or segregation).

CS proximity region is constructed with the expansion parameter t > 0 and centrality parameter c which yields M-vertex regions. More precisely, for a middle interval (y_{(i)},y_{(i+1)}), the center is M=y_{(i)}+c(y_{(i+1)}-y_{(i)}) for the centrality parameter c \in (0,1). If there are duplicates of Yp points, only one point is retained for each duplicate value, and a warning message is printed.

**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are assumed to be fixed (i.e., the test is conditional on Yp points). Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points, say at least 5 times more. This test is more appropriate when supports of Xp and Yp have a substantial overlap. Currently, the Xp points outside the range of Yp points are handled with a range correction (or end-interval correction) factor (see the description below and the function code.) However, in the special case of no Xp points in the range of Yp points, arc density is taken to be 1, as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is an ongoing line of research of the author of the package.

end.int.cor is for end-interval correction, recommended when both Xp and Yp have the same interval support (default is "no end-interval correction", i.e., end.int.cor=FALSE).

Usage

CSarc.dens.test1D(
  Xp,
  Yp,
  t,
  c = 0.5,
  support.int = NULL,
  end.int.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

There are no references for Rd macro ⁠\insertAllCites⁠ on this help page.

See Also

CSarc.dens.test and CSarc.dens.test.int

Examples

tau<-2
c<-.4
a<-0; b<-10; int=c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

CSarc.dens.test1D(Xp,Yp,tau,c,int)
CSarc.dens.test1D(Xp,Yp,tau,c,int,alt="l")
CSarc.dens.test1D(Xp,Yp,tau,c,int,alt="g")

CSarc.dens.test1D(Xp,Yp,tau,c,int,end.int.cor = TRUE)

Yp2<-runif(ny,a,b)+11
CSarc.dens.test1D(Xp,Yp2,tau,c,int)

n<-10  #try also n<-20
Xp<-runif(n,a,b)
CSarc.dens.test1D(Xp,Yp,tau,c,int)

Arc density of Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case

Description

Returns the arc density of CS-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

CS proximity regions is defined with respect to tri with expansion parameter t>0 and 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. The function also provides arc density standardized by the mean and asymptotic variance of the arc density of CS-PCD for uniform data in the triangle tri only when M is the center of mass. For the number of arcs, loops are not allowed.

is a logical argument (default is FALSE) for considering only the points inside the triangle or all the points as the vertices of the digraph. if in.tri.only=TRUE, arc density is computed only for the points inside the triangle (i.e., arc density of the subdigraph induced by the vertices in the triangle is computed), otherwise arc density of the entire digraph (i.e., digraph with all the vertices) is computed.

See (Ceyhan (2005); Ceyhan et al. (2007); Ceyhan (2014)) for more on CS-PCDs.

Usage

CSarc.dens.tri(Xp, tri, t, M = c(1, 1, 1), in.tri.only = FALSE)

Arguments

Value

A list with the elements

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

ASarc.dens.tri, PEarc.dens.tri, and num.arcsCStri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

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)

CSarc.dens.tri(Xp,Tr,t=.5,M)
CSarc.dens.tri(Xp,Tr,t=.5,M, in.tri.only= FALSE)
#try also t=1 and t=1.5 above

The distance between two vectors, matrices, or data frames

Description

Returns the Euclidean distance between x and y which can be vectors or matrices or data frames of any dimension (x and y should be of same dimension).

This function is different from the dist function in the stats package of the standard R distribution. dist requires its argument to be a data matrix and dist computes and returns the distance matrix computed by using the specified distance measure to compute the distances between the rows of a data matrix (Becker et al. (1988)), while Dist needs two arguments to find the distances between. For two data matrices A and B, dist(rbind(as.vector(A), as.vector(B))) and Dist(A,B) yield the same result.

Usage

Dist(x, y)

Arguments

Value

Euclidean distance between x and y

Author(s)

Elvan Ceyhan

References

Becker RA, Chambers JM, Wilks AR (1988). The New S Language. Wadsworth & Brooks/Cole.

See Also

dist from the base package stats

Examples

B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Dist(B,C);
dist(rbind(B,C))

x<-runif(10)
y<-runif(10)
Dist(x,y)

xm<-matrix(x,ncol=2)
ym<-matrix(y,ncol=2)
Dist(xm,ym)
dist(rbind(as.vector(xm),as.vector(ym)))

Dist(xm,xm)

The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case

Description

Returns I(p2 \in N_{AS}(p1)) for points p1 and p2, that is, returns 1 if p2 is in N_{AS}(p1), returns 0 otherwise, where N_{AS}(x) is the AS proximity region for point x.

AS proximity region is constructed in the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)) where c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

Vertex 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 the standard basic triangle T_b or based on circumcenter of T_b; default is M="CC", i.e., circumcenter of T_b. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_b, the function returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010)).

Usage

IarcASbasic.tri(p1, p2, c1, c2, M = "CC", rv = NULL)

Arguments

Value

I(p2 \in N_{AS}(p1)) for points p1 and p2, that is, returns 1 if p2 is in N_{AS}(p1) (i.e., if there is an arc from p1 to p2), returns 0 otherwise.

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

IarcAStri and NAStri

Examples

c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcASbasic.tri(P1,P2,c1,c2,M)

P1<-c(.3,.2)
P2<-c(.6,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

#or try
Rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
IarcASbasic.tri(P1,P2,c1,c2,M,Rv)

P1<-c(.3,.2)
P2<-c(.8,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

P3<-c(.5,.4)
IarcASbasic.tri(P1,P3,c1,c2,M)

P4<-c(1.5,.4)
IarcASbasic.tri(P1,P4,c1,c2,M)
IarcASbasic.tri(P4,P4,c1,c2,M)

c1<-.4; c2<-.6;
P1<-c(.3,.2)
P2<-c(.6,.2)
IarcASbasic.tri(P1,P2,c1,c2,M)

The indicator for the presence of an arc from a point in set S to the point p for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns I(pt \in N_{AS}(x) for some x \in S), that is, returns 1 if p is in \cup_{x \in S}N_{AS}(x), returns 0 otherwise, where N_{AS}(x) is the AS proximity region for point x.

AS proximity regions are constructed with respect to the triangle, tri=T(A,B,C)=(rv=1,rv=2,rv=3), and vertices of tri are also labeled as 1,2, and 3, respectively.

Vertex 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 the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri.

If p is not in S and either p or all points in S are outside tri, it returns 0, but if p is in S, then it always returns 1 (i.e., loops are allowed).

See also (Ceyhan (2005, 2010)).

Usage

IarcASset2pnt.tri(S, p, tri, M = "CC")

Arguments

Value

I(pt \in \cup_{x in S}N_{AS}(x,r)), that is, returns 1 if p is in S or inside N_{AS}(x) for at least one x in S, returns 0 otherwise, where AS proximity region is constructed in tri

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

IarcAStri, IarcASset2pnt.tri, and IarcCSset2pnt.tri

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)$gen.points

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

IarcASset2pnt.tri(S,Xp[3,],Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcASset2pnt.tri(S,Xp[3,],Tr,M)

IarcASset2pnt.tri(S,Xp[6,],Tr,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcASset2pnt.tri(S,Xp[3,],Tr,M)

IarcASset2pnt.tri(c(.2,.5),Xp[2,],Tr,M)
IarcASset2pnt.tri(Xp,c(.2,.5),Tr,M)
IarcASset2pnt.tri(Xp,Xp[2,],Tr,M)
IarcASset2pnt.tri(c(.2,.5),c(.2,.5),Tr,M)
IarcASset2pnt.tri(Xp[5,],Xp[2,],Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
IarcASset2pnt.tri(S,Xp[3,],Tr,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcASset2pnt.tri(Xp,P,Tr,M)
IarcASset2pnt.tri(S,P,Tr,M)

IarcASset2pnt.tri(rbind(S,S),P,Tr,M)

The indicator for the presence of an arc from a point to another for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns I(p2 \in N_{AS}(p1)) for points p1 and p2, that is, returns 1 if p2 is in N_{AS}(p1), returns 0 otherwise, where N_{AS}(x) is the AS proximity region for point x.

AS proximity regions are constructed with respect to the triangle, tri=T(A,B,C)=(rv=1,rv=2,rv=3), and vertex 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 the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside tri, the function returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010)).

Usage

IarcAStri(p1, p2, tri, M = "CC", rv = NULL)

Arguments

Value

I(p2 \in N_{AS}(p1)) for p1, that is, returns 1 if p2 is in N_{AS}(p1), returns 0 otherwise

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

IarcASbasic.tri, IarcPEtri, and IarcCStri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcAStri(P1,P2,Tr,M)

P1<-c(1.3,1.2)
P2<-c(1.8,.5)
IarcAStri(P1,P2,Tr,M)
IarcAStri(P1,P1,Tr,M)

#or try
Rv<-rel.vert.triCC(P1,Tr)$rv
IarcAStri(P1,P2,Tr,M,Rv)

P3<-c(1.6,1.4)
IarcAStri(P1,P3,Tr,M)

P4<-c(1.5,1.0)
IarcAStri(P1,P4,Tr,M)

P5<-c(.5,1.0)
IarcAStri(P1,P5,Tr,M)
IarcAStri(P5,P5,Tr,M)

#or try
Rv<-rel.vert.triCC(P5,Tr)$rv
IarcAStri(P5,P5,Tr,M,Rv)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - first one-sixth of the standard equilateral triangle case

Description

Returns I(p2 is in N_{CS}(p1,t=1)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t=1), returns 0 otherwise, where N_{CS}(x,t=1) is the CS proximity region for point x with expansion parameter t=1.

CS proximity region is defined with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) and edge regions are based on the center of mass CM=(1/2,\sqrt{3}/6). Here p1 must lie in the first one-sixth of T_e, which is the triangle with vertices T(A,D_3,CM)=T((0,0),(1/2,0),CM). If p1 and p2 are distinct and p1 is outside of T(A,D_3,CM) or p2 is outside T_e, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Usage

IarcCS.Te.onesixth(p1, p2)

Arguments

Value

I(p2 is in N_{CS}(p1,t=1)) for p1 in the first one-sixth of T_e, T(A,D_3,CM), that is, returns 1 if p2 is in N_{CS}(p1,t=1), returns 0 otherwise

Author(s)

Elvan Ceyhan

See Also

IarcCSstd.tri

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard basic triangle case

Description

Returns I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise, where N_{CS}(x,t) is the CS proximity region for point x with expansion parameter r \ge 1.

CS proximity region is defined with respect to the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)) where c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

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 the standard basic triangle T_b; default is M=(1,1,1) i.e., the center of mass of T_b. re is the index of the edge region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_b, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation, and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

IarcCSbasic.tri(p1, p2, t, c1, c2, M = c(1, 1, 1), re = NULL)

Arguments

Value

I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise

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, 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

IarcCStri and IarcCSstd.tri

Examples

c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)

tau<-2

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcCSbasic.tri(P1,P2,tau,c1,c2,M)

P1<-c(.4,.2)
P2<-c(.5,.26)
IarcCSbasic.tri(P1,P2,tau,c1,c2,M)
IarcCSbasic.tri(P1,P1,tau,c1,c2,M)

#or try
Re<-rel.edge.basic.tri(P1,c1,c2,M)$re
IarcCSbasic.tri(P1,P2,tau,c1,c2,M,Re)
IarcCSbasic.tri(P1,P1,tau,c1,c2,M,Re)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise, where N_{CS}(x,t) is the CS proximity region for point x with expansion parameter t>0. This function is equivalent to IarcCSstd.tri, except that it computes the indicator using the functions IarcCSstd.triRAB, IarcCSstd.triRBC and IarcCSstd.triRAC which are edge-region specific indicator functions. For example, IarcCSstd.triRAB computes I(p2 is in N_{CS}(p1,t)) for points p1 and p2 when p1 resides in the edge region of edge AB.

CS proximity region 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)) 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. re is the index of the edge region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_e, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

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

Usage

IarcCSedge.reg.std.tri(p1, p2, t, M = c(1, 1, 1), re = NULL)

Arguments

Value

I(p2 is in N_{CS}(p1,t)) for p1, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise

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

IarcCStri and IarcPEstd.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-1
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M)
IarcCSstd.tri(Xp[1,],Xp[2,],t,M)

#or try
re<-rel.edge.std.triCM(Xp[1,])$re
IarcCSedge.reg.std.tri(Xp[1,],Xp[2,],t,M,re=re)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - end-interval case

Description

Returns I(p_2 in N_{CS}(p_1,t)) for points p_1 and p_2, that is, returns 1 if p_2 is in N_{CS}(p_1,t), returns 0 otherwise, where N_{CS}(x,t) is the CS proximity region for point x with expansion parameter t>0 for the region outside the interval (a,b).

rv is the index of the end vertex region p_1 resides, with default=NULL, and rv=1 for left end-interval and rv=2 for the right end-interval. If p_1 and p_2 are distinct and either of them are inside interval int, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2016)).

Usage

IarcCSend.int(p1, p2, int, t, rv = NULL)

Arguments

Value

I(p_2 in N_{CS}(p_1,t)) for points p_1 and p_2, that is, returns 1 if p_2 is in N_{CS}(p_1,t) (i.e., if there is an arc from p_1 to p_2), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

See Also

IarcCSmid.int, IarcPEmid.int, and IarcPEend.int

Examples

a<-0; b<-10; int<-c(a,b)
t<-2

IarcCSend.int(15,17,int,t)
IarcCSend.int(15,15,int,t)

IarcCSend.int(1.5,17,int,t)
IarcCSend.int(1.5,1.5,int,t)

IarcCSend.int(-15,17,int,t)

IarcCSend.int(-15,-17,int,t)

a<-0; b<-10; int<-c(a,b)
t<-.5

IarcCSend.int(15,17,int,t)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one interval case

Description

Returns I(p_2 in N_{CS}(p_1,t,c)) for points p_1 and p_2, that is, returns 1 if p_2 is in N_{CS}(p_1,t,c), returns 0 otherwise, where N_{CS}(x,t,c) is the CS proximity region for point x with expansion parameter t>0 and centrality parameter c \in (0,1).

CS proximity region is constructed with respect to the interval (a,b). This function works whether p_1 and p_2 are inside or outside the interval int.

Vertex regions for middle intervals are based on the center associated with the centrality parameter c \in (0,1). If p_1 and p_2 are identical, then it returns 1 regardless of their locations (i.e., loops are allowed in the digraph).

See also (Ceyhan (2016)).

Usage

IarcCSint(p1, p2, int, t, c = 0.5)

Arguments

Value

I(p_2 in N_{CS}(p_1,t,c)) for p2, that is, returns 1 if p_2 in N_{CS}(p_1,t,c), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

See Also

IarcCSmid.int, IarcCSend.int and IarcPEint

Examples

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

IarcCSint(7,5,int,t,c)
IarcCSint(17,17,int,t,c)
IarcCSint(15,17,int,t,c)
IarcCSint(1,3,int,t,c)

IarcCSint(-17,17,int,t,c)

IarcCSint(3,5,int,t,c)
IarcCSint(3,3,int,t,c)
IarcCSint(4,5,int,t,c)
IarcCSint(a,5,int,t,c)

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

IarcCSint(7,5,int,t,c)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - middle interval case

Description

Returns I(p_2 in N_{CS}(p_1,t,c)) for points p_1 and p_2, that is, returns 1 if p_2 is in N_{CS}(p_1,t,c), returns 0 otherwise, where N_{CS}(x,t,c) is the CS proximity region for point x and is constructed with expansion parameter t>0 and centrality parameter c \in (0,1) for the interval (a,b).

CS proximity regions are defined with respect to the middle interval int and vertex regions are based on the center associated with the centrality parameter c \in (0,1). For the interval, int=(a,b), the parameterized center is M_c=a+c(b-a). rv is the index of the vertex region p_1 resides, with default=NULL.

If p_1 and p_2 are distinct and either of them are outside interval int, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., loops are allowed in the digraph).

See also (Ceyhan (2016)).

Usage

IarcCSmid.int(p1, p2, int, t, c = 0.5, rv = NULL)

Arguments

Value

I(p_2 in N_{CS}(p_1,t,c)) for points p_1 and p_2 that is, returns 1 if p_2 is in N_{CS}(p_1,t,c), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

See Also

IarcCSend.int, IarcPEmid.int, and IarcPEend.int

Examples

c<-.5
t<-2
a<-0; b<-10; int<-c(a,b)

IarcCSmid.int(7,5,int,t,c)
IarcCSmid.int(7,7,int,t,c)
IarcCSmid.int(7,5,int,t,c=.4)

IarcCSmid.int(1,3,int,t,c)

IarcCSmid.int(9,11,int,t,c)

IarcCSmid.int(19,1,int,t,c)
IarcCSmid.int(19,19,int,t,c)

IarcCSmid.int(3,5,int,t,c)

#or try
Rv<-rel.vert.mid.int(3,int,c)$rv
IarcCSmid.int(3,5,int,t,c,rv=Rv)

IarcCSmid.int(7,5,int,t,c)

The indicator for the presence of an arc from a point in set S to the point p for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns I(p in N_{CS}(x,t) for some x in S), that is, returns 1 if p is in \cup_{x in S} N_{CS}(x,t), returns 0 otherwise, CS proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) with the expansion parameter t>0 and 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 T_e; default is M=(1,1,1) i.e., the center of mass of T_e (which is equivalent to circumcenter of T_e).

Edges of T_e, AB, BC, AC, are also labeled as edges 3, 1, and 2, respectively. If p is not in S and either p or all points in S are outside T_e, it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

See also (Ceyhan (2012)).

Usage

IarcCSset2pnt.std.tri(S, p, t, M = c(1, 1, 1))

Arguments

Value

I(p is in \cup_{x in S} N_{CS}(x,t)), that is, returns 1 if p is in S or inside N_{CS}(x,t) for at least one x in S, returns 0 otherwise. CS proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) with M-edge regions.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

IarcCSset2pnt.tri, IarcCSstd.tri, IarcCStri, and IarcPEset2pnt.std.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(.5,.5)
IarcCSset2pnt.std.tri(S,Xp[3,],t,M)
IarcCSset2pnt.std.tri(S,Xp[3,],t=1,M)
IarcCSset2pnt.std.tri(S,Xp[3,],t=1.5,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcCSset2pnt.std.tri(S,Xp[3,],t,M)

The indicator for the presence of an arc from a point in set S to the point p for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case

Description

Returns I(p in N_{CS}(x,t) for some x in S), that is, returns 1 if p in \cup_{x in S} N_{CS}(x,t), returns 0 otherwise.

CS proximity region is constructed with respect to the triangle tri with the 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 the triangle tri; default is M=(1,1,1) i.e., the center of mass of tri.

Edges of tri=T(A,B,C), AB, BC, AC, are also labeled as edges 3, 1, and 2, respectively. If p is not in S and either p or all points in S are outside tri, it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

Usage

IarcCSset2pnt.tri(S, p, tri, t, M = c(1, 1, 1))

Arguments

Value

I(p is in \cup_{x in S} N_{CS}(x,t)), that is, returns 1 if p is in S or inside N_{CS}(x,t) for at least one x in S, returns 0 otherwise where CS proximity region is constructed with respect to the triangle tri

Author(s)

Elvan Ceyhan

See Also

IarcCSset2pnt.std.tri, IarcCStri, IarcCSstd.tri, IarcASset2pnt.tri, and IarcPEset2pnt.tri

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)$gen.points

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

tau<-.5

IarcCSset2pnt.tri(S,Xp[3,],Tr,tau,M)
IarcCSset2pnt.tri(S,Xp[3,],Tr,t=1,M)
IarcCSset2pnt.tri(S,Xp[3,],Tr,t=1.5,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcCSset2pnt.tri(S,Xp[3,],Tr,tau,M)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise, where N_{CS}(x,t) is the CS proximity region for point x with expansion parameter t >0.

CS proximity region 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)) and vertex 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. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_e, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

IarcCSstd.tri(p1, p2, t, M = c(1, 1, 1), re = NULL)

Arguments

Value

I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise

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, 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

IarcCStri, IarcCSbasic.tri, and IarcPEstd.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2) or M=(A+B+C)/3

IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M)
IarcCSstd.tri(c(0,1),Xp[3,],t=2,M)

#or try
Re<-rel.edge.tri(Xp[1,],Te,M) $re
IarcCSstd.tri(Xp[1,],Xp[3,],t=2,M,Re)

The indicator for the presence of an arc from a point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case with t=1

Description

Returns I(p2 is in N_{CS}(p1,t=1)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t=1), returns 0 otherwise, where N_{CS}(x,t=1) is the CS proximity region for point x with expansion parameter t=1.

CS proximity region is defined with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) and edge regions are based on the center of mass CM=(1/2,\sqrt{3}/6).

If p1 and p2 are distinct and either are outside T_e, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Usage

IarcCSt1.std.tri(p1, p2)

Arguments

Value

I(p2 is in N_{CS}(p1,t=1)) for p1 in T_e that is, returns 1 if p2 is in N_{CS}(p1,t=1), returns 0 otherwise

Author(s)

Elvan Ceyhan

See Also

IarcCSstd.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

IarcCSt1.std.tri(Xp[1,],Xp[2,])
IarcCSt1.std.tri(c(.2,.5),Xp[2,])

The indicator for the presence of an arc from one point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs)

Description

Returns I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in NCS(p1,t), returns 0 otherwise, where N_{CS}(x,t) is the CS proximity region for point x with the expansion parameter t>0.

CS proximity region is constructed with respect to the triangle tri 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 tri or based on the circumcenter of tri. re is the index of the edge region p resides, with default=NULL

If p1 and p2 are distinct and either of them are outside tri, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

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

Usage

IarcCStri(p1, p2, tri, t, M, re = NULL)

Arguments

Value

I(p2 is in NCS(p1,t)) for p1, that is, returns 1 if p2 is in NCS(p1,t), returns 0 otherwise

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

IarcAStri, IarcPEtri, IarcCStri, and IarcCSstd.tri

Examples

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

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

n<-10
set.seed(1)
Xp<-runif.tri(n,Tr)$g

IarcCStri(Xp[1,],Xp[2,],Tr,tau,M)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcCStri(P1,P2,Tr,tau,M)

#or try
re<-rel.edges.tri(P1,Tr,M)$re
IarcCStri(P1,P2,Tr,tau,M,re)

An alternative to the function IarcCStri which yields the indicator for the presence of an arc from one point to another for Central Similarity Proximity Catch Digraphs (CS-PCDs)

Description

Returns I(p2 is in N_{CS}(p1,t)) for points p1 and p2, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise, where N_{CS}(x,t) is the CS proximity region for point x with the expansion parameter t>0.

CS proximity region is constructed with respect to the triangle tri and edge regions are based on the center of mass, CM. re is the index of the CM-edge region p resides, with default=NULL but must be provided as vertices (y_1,y_2,y_3) for re=3 as rbind(y2,y3,y1) for re=1 and as rbind(y1,y3,y2) for re=2 for triangle T(y_1,y_2,y_3).

If p1 and p2 are distinct and either of them are outside tri, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

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

Usage

IarcCStri.alt(p1, p2, tri, t, re = NULL)

Arguments

Value

I(p2 is in N_{CS}(p1,t)) for p1, that is, returns 1 if p2 is in N_{CS}(p1,t), returns 0 otherwise.

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

IarcAStri, IarcPEtri, IarcCStri, and IarcCSstd.tri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.6,2);
Tr<-rbind(A,B,C);
t<-1.5

P1<-c(.4,.2)
P2<-c(1.8,.5)
IarcCStri(P1,P2,Tr,t,M=c(1,1,1))
IarcCStri.alt(P1,P2,Tr,t)

IarcCStri(P2,P1,Tr,t,M=c(1,1,1))
IarcCStri.alt(P2,P1,Tr,t)

#or try
re<-rel.edges.triCM(P1,Tr)$re
IarcCStri(P1,P2,Tr,t,M=c(1,1,1),re)
IarcCStri.alt(P1,P2,Tr,t,re)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic triangle case

Description

Returns I(p2 is in N_{PE}(p1,r)) for points p1 and p2 in the standard basic triangle, that is, returns 1 if p2 is in N_{PE}(p1,r), and returns 0 otherwise, where N_{PE}(x,r) is the PE proximity region for point x with expansion parameter r \ge 1.

PE proximity region is defined with respect to the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)) where c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

Vertex 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 the standard basic triangle T_b or based on circumcenter of T_b; default is M=(1,1,1), i.e., the center of mass of T_b. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_b, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010); Ceyhan et al. (2006)).

Usage

IarcPEbasic.tri(p1, p2, r, c1, c2, M = c(1, 1, 1), rv = NULL)

Arguments

Value

I(p2 is in N_{PE}(p1,r)) for points p1 and p2 in the standard basic triangle, that is, returns 1 if p2 is in N_{PE}(p1,r), and returns 0 otherwise.

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, 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

IarcPEtri and IarcPEstd.tri

Examples

c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)

r<-2

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)
P2<-as.numeric(runif.basic.tri(1,c1,c2)$g)
IarcPEbasic.tri(P1,P2,r,c1,c2,M)

P1<-c(.4,.2)
P2<-c(.5,.26)
IarcPEbasic.tri(P1,P2,r,c1,c2,M)
IarcPEbasic.tri(P2,P1,r,c1,c2,M)

#or try
Rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
IarcPEbasic.tri(P1,P2,r,c1,c2,M,Rv)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - end-interval case

Description

Returns I(p_2 \in N_{PE}(p_1,r)) for points p_1 and p_2, that is, returns 1 if p_2 is in N_{PE}(p_1,r), returns 0 otherwise, where N_{PE}(x,r) is the PE proximity region for point x with expansion parameter r \ge 1 for the region outside the interval (a,b).

rv is the index of the end vertex region p_1 resides, with default=NULL, and rv=1 for left end-interval and rv=2 for the right end-interval. If p_1 and p_2 are distinct and either of them are inside interval int, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2012)).

Usage

IarcPEend.int(p1, p2, int, r, rv = NULL)

Arguments

Value

I(p_2 \in N_{PE}(p_1,r)) for points p_1 and p_2, that is, returns 1 if p_2 is in N_{PE}(p_1,r) (i.e., if there is an arc from p_1 to p_2), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

See Also

IarcPEmid.int, IarcCSmid.int, and IarcCSend.int

Examples

a<-0; b<-10; int<-c(a,b)
r<-2

IarcPEend.int(15,17,int,r)
IarcPEend.int(1.5,17,int,r)
IarcPEend.int(-15,17,int,r)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one interval case

Description

Returns I(p_2 \in N_{PE}(p_1,r,c)) for points p_1 and p_2, that is, returns 1 if p_2 is in N_{PE}(p_1,r,c), returns 0 otherwise, where N_{PE}(x,r,c) is the PE proximity region for point x with expansion parameter r \ge 1 and centrality parameter c \in (0,1).

PE proximity region is constructed with respect to the interval (a,b). This function works whether p_1 and p_2 are inside or outside the interval int.

Vertex regions for middle intervals are based on the center associated with the centrality parameter c \in (0,1). If p_1 and p_2 are identical, then it returns 1 regardless of their locations (i.e., loops are allowed in the digraph).

See also (Ceyhan (2012)).

Usage

IarcPEint(p1, p2, int, r, c = 0.5)

Arguments

Value

I(p_2 \in N_{PE}(p_1,r,c)), that is, returns 1 if p_2 in N_{PE}(p_1,r,c), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

See Also

IarcPEmid.int, IarcPEend.int and IarcCSint

Examples

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

IarcPEint(7,5,int,r,c)
IarcPEint(15,17,int,r,c)
IarcPEint(1,3,int,r,c)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case

Description

Returns I(p_2 \in N_{PE}(p_1,r,c)) for points p_1 and p_2, that is, returns 1 if p_2 is in N_{PE}(p_1,r,c), returns 0 otherwise, where N_{PE}(x,r,c) is the PE proximity region for point x and is constructed with expansion parameter r \ge 1 and centrality parameter c \in (0,1) for the interval (a,b).

PE proximity regions are defined with respect to the middle interval int and vertex regions are based on the center associated with the centrality parameter c \in (0,1). For the interval, int=(a,b), the parameterized center is M_c=a+c(b-a). rv is the index of the vertex region p_1 resides, with default=NULL. If p_1 and p_2 are distinct and either of them are outside interval int, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., loops are allowed in the digraph).

See also (Ceyhan (2012, 2016)).

Usage

IarcPEmid.int(p1, x2, int, r, c = 0.5, rv = NULL)

Arguments

Value

I(p_2 \in N_{PE}(p_1,r,c)) for points p_1 and p_2 that is, returns 1 if p_2 is in N_{PE}(p_1,r,c), returns 0 otherwise

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

See Also

IarcPEend.int, IarcCSmid.int, and IarcCSend.int

Examples

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

IarcPEmid.int(7,5,int,r,c)
IarcPEmid.int(1,3,int,r,c)

The indicator for the presence of an arc from a point in set S to the point p or Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns I(p in N_{PE}(x,r) for some x in S) for S, in the standard equilateral triangle, that is, returns 1 if p is in \cup_{x in S}N_{PE}(x,r), and returns 0 otherwise.

PE proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) with the expansion parameter r \ge 1 and 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 T_e; default is M=(1,1,1), i.e., the center of mass of T_e (which is equivalent to the circumcenter for T_e).

Vertices of T_e are also labeled as 1, 2, and 3, respectively. If p is not in S and either p or all points in S are outside T_e, it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

Usage

IarcPEset2pnt.std.tri(S, p, r, M = c(1, 1, 1))

Arguments

Value

I(p is in U_{x in S} N_{PE}(x,r)) for S in the standard equilateral triangle, that is, returns 1 if p is in S or inside N_{PE}(x,r) for at least one x in S, and returns 0 otherwise. PE proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) with M-vertex regions

Author(s)

Elvan Ceyhan

See Also

IarcPEset2pnt.tri, IarcPEstd.tri, IarcPEtri, and IarcCSset2pnt.std.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(.5,.5)
IarcPEset2pnt.std.tri(S,Xp[3,],r,M)
IarcPEset2pnt.std.tri(S,Xp[3,],r=1,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcPEset2pnt.std.tri(S,Xp[3,],r,M)

IarcPEset2pnt.std.tri(S,Xp[6,],r,M)
IarcPEset2pnt.std.tri(S,Xp[6,],r=1.25,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcPEset2pnt.std.tri(Xp,P,r,M)

The indicator for the presence of an arc from a point in set S to the point p for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns I(p in N_{PE}(x,r) for some x in S), that is, returns 1 if p is in \cup_{x in S}N_{PE}(x,r), and returns 0 otherwise.

PE proximity region is constructed with respect to the triangle tri with the expansion parameter r \ge 1 and vertex 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 the triangle tri or based on the circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri. Vertices of tri are also labeled as 1, 2, and 3, respectively.

If p is not in S and either p or all points in S are outside tri, it returns 0, but if p is in S, then it always returns 1 regardless of its location (i.e., loops are allowed).

Usage

IarcPEset2pnt.tri(S, p, tri, r, M = c(1, 1, 1))

Arguments

Value

I(p is in U_{x in S} N_{PE}(x,r)), that is, returns 1 if p is in S or inside N_{PE}(x,r) for at least one x in S, and returns 0 otherwise, where PE proximity region is constructed with respect to the triangle tri

Author(s)

Elvan Ceyhan

See Also

IarcPEset2pnt.std.tri, IarcPEtri, IarcPEstd.tri, IarcASset2pnt.tri, and IarcCSset2pnt.tri

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)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])  #try also S<-c(1.5,1)

IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)
IarcPEset2pnt.tri(S,Xp[3,],r=1,Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
IarcPEset2pnt.tri(S,Xp[3,],Tr,r,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
IarcPEset2pnt.tri(Xp,P,Tr,r,M)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case

Description

Returns I(p2 is in N_{PE}(p1,r)) for points p1 and p2, that is, returns 1 if p2 is in N_{PE}(p1,r), returns 0 otherwise, where N_{PE}(x,r) is the PE proximity region for point x with expansion parameter r \ge 1.

PE proximity region is defined with respect to the standard regular tetrahedron T_h=T(v=1,v=2,v=3,v=4)=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3)) and vertex regions are based on the circumcenter (which is equivalent to the center of mass for standard regular tetrahedron) of T_h. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_h, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010)).

Usage

IarcPEstd.tetra(p1, p2, r, rv = NULL)

Arguments

Value

I(p2 is in N_{PE}(p1,r)) for points p1 and p2, that is, returns 1 if p2 is in N_{PE}(p1,r), returns 0 otherwise

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.

See Also

IarcPEtetra, IarcPEtri and IarcPEint

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-3  #try also n<-20
Xp<-runif.std.tetra(n)$g
r<-1.5
IarcPEstd.tetra(Xp[1,],Xp[3,],r)
IarcPEstd.tetra(c(.4,.4,.4),c(.5,.5,.5),r)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
IarcPEstd.tetra(Xp[1,],Xp[3,],r,rv=RV)

P1<-c(.1,.1,.1)
P2<-c(.5,.5,.5)
IarcPEstd.tetra(P1,P2,r)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns I(p2 is in N_{PE}(p1,r)) for points p1 and p2 in the standard equilateral triangle, that is, returns 1 if p2 is in N_{PE}(p1,r), and returns 0 otherwise, where N_{PE}(x,r) is the PE proximity region for point x with expansion parameter r \ge 1.

PE proximity region 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)) and vertex 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. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside T_e, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

IarcPEstd.tri(p1, p2, r, M = c(1, 1, 1), rv = NULL)

Arguments

Value

I(p2 is in N_{PE}(p1,r)) for points p1 and p2 in the standard equilateral triangle, that is, returns 1 if p2 is in N_{PE}(p1,r), and returns 0 otherwise.

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, 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

IarcPEtri, IarcPEbasic.tri, and IarcCSstd.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C)
n<-3

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

IarcPEstd.tri(Xp[1,],Xp[3,],r=1.5,M)
IarcPEstd.tri(Xp[1,],Xp[3,],r=2,M)

#or try
Rv<-rel.vert.std.triCM(Xp[1,])$rv
IarcPEstd.tri(Xp[1,],Xp[3,],r=2,rv=Rv)

P1<-c(.4,.2)
P2<-c(.5,.26)
r<-2
IarcPEstd.tri(P1,P2,r,M)

The indicator for the presence of an arc from one 3D point to another 3D point for Proportional Edge Proximity Catch Digraphs (PE-PCDs)

Description

Returns I(p2 is in N_{PE}(p1,r)) for 3D points p1 and p2, that is, returns 1 if p2 is in N_{PE}(p1,r), returns 0 otherwise, where N_{PE}(x,r) is the PE proximity region for point x with the expansion parameter r \ge 1.

PE proximity region is constructed with respect to the tetrahedron th and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM". rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside th, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005, 2010)).

Usage

IarcPEtetra(p1, p2, th, r, M = "CM", rv = NULL)

Arguments

Value

I(p2 is in N_{PE}(p1,r)) for p1, that is, returns 1 if p2 is in N_{PE}(p1,r), returns 0 otherwise

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.

See Also

IarcPEstd.tetra, IarcPEtri and IarcPEint

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-3  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.5

IarcPEtetra(Xp[1,],Xp[2,],tetra,r)  #uses the default M="CM"
IarcPEtetra(Xp[1,],Xp[2,],tetra,r,M)

IarcPEtetra(c(.4,.4,.4),c(.5,.5,.5),tetra,r,M)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
IarcPEtetra(Xp[1,],Xp[3,],tetra,r,M,rv=RV)

P1<-c(.1,.1,.1)
P2<-c(.5,.5,.5)
IarcPEtetra(P1,P2,tetra,r,M)

The indicator for the presence of an arc from a point to another for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns I(p2 is in N_{PE}(p1,r)) for points p1 and p2, that is, returns 1 if p2 is in N_{PE}(p1,r), and returns 0 otherwise, where N_{PE}(x,r) is the PE proximity region for point x with the expansion parameter r \ge 1.

PE proximity region is constructed with respect to the triangle tri and vertex 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 tri or based on the circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri. rv is the index of the vertex region p1 resides, with default=NULL.

If p1 and p2 are distinct and either of them are outside tri, it returns 0, but if they are identical, then it returns 1 regardless of their locations (i.e., it allows loops).

See also (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)).

Usage

IarcPEtri(p1, p2, tri, r, M = c(1, 1, 1), rv = NULL)

Arguments

Value

I(p2 is in N_{PE}(p1,r)) for points p1 and p2, that is, returns 1 if p2 is in N_{PE}(p1,r), and returns 0 otherwise.

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

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

IarcPEbasic.tri, IarcPEstd.tri, IarcAStri, and IarcCStri

Examples

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

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0);

r<-1.5

n<-3
set.seed(1)
Xp<-runif.tri(n,Tr)$g

IarcPEtri(Xp[1,],Xp[2,],Tr,r,M)

P1<-as.numeric(runif.tri(1,Tr)$g)
P2<-as.numeric(runif.tri(1,Tr)$g)
IarcPEtri(P1,P2,Tr,r,M)

P1<-c(.4,.2)
P2<-c(1.8,.5)
IarcPEtri(P1,P2,Tr,r,M)
IarcPEtri(P2,P1,Tr,r,M)

M<-c(1.3,1.3)
r<-2

#or try
Rv<-rel.vert.tri(P1,Tr,M)$rv
IarcPEtri(P1,P2,Tr,r,M,Rv)

Indicator for an upper bound for the domination number by the exact algorithm

Description

Returns 1 if the domination number is less than or equal to the prespecified value k and also the indices (i.e., row numbers) of a dominating set of size k based on the incidence matrix Inc.Mat of a graph or a digraph. Here the row number in the incidence matrix corresponds to the index of the vertex (i.e., index of the data point). The function works whether loops are allowed or not (i.e., whether the first diagonal is all 1 or all 0). It takes a rather long time for large number of vertices (i.e., large number of row numbers).

Usage

Idom.num.up.bnd(Inc.Mat, k)

Arguments

Value

A list with two elements

Author(s)

Elvan Ceyhan

See Also

dom.num.exact and dom.num.greedy

Examples

n<-10
M<-matrix(sample(c(0,1),n^2,replace=TRUE),nrow=n)
diag(M)<-1

dom.num.greedy(M)
Idom.num.up.bnd(M,2)

for (k in 1:n)
print(c(k,Idom.num.up.bnd(M,k)))

The indicator for a point being a dominating point for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case

Description

Returns I(p is a dominating point of the AS-PCD) where the vertices of the AS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point of AS-PCD, returns 0 otherwise. AS proximity regions are defined with respect to the standard basic triangle, T_b, c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

Vertex 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 the standard basic triangle T_b or based on circumcenter of T_b; default is M="CC", i.e., circumcenter of T_b. Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as 1,2,3 in the order they are stacked row-wise.

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1ASbasic.tri(p, Xp, c1, c2, M = "CC", rv = NULL, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of the AS-PCD) where the vertices of the AS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.num1AStri and Idom.num1PEbasic.tri

Examples

c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Idom.num1ASbasic.tri(Xp[1,],Xp,c1,c2,M)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1ASbasic.tri(Xp[i,],Xp,c1,c2,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
Idom.num1ASbasic.tri(Xp[1,],Xp,c1,c2,M,Rv)

Idom.num1ASbasic.tri(c(.2,.4),Xp,c1,c2,M)
Idom.num1ASbasic.tri(c(.2,.4),c(.2,.4),c1,c2,M)

Xp2<-rbind(Xp,c(.2,.4))
Idom.num1ASbasic.tri(Xp[1,],Xp2,c1,c2,M)

CC<-circumcenter.basic.tri(c1,c2)  #the circumcenter

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

if (isTRUE(all.equal(M,CC)) || identical(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.basic.tri(c1,c2,M)
}

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(Xp)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)

txt<-rbind(Tb,cent,Ds)
xc<-txt[,1]+c(-.03,.03,.02,.06,.06,-0.05,.01)
yc<-txt[,2]+c(.02,.02,.03,.0,.03,.03,-.03)
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1ASbasic.tri(c(.4,.2),Xp,c1,c2,M)

Idom.num1ASbasic.tri(c(.5,.11),Xp,c1,c2,M)

Idom.num1ASbasic.tri(c(.5,.11),Xp,c1,c2,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since the point is not in the standard basic triangle

The indicator for a point being a dominating point for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns I(p is a dominating point of the AS-PCD whose vertices are the 2D data set Xp), that is, returns 1 if p is a dominating point of AS-PCD, returns 0 otherwise. Point, p, is in the region of vertex rv (default is NULL); vertices are labeled as 1,2,3 in the order they are stacked row-wise in tri.

AS proximity regions are defined with respect to the triangle tri and vertex 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 the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri.

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1AStri(p, Xp, tri, M = "CC", rv = NULL, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of the AS-PCD whose vertices are the 2D data set Xp), that is, returns 1 if p is a dominating point of the AS-PCD, returns 0 otherwise

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.num1ASbasic.tri

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(1.6,1.2)

Idom.num1AStri(Xp[1,],Xp,Tr,M)
Idom.num1AStri(Xp[1,],Xp[1,],Tr,M)
Idom.num1AStri(c(1.5,1.5),c(1.6,1),Tr,M)
Idom.num1AStri(c(1.6,1),c(1.5,1.5),Tr,M)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1AStri(Xp[i,],Xp,Tr,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.triCC(Xp[1,],Tr)$rv
Idom.num1AStri(Xp[1,],Xp,Tr,M,Rv)

Idom.num1AStri(c(.2,.4),Xp,Tr,M)
Idom.num1AStri(c(.2,.4),c(.2,.4),Tr,M)

Xp2<-rbind(Xp,c(.2,.4))
Idom.num1AStri(Xp[1,],Xp2,Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

CC<-circumcenter.tri(Tr)  #the circumcenter

if (isTRUE(all.equal(M,CC)) || identical(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)
}

Xlim<-range(Tr[,1],Xp[,1])
Ylim<-range(Tr[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp)
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)

txt<-rbind(Tr,cent,Ds)
xc<-txt[,1]
yc<-txt[,2]
txt.str<-c("A","B","C",cent.name,"D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1AStri(c(1.5,1.1),Xp,Tr,M)

Idom.num1AStri(c(1.5,1.1),Xp,Tr,M)

Idom.num1AStri(c(1.5,1.1),Xp,Tr,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since point p is not a data point in Xp

The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - first one-sixth of the standard equilateral triangle case

Description

Returns I(p is a dominating point of the 2D data set Xp of CS-PCD) in the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)), that is, returns 1 if p is a dominating point of CS-PCD, returns 0 otherwise.

Point, p, must lie in the first one-sixth of T_e, which is the triangle with vertices T(A,D_3,CM)=T((0,0),(1/2,0),CM).

CS proximity region is constructed with respect to T_e with expansion parameter t=1.

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005)).

Usage

Idom.num1CS.Te.onesixth(p, Xp, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of the CS-PCD) where the vertices of the CS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

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.

See Also

Idom.num1CSstd.tri and Idom.num1CSt1std.tri

The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) for an interval

Description

Returns I(p is a dominating point of CS-PCD) where the vertices of the CS-PCD are the 1D data set Xp).

CS proximity region is defined with respect to the interval int with an expansion parameter, t>0, and a centrality parameter, c \in (0,1), so arcs may exist for Xp points inside the interval int=(a,b).

Vertex regions are based on the center associated with the centrality parameter c \in (0,1). rv is the index of the vertex region p resides, with default=NULL.

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

Usage

Idom.num1CSint(p, Xp, int, t, c = 0.5, rv = NULL, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of CS-PCD) where the vertices of the CS-PCD are the 1D data set Xp), that is, returns 1 if p is a dominating point, returns 0 otherwise

Author(s)

Elvan Ceyhan

See Also

Idom.num1PEint

Examples

t<-2
c<-.4
a<-0; b<-10; int<-c(a,b)

Mc<-centerMc(int,c)
n<-10

set.seed(1)
Xp<-runif(n,a,b)

Idom.num1CSint(Xp[5],Xp,int,t,c)

Idom.num1CSint(2,Xp,int,t,c,ch.data.pnt = FALSE)
#gives an error if ch.data.pnt = TRUE since p is not a data point in Xp

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSint(Xp[i],Xp,int,t,c))}

ind.gam1<-which(gam.vec==1)
ind.gam1

domset<-Xp[ind.gam1]
if (length(ind.gam1)==0)
{domset<-NA}

#or try
Rv<-rel.vert.mid.int(Xp[5],int,c)$rv
Idom.num1CSint(Xp[5],Xp,int,t,c,Rv)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",xlim=Xlim+xd*c(-.05,.05))
abline(h=0)
abline(v=c(a,b,Mc),col=c(1,1,2),lty=2)
points(cbind(Xp,0))
points(cbind(domset,0),pch=4,col=2)
text(cbind(c(a,b,Mc),-0.1),c("a","b","Mc"))

Idom.num1CSint(Xp[5],Xp,int,t,c)

n<-10
Xp2<-runif(n,a+b,b+10)
Idom.num1CSint(5,Xp2,int,t,c)

The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns I(p is a dominating point of the CS-PCD) where the vertices of the CS-PCD are the 2D data set Xp in the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)), that is, returns 1 if p is a dominating point of CS-PCD, returns 0 otherwise.

CS proximity region is constructed with respect to T_e with expansion parameter t>0 and edge regions are based on center of mass CM=(1/2,\sqrt{3}/6).

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1CSstd.tri(p, Xp, t, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of the CS-PCD) where the vertices of the CS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.num1CSt1std.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
Te<-rbind(A,B,C);
t<-1.5
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num1CSstd.tri(Xp[3,],Xp,t)
Idom.num1CSstd.tri(c(1,2),c(1,2),t)
Idom.num1CSstd.tri(c(1,2),c(1,2),t,ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSstd.tri(Xp[i,],Xp,t))}

ind.gam1<-which(gam.vec==1)
ind.gam1

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE);
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.02,.01,.05)
yc<-txt[,2]+c(.02,.02,.03,.02)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

Idom.num1CSstd.tri(c(1,2),Xp,t,ch.data.pnt = FALSE)
#gives an error if ch.data.pnt = TRUE message since p is not a data point

The indicator for a point being a dominating point for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case with t=1

Description

Returns I(p is a dominating point of the CS-PCD) where the vertices of the CS-PCD are the 2D data set Xp in the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)), that is, returns 1 if p is a dominating point of CS-PCD, returns 0 otherwise.

Point, p, is in the edge region of edge re (default is NULL); vertices are labeled as 1,2,3 in the order they are stacked row-wise in T_e, and the opposite edges are labeled with label of the vertices (that is, edge numbering is 1,2, and 3 for edges AB, BC, and AC).

CS proximity region is constructed with respect to T_e with expansion parameter t=1 and edge regions are based on center of mass CM=(1/2,\sqrt{3}/6).

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1CSt1std.tri(p, Xp, re = NULL, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of the CS-PCD) where the vertices of the CS-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise.

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.num1CSstd.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
CM<-(A+B+C)/3
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

Idom.num1CSt1std.tri(Xp[3,],Xp)

Idom.num1CSt1std.tri(c(1,2),c(1,2))
Idom.num1CSt1std.tri(c(1,2),c(1,2),ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1CSt1std.tri(Xp[i,],Xp))}

ind.gam1<-which(gam.vec==1)
ind.gam1

Xlim<-range(Te[,1],Xp[,1])
Ylim<-range(Te[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Te,pch=".",xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Te)
points(Xp)
L<-Te; R<-matrix(rep(CM,3),ncol=2,byrow=TRUE);
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Te,CM)
xc<-txt[,1]+c(-.02,.02,.01,.05)
yc<-txt[,2]+c(.02,.02,.03,.02)
txt.str<-c("A","B","C","CM")
text(xc,yc,txt.str)

The indicator for a point being a dominating point or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic triangle case

Description

Returns I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp for data in the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)), that is, returns 1 if p is a dominating point of PE-PCD, and returns 0 otherwise.

PE proximity regions are defined with respect to the standard basic triangle T_b. In the standard basic triangle, T_b, c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle is useful for simulation studies under the uniformity hypothesis.

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 a standard basic triangle to the edges on the extension of the lines joining M to the vertices or based on the circumcenter of T_b; default is M=(1,1,1), i.e., the center of mass of T_b. Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as 1,2,3 in the order they are stacked row-wise.

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005, 2011)).

Usage

Idom.num1PEbasic.tri(
  p,
  Xp,
  r,
  c1,
  c2,
  M = c(1, 1, 1),
  rv = NULL,
  ch.data.pnt = FALSE
)

Arguments

Value

I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, and returns 0 otherwise.

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

See Also

Idom.num1ASbasic.tri and Idom.num1AStri

Examples

c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)
r<-2

P<-c(.4,.2)
Idom.num1PEbasic.tri(P,Xp,r,c1,c2,M)
Idom.num1PEbasic.tri(Xp[1,],Xp,r,c1,c2,M)

Idom.num1PEbasic.tri(c(1,1),Xp,r,c1,c2,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since point p=c(1,1) is not a data point in Xp

#or try
Rv<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv
Idom.num1PEbasic.tri(Xp[1,],Xp,r,c1,c2,M,Rv)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEbasic.tri(Xp[i,],Xp,r,c1,c2,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

Xlim<-range(Tb[,1],Xp[,1])
Ylim<-range(Tb[,2],Xp[,2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

if (dimension(M)==3) {M<-bary2cart(M,Tb)}
#need to run this when M is given in barycentric coordinates

if (identical(M,circumcenter.tri(Tb)))
{
  plot(Tb,pch=".",asp=1,xlab="",ylab="",axes=TRUE,
  xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
  polygon(Tb)
  points(Xp,pch=1,col=1)
  Ds<-rbind((B+C)/2,(A+C)/2,(A+B)/2)
} else
{plot(Tb,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
  polygon(Tb)
  points(Xp,pch=1,col=1)
  Ds<-prj.cent2edges.basic.tri(c1,c2,M)}
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)

txt<-rbind(Tb,M,Ds)
xc<-txt[,1]+c(-.02,.02,.02,-.02,.03,-.03,.01)
yc<-txt[,2]+c(.02,.02,.02,-.02,.02,.02,-.03)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

Idom.num1PEbasic.tri(c(.2,.1),Xp,r,c1,c2,M,ch.data.pnt=FALSE)
#gives an error message if ch.data.pnt=TRUE since point p is not a data point in Xp

The indicator for a point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) for an interval

Description

Returns I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 1D data set Xp.

PE proximity region is defined with respect to the interval int with an expansion parameter, r \ge 1, and a centrality parameter, c \in (0,1), so arcs may exist for Xp points inside the interval int=(a,b).

Vertex regions are based on the center associated with the centrality parameter c \in (0,1). rv is the index of the vertex region p resides, with default=NULL.

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

Usage

Idom.num1PEint(p, Xp, int, r, c = 0.5, rv = NULL, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 1D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

Author(s)

Elvan Ceyhan

See Also

Idom.num1PEtri

Examples

r<-2
c<-.4
a<-0; b<-10
int=c(a,b)

Mc<-centerMc(int,c)

n<-10

set.seed(1)
Xp<-runif(n,a,b)

Idom.num1PEint(Xp[5],Xp,int,r,c)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEint(Xp[i],Xp,int,r,c))}

ind.gam1<-which(gam.vec==1)
ind.gam1

domset<-Xp[ind.gam1]
if (length(ind.gam1)==0)
{domset<-NA}

#or try
Rv<-rel.vert.mid.int(Xp[5],int,c)$rv
Idom.num1PEint(Xp[5],Xp,int,r,c,Rv)

Xlim<-range(a,b,Xp)
xd<-Xlim[2]-Xlim[1]

plot(cbind(a,0),xlab="",pch=".",xlim=Xlim+xd*c(-.05,.05))
abline(h=0)
points(cbind(Xp,0))
abline(v=c(a,b,Mc),col=c(1,1,2),lty=2)
points(cbind(domset,0),pch=4,col=2)
text(cbind(c(a,b,Mc),-0.1),c("a","b","Mc"))

Idom.num1PEint(2,Xp,int,r,c,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since point p is not a data point in Xp

The indicator for a 3D point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case

Description

Returns I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp in the standard regular tetrahedron T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3)), that is, returns 1 if p is a dominating point of PE-PCD, returns 0 otherwise.

Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as 1,2,3,4 in the order they are stacked row-wise in T_h.

PE proximity region is constructed with respect to the tetrahedron T_h with expansion parameter r \ge 1 and vertex regions are based on center of mass CM (equivalent to circumcenter in this case).

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1PEstd.tetra(p, Xp, r, rv = NULL, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

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.

See Also

Idom.num1PEtetra, Idom.num1PEtri and Idom.num1PEbasic.tri

Examples

set.seed(123)
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-5 #try also n<-20
Xp<-runif.std.tetra(n)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))
r<-1.5

P<-c(.4,.1,.2)
Idom.num1PEstd.tetra(Xp[1,],Xp,r)
Idom.num1PEstd.tetra(P,Xp,r)

Idom.num1PEstd.tetra(Xp[1,],Xp,r)
Idom.num1PEstd.tetra(Xp[1,],Xp[1,],r)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
Idom.num1PEstd.tetra(Xp[1,],Xp,r,rv=RV)

Idom.num1PEstd.tetra(c(-1,-1,-1),Xp,r)
Idom.num1PEstd.tetra(c(-1,-1,-1),c(-1,-1,-1),r)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEstd.tetra(Xp[i,],Xp,r))}

ind.gam1<-which(gam.vec==1)
ind.gam1
g1.pts<-Xp[ind.gam1,]

Xlim<-range(tetra[,1],Xp[,1])
Ylim<-range(tetra[,2],Xp[,2])
Zlim<-range(tetra[,3],Xp[,3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi =0,theta=40, bty = "g",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
         pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
if (length(g1.pts)!=0)
{
  if (length(g1.pts)==3) g1.pts<-matrix(g1.pts,nrow=1)
  plot3D::points3D(g1.pts[,1],g1.pts[,2],g1.pts[,3], pch=4,col="red", add=TRUE)}

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), add=TRUE)

CM<-apply(tetra,2,mean)
D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-matrix(rep(CM,6),ncol=3,byrow=TRUE)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2)

P<-c(.4,.1,.2)
Idom.num1PEstd.tetra(P,Xp,r)

Idom.num1PEstd.tetra(c(-1,-1,-1),Xp,r,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE

The indicator for a 3D point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp in the tetrahedron th, that is, returns 1 if p is a dominating point of PE-PCD, returns 0 otherwise.

Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as 1,2,3,4 in the order they are stacked row-wise in th.

PE proximity region is constructed with respect to the tetrahedron th with expansion parameter r \ge 1 and vertex regions are based on center of mass (M="CM") or circumcenter (M="CC") only. and vertex regions are based on center of mass CM (equivalent to circumcenter in this case).

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num1PEtetra(p, Xp, th, r, M = "CM", rv = NULL, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, returns 0 otherwise

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.

See Also

Idom.num1PEstd.tetra, Idom.num1PEtri and Idom.num1PEbasic.tri

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5 #try also n<-20

Xp<-runif.tetra(n,tetra)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))

M<-"CM"; cent<-apply(tetra,2,mean)  #center of mass
#try also M<-"CC"; cent<-circumcenter.tetra(tetra)  #circumcenter

r<-2

P<-c(.4,.1,.2)
Idom.num1PEtetra(Xp[1,],Xp,tetra,r,M)
Idom.num1PEtetra(P,Xp,tetra,r,M)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
Idom.num1PEtetra(Xp[1,],Xp,tetra,r,M,rv=RV)

Idom.num1PEtetra(c(-1,-1,-1),Xp,tetra,r,M)
Idom.num1PEtetra(c(-1,-1,-1),c(-1,-1,-1),tetra,r,M)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEtetra(Xp[i,],Xp,tetra,r,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1
g1.pts<-Xp[ind.gam1,]

Xlim<-range(tetra[,1],Xp[,1],cent[1])
Ylim<-range(tetra[,2],Xp[,2],cent[2])
Zlim<-range(tetra[,3],Xp[,3],cent[3])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi =0,theta=40, bty = "g",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05),
         pch = 20, cex = 1, ticktype = "detailed")
#add the vertices of the tetrahedron
plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE)
L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2)
if (length(g1.pts)!=0)
{plot3D::points3D(g1.pts[,1],g1.pts[,2],g1.pts[,3], pch=4,col="red", add=TRUE)}

plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), add=TRUE)

D1<-(A+B)/2; D2<-(A+C)/2; D3<-(A+D)/2; D4<-(B+C)/2; D5<-(B+D)/2; D6<-(C+D)/2;
L<-rbind(D1,D2,D3,D4,D5,D6); R<-rbind(cent,cent,cent,cent,cent,cent)
plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lty=2)

P<-c(.4,.1,.2)
Idom.num1PEtetra(P,Xp,tetra,r,M)

Idom.num1PEtetra(c(-1,-1,-1),Xp,tetra,r,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since p is not a data point

The indicator for a point being a dominating point for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp in the triangle tri, that is, returns 1 if p is a dominating point of PE-PCD, and returns 0 otherwise.

Point, p, is in the vertex region of vertex rv (default is NULL); vertices are labeled as 1,2,3 in the order they are stacked row-wise in tri.

PE proximity region is constructed with respect to the triangle tri with expansion parameter r \ge 1 and 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.

ch.data.pnt is for checking whether point p is a data point in Xp or not (default is FALSE), so by default this function checks whether the point p would be a dominating point if it actually were in the data set.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

Idom.num1PEtri(p, Xp, tri, r, M = c(1, 1, 1), rv = NULL, ch.data.pnt = FALSE)

Arguments

Value

I(p is a dominating point of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if p is a dominating point, and returns 0 otherwise.

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Ceyhan E, Priebe CE (2007). “On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs.” Model Assisted Statistics and Applications, 1(4), 231-255.

See Also

Idom.num1PEbasic.tri and Idom.num1AStri

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

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

Idom.num1PEtri(Xp[1,],Xp,Tr,r,M)
Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M)
Idom.num1PEtri(c(1,2),c(1,2),Tr,r,M,ch.data.pnt = TRUE)

gam.vec<-vector()
for (i in 1:n)
{gam.vec<-c(gam.vec,Idom.num1PEtri(Xp[i,],Xp,Tr,r,M))}

ind.gam1<-which(gam.vec==1)
ind.gam1

#or try
Rv<-rel.vert.tri(Xp[1,],Tr,M)$rv
Idom.num1PEtri(Xp[1,],Xp,Tr,r,M,Rv)

Ds<-prj.cent2edges(Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

Xlim<-range(Tr[,1],Xp[,1],M[1])
Ylim<-range(Tr[,2],Xp[,2],M[2])
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(Tr,pch=".",xlab="",ylab="",axes=TRUE,
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tr)
points(Xp,pch=1,col=1)
L<-rbind(M,M,M); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
points(rbind(Xp[ind.gam1,]),pch=4,col=2)
#rbind is to insert the points correctly if there is only one dominating point

txt<-rbind(Tr,M,Ds)
xc<-txt[,1]+c(-.02,.03,.02,-.02,.04,-.03,.0)
yc<-txt[,2]+c(.02,.02,.05,-.03,.04,.06,-.07)
txt.str<-c("A","B","C","M","D1","D2","D3")
text(xc,yc,txt.str)

P<-c(1.4,1)
Idom.num1PEtri(P,P,Tr,r,M)
Idom.num1PEtri(Xp[1,],Xp,Tr,r,M)

Idom.num1PEtri(c(1,2),Xp,Tr,r,M,ch.data.pnt = FALSE)
#gives an error message if ch.data.pnt = TRUE since p is not a data point

The indicator for two points being a dominating set for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case

Description

Returns I({p1,p2} is a dominating set of AS-PCD) where vertices of AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise.

AS proximity regions are defined with respect to the standard basic triangle T_b=T(c(0,0),c(1,0),c(c1,c2)), In the standard basic triangle, T_b, c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

Point, p1, is in the vertex region of vertex rv1 (default is NULL) and point, p2, is in the vertex region of vertex rv2 (default is NULL); vertices are labeled as 1,2,3 in the order they are stacked row-wise.

Vertex 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 the standard basic triangle T_b or based on circumcenter of T_b; default is M="CC", i.e., circumcenter of T_b.

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would be a dominating set if they actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2ASbasic.tri(
  p1,
  p2,
  Xp,
  c1,
  c2,
  M = "CC",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

Value

I({p1,p2} is a dominating set of the AS-PCD) where the vertices of AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.num2AStri

Examples

c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M)
Idom.num2ASbasic.tri(Xp[1,],Xp[1,],Xp,c1,c2,M)  #one point can not a dominating set of size two

Idom.num2ASbasic.tri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),c1,c2,M)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2ASbasic.tri(Xp[i,],Xp[j,],Xp,c1,c2,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
rv2<-rel.vert.basic.triCC(Xp[2,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv1,rv2)
Idom.num2ASbasic.tri(c(.2,.4),Xp[2,],Xp,c1,c2,M,rv1,rv2)

#or try
rv1<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv1)

#or try
Rv2<-rel.vert.basic.triCC(Xp[2,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv2=Rv2)

Idom.num2ASbasic.tri(c(.3,.2),c(.35,.25),Xp,c1,c2,M)

The indicator for two points constituting a dominating set for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns I({p1,p2} is a dominating set of the AS-PCD) where vertices of the AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise.

AS proximity regions are defined with respect to the triangle tri. Point, p1, is in the region of vertex rv1 (default is NULL) and point, p2, is in the region of vertex rv2 (default is NULL); vertices (and hence rv1 and rv2) are labeled as 1,2,3 in the order they are stacked row-wise in tri.

Vertex regions are based on the center M="CC" for circumcenter of tri; or 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="CC" the circumcenter of tri.

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would constitute dominating set if they actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2AStri(
  p1,
  p2,
  Xp,
  tri,
  M = "CC",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

Value

I({p1,p2} is a dominating set of the AS-PCD) where vertices of the AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.num2ASbasic.tri

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(1.6,1.2)

Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M)
Idom.num2AStri(Xp[1,],Xp[1,],Xp,Tr,M)  #same two points cannot be a dominating set of size 2

Idom.num2AStri(c(.2,.4),Xp[2,],Xp,Tr,M)
Idom.num2AStri(c(.2,.4),c(.2,.5),Xp,Tr,M)
Idom.num2AStri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),Tr,M)

#or try
rv1<-rel.vert.triCC(c(.2,.4),Tr)$rv
rv2<-rel.vert.triCC(c(.2,.5),Tr)$rv
Idom.num2AStri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),Tr,M,rv1,rv2)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2AStri(Xp[i,],Xp[j,],Xp,Tr,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.triCC(Xp[1,],Tr)$rv
rv2<-rel.vert.triCC(Xp[2,],Tr)$rv
Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv1,rv2)

#or try
rv1<-rel.vert.triCC(Xp[1,],Tr)$rv
Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv1)

#or try
Rv2<-rel.vert.triCC(Xp[2,],Tr)$rv
Idom.num2AStri(Xp[1,],Xp[2,],Xp,Tr,M,rv2=Rv2)

Idom.num2AStri(c(1.3,1.2),c(1.35,1.25),Xp,Tr,M)

The indicator for two points constituting a dominating set for Central Similarity Proximity Catch Digraphs (CS-PCDs) - first one-sixth of the standard equilateral triangle case

Description

Returns I({p1,p2} is a dominating set of the CS-PCD) where the vertices of the CS-PCD are the 2D data set Xp), that is, returns 1 if p is a dominating point of CS-PCD, returns 0 otherwise.

CS proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) and with expansion parameter t=1. Point, p1, must lie in the first one-sixth of T_e, which is the triangle with vertices T(A,D_3,CM)=T((0,0),(1/2,0),CM).

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would be a dominating set if they actually were in the data set.

See also (Ceyhan (2005)).

Usage

Idom.num2CS.Te.onesixth(p1, p2, Xp, ch.data.pnts = FALSE)

Arguments

Value

I({p1,p2} is a dominating set of the CS-PCD) where the vertices of the CS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of CS-PCD, returns 0 otherwise

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.

See Also

Idom.num2CSstd.tri

The indicator for two points being a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard basic triangle case

Description

Returns I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp in the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)), that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, and returns 0 otherwise.

PE proximity regions are defined with respect to T_b. In the standard basic triangle, T_b, c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence, standard basic triangle is useful for simulation studies under the uniformity hypothesis.

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 a standard basic triangle T_b; default is M=(1,1,1), i.e., the center of mass of T_b. Point, p1, is in the vertex region of vertex rv1 (default is NULL); and point, p2, is in the vertex region of vertex rv2 (default is NULL); vertices are labeled as 1,2,3 in the order they are stacked row-wise.

ch.data.pnts is for checking whether points p1 and p2 are both data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would constitute a dominating set if they both were actually in the data set.

See also (Ceyhan (2005, 2011)).

Usage

Idom.num2PEbasic.tri(
  p1,
  p2,
  Xp,
  r,
  c1,
  c2,
  M = c(1, 1, 1),
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

Value

I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, and returns 0 otherwise.

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

See Also

Idom.num2PEtri, Idom.num2ASbasic.tri, and Idom.num2AStri

Examples

c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.3)

r<-2

Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M)

Idom.num2PEbasic.tri(c(1,2),c(1,3),rbind(c(1,2),c(1,3)),r,c1,c2,M)
Idom.num2PEbasic.tri(c(1,2),c(1,3),rbind(c(1,2),c(1,3)),r,c1,c2,M,
ch.data.pnts = TRUE)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2PEbasic.tri(Xp[i,],Xp[j,],Xp,r,c1,c2,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv;
rv2<-rel.vert.basic.tri(Xp[2,],c1,c2,M)$rv;
Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M,rv1,rv2)

#or try
rv1<-rel.vert.basic.tri(Xp[1,],c1,c2,M)$rv;
Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M,rv1)

#or try
rv2<-rel.vert.basic.tri(Xp[2,],c1,c2,M)$rv;
Idom.num2PEbasic.tri(Xp[1,],Xp[2,],Xp,r,c1,c2,M,rv2=rv2)

Idom.num2PEbasic.tri(c(1,2),Xp[2,],Xp,r,c1,c2,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not both points are data points in Xp

The indicator for two 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case

Description

Returns I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp in the standard regular tetrahedron T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3)), that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL) and point, p2, is in the region of vertex rv2 (default is NULL); vertices (and hence rv1 and rv2) are labeled as 1,2,3,4 in the order they are stacked row-wise in T_h.

PE proximity region is constructed with respect to the tetrahedron T_h with expansion parameter r \ge 1 and vertex regions are based on center of mass CM (equivalent to circumcenter in this case).

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would constitute a dominating set if they actually were both in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2PEstd.tetra(
  p1,
  p2,
  Xp,
  r,
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

Value

I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise

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.

See Also

Idom.num2PEtetra, Idom.num2PEtri and Idom.num2PEbasic.tri

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-5 #try also n<-20
Xp<-runif.std.tetra(n)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))
r<-1.5

Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r)

ind.gam2<-vector()
for (i in 1:(n-1))
 for (j in (i+1):n)
 {if (Idom.num2PEstd.tetra(Xp[i,],Xp[j,],Xp,r)==1)
  ind.gam2<-rbind(ind.gam2,c(i,j))}

ind.gam2

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r,rv1,rv2)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEstd.tetra(Xp[1,],Xp[2,],Xp,r,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.4,.1,.2)
Idom.num2PEstd.tetra(P1,P2,Xp,r)

Idom.num2PEstd.tetra(c(-1,-1,-1),Xp[2,],Xp,r,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp

The indicator for two 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp in the tetrahedron th, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL) and point, p2, is in the region of vertex rv2 (default is NULL); vertices (and hence rv1 and rv2) are labeled as 1,2,3,4 in the order they are stacked row-wise in th.

PE proximity region is constructed with respect to the tetrahedron th with expansion parameter r \ge 1 and vertex regions are based on center of mass (M="CM") or circumcenter (M="CC") only.

ch.data.pnts is for checking whether points p1 and p2 are both data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would constitute a dominating set if they actually were both in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2PEtetra(
  p1,
  p2,
  Xp,
  th,
  r,
  M = "CM",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

Value

I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, returns 0 otherwise

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.

See Also

Idom.num2PEstd.tetra, Idom.num2PEtri and Idom.num2PEbasic.tri

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5

set.seed(1)
Xp<-runif.tetra(n,tetra)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))

M<-"CM";  #try also M<-"CC";
r<-1.5

Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M)
Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M)

ind.gam2<-ind.gamn2<-vector()
for (i in 1:(n-1))
 for (j in (i+1):n)
 {if (Idom.num2PEtetra(Xp[i,],Xp[j,],Xp,tetra,r,M)==1)
 {ind.gam2<-rbind(ind.gam2,c(i,j))
 }
 }
ind.gam2

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1,rv2)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num2PEtetra(Xp[1,],Xp[2,],Xp,tetra,r,M,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.4,.1,.2)
Idom.num2PEtetra(P1,P2,Xp,tetra,r,M)

Idom.num2PEtetra(c(-1,-1,-1),Xp[2,],Xp,tetra,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp

The indicator for two points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, and returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL) and point, p2, is in the region of vertex rv2 (default is NULL); vertices (and hence rv1 and rv2) are labeled as 1,2,3 in the order they are stacked row-wise in tri.

PE proximity regions are defined with respect to the triangle tri and 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 circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri.

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would be a dominating set if they actually were in the data set.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

Idom.num2PEtri(
  p1,
  p2,
  Xp,
  tri,
  r,
  M = c(1, 1, 1),
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

Value

I({p1,p2} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 2D data set Xp, that is, returns 1 if {p1,p2} is a dominating set of PE-PCD, and returns 0 otherwise.

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Ceyhan E, Priebe CE (2007). “On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs.” Model Assisted Statistics and Applications, 1(4), 231-255.

See Also

Idom.num2PEbasic.tri, Idom.num2AStri, and Idom.num2PEtetra

Examples

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C);
n<-10  #try also n<-20

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

Idom.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2PEtri(Xp[i,],Xp[j,],Xp,Tr,r,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.tri(Xp[1,],Tr,M)$rv;
rv2<-rel.vert.tri(Xp[2,],Tr,M)$rv
Idom.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M,rv1,rv2)

Idom.num2PEtri(Xp[1,],c(1,2),Xp,Tr,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE
#since not both points, p1 and p2, are data points in Xp

The indicator for three 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard regular tetrahedron case

Description

Returns I(\{p1,p2,pt3} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp in the standard regular tetrahedron T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3)), that is, returns 1 if {p1,p2,pt3} is a dominating set of PE-PCD, returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL), point, p2, is in the region of vertex rv2 (default is NULL); point, pt3), is in the region of vertex rv3) (default is NULL); vertices (and hence rv1, rv2 and rv3) are labeled as 1,2,3,4 in the order they are stacked row-wise in T_h.

PE proximity region is constructed with respect to the tetrahedron T_h with expansion parameter r \ge 1 and vertex regions are based on center of mass CM (equivalent to circumcenter in this case).

ch.data.pnts is for checking whether points p1, p2 and pt3 are all data points in Xp or not (default is FALSE), so by default this function checks whether the points p1, p2 and pt3 would constitute a dominating set if they actually were all in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num3PEstd.tetra(
  p1,
  p2,
  pt3,
  Xp,
  r,
  rv1 = NULL,
  rv2 = NULL,
  rv3 = NULL,
  ch.data.pnts = FALSE
)

Arguments

Value

I(\{p1,p2,pt3} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp), that is, returns 1 if {p1,p2,pt3} is a dominating set of PE-PCD, returns 0 otherwise

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.

See Also

Idom.num3PEtetra

Examples

set.seed(123)
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5 #try 20, 40, 100 (larger n may take a long time)
Xp<-runif.std.tetra(n)$g  #try also Xp<-cbind(runif(n),runif(n),runif(n))
r<-1.25

Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r)

ind.gam3<-vector()
for (i in 1:(n-2))
 for (j in (i+1):(n-1))
   for (k in (j+1):n)
 {if (Idom.num3PEstd.tetra(Xp[i,],Xp[j,],Xp[k,],Xp,r)==1)
  ind.gam3<-rbind(ind.gam3,c(i,j,k))}

ind.gam3

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv; rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv;
rv3<-rel.vert.tetraCC(Xp[3,],tetra)$rv
Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv1,rv2,rv3)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num3PEstd.tetra(Xp[1,],Xp[2,],Xp[3,],Xp,r,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.3,.3,.3)
P3<-c(.4,.1,.2)
Idom.num3PEstd.tetra(P1,P2,P3,Xp,r)

Idom.num3PEstd.tetra(Xp[1,],c(1,1,1),Xp[3,],Xp,r,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

The indicator for three 3D points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns I(\{p1,p2,pt3} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp in the tetrahedron th, that is, returns 1 if {p1,p2,pt3} is a dominating set of PE-PCD, returns 0 otherwise.

Point, p1, is in the region of vertex rv1 (default is NULL), point, p2, is in the region of vertex rv2 (default is NULL); point, pt3), is in the region of vertex rv3) (default is NULL); vertices (and hence rv1, rv2 and rv3) are labeled as 1,2,3,4 in the order they are stacked row-wise in th.

PE proximity region is constructed with respect to the tetrahedron th with expansion parameter r \ge 1 and vertex regions are based on center of mass CM (equivalent to circumcenter in this case).

ch.data.pnts is for checking whether points p1, p2 and pt3 are all data points in Xp or not (default is FALSE), so by default this function checks whether the points p1, p2 and pt3 would constitute a dominating set if they actually were all in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num3PEtetra(
  p1,
  p2,
  pt3,
  Xp,
  th,
  r,
  M = "CM",
  rv1 = NULL,
  rv2 = NULL,
  rv3 = NULL,
  ch.data.pnts = FALSE
)

Arguments

Value

I(\{p1,p2,pt3} is a dominating set of the PE-PCD) where the vertices of the PE-PCD are the 3D data set Xp), that is, returns 1 if {p1,p2,pt3} is a dominating set of PE-PCD, returns 0 otherwise

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.

See Also

Idom.num3PEstd.tetra

Examples

set.seed(123)
A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-5 #try 20, 40, 100 (larger n may take a long time)
Xp<-runif.tetra(n,tetra)$g

M<-"CM";  #try also M<-"CC";
r<-1.25

Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M)

ind.gam3<-vector()
for (i in 1:(n-2))
 for (j in (i+1):(n-1))
   for (k in (j+1):n)
   {if (Idom.num3PEtetra(Xp[i,],Xp[j,],Xp[k,],Xp,tetra,r,M)==1)
    ind.gam3<-rbind(ind.gam3,c(i,j,k))}

ind.gam3

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv; rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv;
rv3<-rel.vert.tetraCC(Xp[3,],tetra)$rv
Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M,rv1,rv2,rv3)

#or try
rv1<-rel.vert.tetraCC(Xp[1,],tetra)$rv;
Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M,rv1)

#or try
rv2<-rel.vert.tetraCC(Xp[2,],tetra)$rv
Idom.num3PEtetra(Xp[1,],Xp[2,],Xp[3,],Xp,tetra,r,M,rv2=rv2)

P1<-c(.1,.1,.1)
P2<-c(.3,.3,.3)
P3<-c(.4,.1,.2)
Idom.num3PEtetra(P1,P2,P3,Xp,tetra,r,M)

Idom.num3PEtetra(Xp[1,],c(1,1,1),Xp[3,],Xp,tetra,r,M,ch.data.pnts = FALSE)
#gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp

Indicator for an upper bound for the domination number of Arc Slice Proximity Catch Digraph (AS-PCD) by the exact algorithm - one triangle case

Description

Returns I(domination number of AS-PCD whose vertices are the data points Xp is less than or equal to k), that is, returns 1 if the domination number of AS-PCD is less than the prespecified value k, returns 0 otherwise. It also provides the vertices (i.e., data points) in a dominating set of size k of AS-PCD.

AS proximity regions are constructed with respect to the triangle tri and vertex 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 the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri.

The vertices of triangle, tri, are labeled as 1,2,3 according to the row number the vertex is recorded in tri. Loops are allowed in the digraph. It takes a long time for large number of vertices (i.e., large number of row numbers).

Usage

Idom.numASup.bnd.tri(Xp, k, tri, M = "CC")

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

See Also

Idom.numCSup.bnd.tri, Idom.numCSup.bnd.std.tri, Idom.num.up.bnd, and dom.num.exact

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)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

Idom.numASup.bnd.tri(Xp,1,Tr)

for (k in 1:n)
  print(c(k,Idom.numASup.bnd.tri(Xp,k,Tr,M)))

Idom.numASup.bnd.tri(Xp,k=4,Tr,M)

P<-c(.4,.2)
Idom.numASup.bnd.tri(P,1,Tr,M)

Idom.numASup.bnd.tri(rbind(Xp,Xp),k=2,Tr,M)

The indicator for k being an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - standard equilateral triangle case

Description

Returns I(domination number of CS-PCD is less than or equal to k) where the vertices of the CS-PCD are the data points Xp, that is, returns 1 if the domination number of CS-PCD is less than the prespecified value k, returns 0 otherwise. It also provides the vertices (i.e., data points) in a dominating set of size k of CS-PCD.

CS proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=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 (which is equivalent to the circumcenter of T_e).

Edges of T_e, AB, BC, AC, are also labeled as 3, 1, and 2, respectively. Loops are allowed in the digraph. It takes a long time for large number of vertices (i.e., large number of row numbers).

See also (Ceyhan (2012)).

Usage

Idom.numCSup.bnd.std.tri(Xp, k, t, M = c(1, 1, 1))

Arguments

Value

A list with two elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.numCSup.bnd.tri, Idom.num.up.bnd, Idom.numASup.bnd.tri, and dom.num.exact

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

Idom.numCSup.bnd.std.tri(Xp,1,t,M)

for (k in 1:n)
  print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$Idom.num.up.bnd))
  print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$domUB))

Indicator for an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - one triangle case

Description

Returns I(domination number of CS-PCD is less than or equal to k) where the vertices of the CS-PCD are the data points Xp, that is, returns 1 if the domination number of CS-PCD is less than the prespecified value k, returns 0 otherwise. It also provides the vertices (i.e., data points) in a dominating set of size k of CS-PCD.

CS proximity region is constructed with respect to the triangle tri=T(A,B,C) 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 tri; default is M=(1,1,1) i.e., the center of mass of tri.

Edges of tri, AB, BC, AC, are also labeled as 3, 1, and 2, respectively. Loops are allowed in the digraph.

See also (Ceyhan (2012)).

Caveat: It takes a long time for large number of vertices (i.e., large number of row numbers).

Usage

Idom.numCSup.bnd.tri(Xp, k, tri, t, M = c(1, 1, 1))

Arguments

Value

A list with two elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.numCSup.bnd.std.tri, Idom.num.up.bnd, Idom.numASup.bnd.tri, and dom.num.exact

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)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

t<-.5

Idom.numCSup.bnd.tri(Xp,1,Tr,t,M)

for (k in 1:n)
  print(c(k,Idom.numCSup.bnd.tri(Xp,k,Tr,t,M)))

The indicator for the set of points S being a dominating set or not for Arc Slice Proximity Catch Digraphs (AS-PCDs) - one triangle case

Description

Returns I(S a dominating set of AS-PCD), that is, returns 1 if S is a dominating set of AS-PCD, returns 0 otherwise.

AS-PCD has vertex set Xp and AS proximity region is constructed with vertex 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 the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri whose vertices are also labeled as edges 1, 2, and 3, respectively.

See also (Ceyhan (2005, 2010)).

Usage

Idom.setAStri(S, Xp, tri, M = "CC")

Arguments

Value

I(S a dominating set of AS-PCD), that is, returns 1 if S is a dominating set of AS-PCD whose vertices are the data points in Xp; returns 0 otherwise, where AS proximity region is constructed in the triangle tri.

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

IarcASset2pnt.tri, Idom.setPEtri and Idom.setCStri

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)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

S<-rbind(Xp[1,],Xp[2,])
Idom.setAStri(S,Xp,Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setAStri(S,Xp,Tr,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
Idom.setAStri(S,Xp,Tr,M)

Idom.setAStri(c(.2,.5),Xp,Tr,M)
Idom.setAStri(c(.2,.5),c(.2,.5),Tr,M)
Idom.setAStri(Xp[5,],Xp[2,],Tr,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
Idom.setAStri(S,Xp[3,],Tr,M)

Idom.setAStri(Xp,Xp,Tr,M)

P<-c(.4,.2)
S<-Xp[c(1,3,4),]
Idom.setAStri(Xp,P,Tr,M)
Idom.setAStri(S,P,Tr,M)
Idom.setAStri(S,Xp,Tr,M)

Idom.setAStri(rbind(S,S),Xp,Tr,M)

The indicator for the set of points S being a dominating set or not for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case

Description

Returns I(S a dominating set of the CS-PCD) where the vertices of the CS-PCD are the data set Xp), that is, returns 1 if S is a dominating set of CS-PCD, returns 0 otherwise.

CS proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=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 (which is equivalent to the circumcenter of T_e).

Edges of T_e, AB, BC, AC, are also labeled as 3, 1, and 2, respectively.

See also (Ceyhan (2012)).

Usage

Idom.setCSstd.tri(S, Xp, t, M = c(1, 1, 1))

Arguments

Value

I(S a dominating set of the CS-PCD), that is, returns 1 if S is a dominating set of CS-PCD, returns 0 otherwise, where CS proximity region is constructed in the standard equilateral triangle T_e

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.setCStri and Idom.setPEstd.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setCSstd.tri(S,Xp,t,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setCSstd.tri(S,Xp,t,M)

The indicator for the set of points S being a dominating set or not for Central Similarity Proximity Catch Digraphs (CS-PCDs) - one triangle case

Description

Returns I(S a dominating set of CS-PCD whose vertices are the data set Xp), that is, returns 1 if S is a dominating set of CS-PCD, returns 0 otherwise.

CS proximity region is constructed with respect to the triangle tri with the 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 the triangle tri; default is M=(1,1,1) i.e., the center of mass of tri.

The triangle tri=T(A,B,C) has edges AB, BC, AC which are also labeled as edges 3, 1, and 2, respectively.

See also (Ceyhan (2012)).

Usage

Idom.setCStri(S, Xp, tri, t, M = c(1, 1, 1))

Arguments

Value

I(S a dominating set of the CS-PCD), that is, returns 1 if S is a dominating set of CS-PCD whose vertices are the data points in Xp; returns 0 otherwise, where CS proximity region is constructed in the triangle tri

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.setCSstd.tri, Idom.setPEtri and Idom.setAStri

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)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

tau<-.5
S<-rbind(Xp[1,],Xp[2,])
Idom.setCStri(S,Xp,Tr,tau,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setCStri(S,Xp,Tr,tau,M)

The indicator for the set of points S being a dominating set or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns I(S a dominating set of PE-PCD whose vertices are the data points Xp) for S in the standard equilateral triangle, that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise.

PE proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) with expansion parameter r \ge 1 and vertex 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 (which is also equivalent to the circumcenter of T_e). Vertices of T_e are also labeled as 1, 2, and 3, respectively.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

Idom.setPEstd.tri(S, Xp, r, M = c(1, 1, 1))

Arguments

Value

I(S a dominating set of PE-PCD) for S in the standard equilateral triangle, that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise, where PE proximity region is constructed in the standard equilateral triangle T_e.

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Ceyhan E, Priebe CE (2007). “On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs.” Model Assisted Statistics and Applications, 1(4), 231-255.

See Also

Idom.setPEtri and Idom.setCSstd.tri

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.std.tri(n)$gen.points

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setPEstd.tri(S,Xp,r,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
Idom.setPEstd.tri(S,Xp[3,],r,M)

The indicator for the set of points S being a dominating set or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns I(S a dominating set of PE-PCD whose vertices are the data set Xp), that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise.

PE proximity region is constructed with respect to the triangle tri with the expansion parameter r \ge 1 and vertex 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 the triangle tri or based on the circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri. The triangle tri=T(A,B,C) has edges AB, BC, AC which are also labeled as edges 3, 1, and 2, respectively.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

Idom.setPEtri(S, Xp, tri, r, M = c(1, 1, 1))

Arguments

Value

I(S a dominating set of PE-PCD), that is, returns 1 if S is a dominating set of PE-PCD whose vertices are the data points in Xp; and returns 0 otherwise, where PE proximity region is constructed in the triangle tri.

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Ceyhan E, Priebe CE (2007). “On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs.” Model Assisted Statistics and Applications, 1(4), 231-255.

See Also

Idom.setPEstd.tri, IarcPEset2pnt.tri, Idom.setCStri, and Idom.setAStri

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)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setPEtri(S,Xp,Tr,r,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setPEtri(S,Xp,Tr,r,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
Idom.setPEtri(S,Xp,Tr,r,M)

The line joining two distinct 2D points a and b

Description

An object of class "Lines". Returns the equation, slope, intercept, and y-coordinates of the line crossing two distinct 2D points a and b with x-coordinates provided in vector x.

This function is different from the line function in the standard stats package in R in the sense that Line(a,b,x) fits the line passing through points a and b and returns various quantities (see below) for this line and x is the x-coordinates of the points we want to find on the Line(a,b,x) while line(a,b) fits the line robustly whose x-coordinates are in a and y-coordinates are in b.

Line(a,b,x) and line(x,Line(A,B,x)$y) would yield the same straight line (i.e., the line with the same coefficients.)

Usage

Line(a, b, x)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

See Also

slope, paraline, perpline, line in the generic stats package and and Line3D

Examples

A<-c(-1.22,-2.33); B<-c(2.55,3.75)

xr<-range(A,B);
xf<-(xr[2]-xr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100

lnAB<-Line(A,B,x)
lnAB
summary(lnAB)
plot(lnAB)

line(A,B)
#this takes vector A as the x points and vector B as the y points and fits the line
#for example, try
x=runif(100); y=x+(runif(100,-.05,.05))
plot(x,y)
line(x,y)

x<-lnAB$x
y<-lnAB$y
Xlim<-range(x,A,B)
if (!is.na(y[1])) {Ylim<-range(y,A,B)} else {Ylim<-range(A,B)}
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
pf<-c(xd,-yd)*.025

#plot of the line joining A and B
plot(rbind(A,B),pch=1,xlab="x",ylab="y",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
if (!is.na(y[1])) {lines(x,y,lty=1)} else {abline(v=A[1])}
text(rbind(A+pf,B+pf),c("A","B"))
int<-round(lnAB$intercep,2)  #intercept
sl<-round(lnAB$slope,2)  #slope
text(rbind((A+B)/2+pf*3),ifelse(is.na(int),paste("x=",A[1]),
ifelse(sl==0,paste("y=",int),
ifelse(sl==1,ifelse(sign(int)<0,paste("y=x",int),paste("y=x+",int)),
ifelse(sign(int)<0,paste("y=",sl,"x",int),paste("y=",sl,"x+",int))))))

The line crossing 3D point p in the direction of vector v (or if v is a point, in direction of v-r_0)

Description

An object of class "Lines3D". Returns the equation, x-, y-, and z-coordinates of the line crossing 3D point r_0 in the direction of vector v (of if v is a point, in the direction of v-r_0) with the parameter t being provided in vector t.

Usage

Line3D(p, v, t, dir.vec = TRUE)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

See Also

line, paraline3D, and Plane

Examples

A<-c(1,10,3); B<-c(1,1,3);

vecs<-rbind(A,B)

Line3D(A,B,.1)
Line3D(A,B,.1,dir.vec=FALSE)

tr<-range(vecs);
tf<-(tr[2]-tr[1])*.1
#how far to go at the lower and upper ends in the x-coordinate
tsq<-seq(-tf*10-tf,tf*10+tf,l=5)  #try also l=10, 20, or 100

lnAB3D<-Line3D(A,B,tsq)
#try also lnAB3D<-Line3D(A,B,tsq,dir.vec=FALSE)
lnAB3D
summary(lnAB3D)
plot(lnAB3D)

x<-lnAB3D$x
y<-lnAB3D$y
z<-lnAB3D$z

zr<-range(z)
zf<-(zr[2]-zr[1])*.2
Bv<-B*tf*5

Xlim<-range(x)
Ylim<-range(y)
Zlim<-range(z)

xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]
zd<-Zlim[2]-Zlim[1]

Dr<-A+min(tsq)*B

plot3D::lines3D(x, y, z, phi = 0, bty = "g",
main="Line Crossing A \n in the Direction of OB",
xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05),
zlim=Zlim+zd*c(-.1,.1),
        pch = 20, cex = 2, ticktype = "detailed")
plot3D::arrows3D(Dr[1],Dr[2],Dr[3]+zf,Dr[1]+Bv[1],
Dr[2]+Bv[2],Dr[3]+zf+Bv[3], add=TRUE)
plot3D::points3D(A[1],A[2],A[3],add=TRUE)
plot3D::arrows3D(A[1],A[2],A[3]-2*zf,A[1],A[2],A[3],lty=2, add=TRUE)
plot3D::text3D(A[1],A[2],A[3]-2*zf,labels="initial point",add=TRUE)
plot3D::text3D(A[1],A[2],A[3]+zf/2,labels=expression(r[0]),add=TRUE)
plot3D::arrows3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+3*zf+Bv[3]/2,
Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+zf+Bv[3]/2,lty=2, add=TRUE)
plot3D::text3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+3*zf+Bv[3]/2,
labels="direction vector",add=TRUE)
plot3D::text3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,
Dr[3]+zf+Bv[3]/2,labels="v",add=TRUE)
plot3D::text3D(0,0,0,labels="O",add=TRUE)

The vertices of the Arc Slice (AS) Proximity Region in the standard basic triangle

Description

Returns the end points of the line segments and arc-slices that constitute the boundary of AS proximity region for a point in the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2)) where c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

Vertex 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 the standard basic triangle T_b or based on circumcenter of T_b; default is M="CC", i.e., circumcenter of T_b. rv is the index of the vertex region p resides, with default=NULL.

If p is outside T_b, it returns NULL for the proximity region. dec is the number of decimals (default is 4) to round the barycentric coordinates when checking whether the end points fall on the boundary of the triangle T_b or not (so as not to miss the intersection points due to precision in the decimals).

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

See also (Ceyhan (2005, 2010)).

Usage

NASbasic.tri(p, c1, c2, M = "CC", rv = NULL, dec = 4)

Arguments

Value

A list with the elements

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

NAStri and IarcASbasic.tri

Examples

c1<-.4; c2<-.6  #try also c1<-.2; c2<-.2;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)

set.seed(1)
M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g);  #try also P1<-c(.3,.2)
NASbasic.tri(P1,c1,c2)  #default with M="CC"
NASbasic.tri(P1,c1,c2,M)

#or try
Rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
NASbasic.tri(P1,c1,c2,M,Rv)

NASbasic.tri(c(3,5),c1,c2,M)

P2<-c(.5,.4)
NASbasic.tri(P2,c1,c2,M)

P3<-c(1.5,.4)
NASbasic.tri(P3,c1,c2,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

#plot of the NAS region
P1<-as.numeric(runif.basic.tri(1,c1,c2)$g);
CC<-circumcenter.basic.tri(c1,c2)

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
rv<-rel.vert.basic.triCC(P1,c1,c2)$rv
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges.basic.tri(c1,c2,M)
rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
}
RV<-Tb[rv,]
rad<-Dist(P1,RV)

Int.Pts<-NASbasic.tri(P1,c1,c2,M)

Xlim<-range(Tb[,1],P1[1]+rad,P1[1]-rad)
Ylim<-range(Tb[,2],P1[2]+rad,P1[2]-rad)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
polygon(Tb)
points(rbind(Tb,P1,rbind(Int.Pts$L,Int.Pts$R)))
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
interp::circles(P1[1],P1[2],rad,lty=2)
L<-Int.Pts$L; R<-Int.Pts$R
segments(L[,1], L[,2], R[,1], R[,2], lty=1,col=2)
Arcs<-Int.Pts$a;
if (!is.null(Arcs))
{
  K<-nrow(Arcs)/2
  for (i in 1:K)
  {A1<-Arcs[2*i-1,]; A2<-Arcs[2*i,];
  angles<-angle.str2end(A1,P1,A2)$c

  plotrix::draw.arc(P1[1],P1[2],rad,angle1=angles[1],angle2=angles[2],col=2)
  }
}

#proximity region with the triangle (i.e., for labeling the vertices of the NAS)
IP.txt<-intpts<-c()
if (!is.null(Int.Pts$a))
{
 intpts<-unique(round(Int.Pts$a,7))
  #this part is for labeling the intersection points of the spherical
  for (i in 1:(length(intpts)/2))
    IP.txt<-c(IP.txt,paste("I",i+1, sep = ""))
}
txt<-rbind(Tb,P1,cent,intpts)
txt.str<-c("A","B","C","P1",cent.name,IP.txt)
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.03,nrow(txt))),txt.str)

c1<-.4; c2<-.6;
P1<-c(.3,.2)
NASbasic.tri(P1,c1,c2,M)

The vertices of the Arc Slice (AS) Proximity Region in a general triangle

Description

Returns the end points of the line segments and arc-slices that constitute the boundary of AS proximity region for a point in the triangle tri=T(A,B,C)=(rv=1,rv=2,rv=3).

Vertex 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 the triangle tri or based on circumcenter of tri; default is M="CC", i.e., circumcenter of tri. rv is the index of the vertex region p1 resides, with default=NULL.

If p is outside of tri, it returns NULL for the proximity region. dec is the number of decimals (default is 4) to round the barycentric coordinates when checking the points fall on the boundary of the triangle tri or not (so as not to miss the intersection points due to precision in the decimals).

See also (Ceyhan (2005, 2010)).

Usage

NAStri(p, tri, M = "CC", rv = NULL, dec = 4)

Arguments

Value

A list with the elements

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

NASbasic.tri, NPEtri, NCStri and IarcAStri

Examples

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

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(.6,.2)

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(1.3,1.2)
NAStri(P1,Tr,M)

#or try
Rv<-rel.vert.triCC(P1,Tr)$rv
NAStri(P1,Tr,M,Rv)

NAStri(c(3,5),Tr,M)

P2<-c(1.5,1.4)
NAStri(P2,Tr,M)

P3<-c(1.5,.4)
NAStri(P3,Tr,M)

if (dimension(M)==3) {M<-bary2cart(M,Tr)}
#need to run this when M is given in barycentric coordinates

CC<-circumcenter.tri(Tr)  #the circumcenter

if (isTRUE(all.equal(M,CC)) || identical(M,"CC"))
{cent<-CC
D1<-(B+C)/2; D2<-(A+C)/2; D3<-(A+B)/2;
Ds<-rbind(D1,D2,D3)
cent.name<-"CC"
rv<-rel.vert.triCC(P1,Tr)$rv
} else
{cent<-M
cent.name<-"M"
Ds<-prj.cent2edges(Tr,M)
rv<-rel.vert.tri(P1,Tr,M)$rv
}
RV<-Tr[rv,]
rad<-Dist(P1,RV)

Int.Pts<-NAStri(P1,Tr,M)

#plot of the NAS region
Xlim<-range(Tr[,1],P1[1]+rad,P1[1]-rad)
Ylim<-range(Tr[,2],P1[2]+rad,P1[2]-rad)
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

plot(A,pch=".",asp=1,xlab="",ylab="",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
#asp=1 must be the case to have the arc properly placed in the figure
polygon(Tr)
points(rbind(Tr,P1,rbind(Int.Pts$L,Int.Pts$R)))
L<-rbind(cent,cent,cent); R<-Ds
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
interp::circles(P1[1],P1[2],rad,lty=2)
L<-Int.Pts$L; R<-Int.Pts$R
segments(L[,1], L[,2], R[,1], R[,2], lty=1,col=2)
Arcs<-Int.Pts$a;
if (!is.null(Arcs))
{
  K<-nrow(Arcs)/2
  for (i in 1:K)
  {A1<-Int.Pts$arc[2*i-1,]; A2<-Int.Pts$arc[2*i,];
  angles<-angle.str2end(A1,P1,A2)$c

  test.ang1<-angles[1]+(.01)*(angles[2]-angles[1])
  test.Pnt<-P1+rad*c(cos(test.ang1),sin(test.ang1))
  if (!in.triangle(test.Pnt,Tr,boundary = TRUE)$i) {angles<-c(min(angles),max(angles)-2*pi)}
  plotrix::draw.arc(P1[1],P1[2],rad,angle1=angles[1],angle2=angles[2],col=2)
  }
}

#proximity region with the triangle (i.e., for labeling the vertices of the NAS)
IP.txt<-intpts<-c()
if (!is.null(Int.Pts$a))
{
 intpts<-unique(round(Int.Pts$a,7))
  #this part is for labeling the intersection points of the spherical
  for (i in 1:(length(intpts)/2))
    IP.txt<-c(IP.txt,paste("I",i+1, sep = ""))
}
txt<-rbind(Tr,P1,cent,intpts)
txt.str<-c("A","B","C","P1",cent.name,IP.txt)
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.03,nrow(txt))),txt.str)

P1<-c(.3,.2)
NAStri(P1,Tr,M)

The end points of the Central Similarity (CS) Proximity Region for a point - one interval case

Description

Returns the end points of the interval which constitutes the CS proximity region for a point in the interval int=(a,b)=(rv=1,rv=2).

CS proximity region is constructed with respect to the interval int with expansion parameter t>0 and centrality parameter c \in (0,1).

Vertex regions are based on the (parameterized) center, M_c, which is M_c=a+c(b-a) for the interval, int=(a,b). The CS proximity region is constructed whether x is inside or outside the interval int.

See also (Ceyhan (2016)).

Usage

NCSint(x, int, t, c = 0.5)

Arguments

Value

The interval which constitutes the CS proximity region for the point x

Author(s)

Elvan Ceyhan

References

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

See Also

NPEint and NCStri

Examples

c<-.4
t<-2
a<-0; b<-10; int<-c(a,b)

NCSint(7,int,t,c)
NCSint(17,int,t,c)
NCSint(1,int,t,c)
NCSint(-1,int,t,c)

NCSint(3,int,t,c)
NCSint(4,int,t,c)
NCSint(a,int,t,c)

The vertices of the Central Similarity (CS) Proximity Region in a general triangle

Description

Returns the vertices of the CS proximity region (which is itself a triangle) for a point in the triangle tri=T(A,B,C)=(rv=1,rv=2,rv=3).

CS proximity region is defined with respect to the triangle tri with expansion parameter t>0 and edge regions 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.

Edge regions are labeled as 1,2,3 rowwise for the corresponding vertices of the triangle tri. re is the index of the edge region p resides, with default=NULL. If p is outside of tri, it returns NULL for the proximity region.

See also (Ceyhan (2005, 2010); Ceyhan et al. (2007)).

Usage

NCStri(p, tri, t, M = c(1, 1, 1), re = NULL)

Arguments

Value

Vertices of the triangular region which constitutes the CS proximity region with expansion parameter t>0 and center M for a point p

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, 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

NPEtri, NAStri, and IarcCStri

Examples

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

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

n<-3
set.seed(1)
Xp<-runif.tri(n,Tr)$g

NCStri(Xp[1,],Tr,tau,M)

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(.4,.2)
NCStri(P1,Tr,tau,M)

#or try
re<-rel.edges.tri(P1,Tr,M)$re
NCStri(P1,Tr,tau,M,re)

The vertices of the Proportional Edge (PE) Proximity Region in a standard basic triangle

Description

Returns the vertices of the PE proximity region (which is itself a triangle) for a point in the standard basic triangle T_b=T((0,0),(1,0),(c_1,c_2))=(rv=1,rv=2,rv=3).

PE proximity region is defined with respect to the standard basic triangle T_b with expansion parameter r \ge 1 and vertex regions based on center M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of the basic triangle T_b or based on the circumcenter of T_b; default is M=(1,1,1), i.e., the center of mass of T_b.

Vertex regions are labeled as 1,2,3 rowwise for the vertices of the triangle T_b. rv is the index of the vertex region p resides, with default=NULL. If p is outside of tri, it returns NULL for the proximity region.

See also (Ceyhan (2005, 2010)).

Usage

NPEbasic.tri(p, r, c1, c2, M = c(1, 1, 1), rv = NULL)

Arguments

Value

Vertices of the triangular region which constitutes the PE proximity region with expansion parameter r and center M for a point p

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 (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

NPEtri, NAStri, NCStri, and IarcPEbasic.tri

Examples

c1<-.4; c2<-.6
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C);

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

r<-2

P1<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also P1<-c(.4,.2)
NPEbasic.tri(P1,r,c1,c2,M)

#or try
Rv<-rel.vert.basic.tri(P1,c1,c2,M)$rv
NPEbasic.tri(P1,r,c1,c2,M,Rv)

P1<-c(1.4,1.2)
P2<-c(1.5,1.26)
NPEbasic.tri(P1,r,c1,c2,M) #gives an error if M=c(1.3,1.3)
#since center is not the circumcenter or not in the interior of the triangle

The end points of the Proportional Edge (PE) Proximity Region for a point - one interval case

Description

Returns the end points of the interval which constitutes the PE proximity region for a point in the interval int=(a,b)=(rv=1,rv=2). PE proximity region is constructed with respect to the interval int with expansion parameter r \ge 1 and centrality parameter c \in (0,1).

Vertex regions are based on the (parameterized) center, M_c, which is M_c=a+c(b-a) for the interval, int=(a,b). The PE proximity region is constructed whether x is inside or outside the interval int.

See also (Ceyhan (2012)).

Usage

NPEint(x, int, r, c = 0.5)

Arguments

Value

The interval which constitutes the PE proximity region for the point x

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

See Also

NCSint, NPEtri and NPEtetra

Examples

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

NPEint(7,int,r,c)
NPEint(17,int,r,c)
NPEint(1,int,r,c)
NPEint(-1,int,r,c)

The vertices of the Proportional Edge (PE) Proximity Region in the standard regular tetrahedron

Description

Returns the vertices of the PE proximity region (which is itself a tetrahedron) for a point in the standard regular tetrahedron T_h=T((0,0,0),(1,0,0),(1/2,\sqrt{3}/2,0),(1/2,\sqrt{3}/6,\sqrt{6}/3))= (rv=1,rv=2,rv=3,rv=4).

PE proximity region is defined with respect to the tetrahedron T_h with expansion parameter r \ge 1 and vertex regions based on the circumcenter of T_h (which is equivalent to the center of mass in the standard regular tetrahedron).

Vertex regions are labeled as 1,2,3,4 rowwise for the vertices of the tetrahedron T_h. rv is the index of the vertex region p resides, with default=NULL. If p is outside of T_h, it returns NULL for the proximity region.

See also (Ceyhan (2005, 2010)).

Usage

NPEstd.tetra(p, r, rv = NULL)

Arguments

Value

Vertices of the tetrahedron which constitutes the PE proximity region with expansion parameter r and circumcenter (or center of mass) for a point p in the standard regular tetrahedron

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.

See Also

NPEtetra, NPEtri and NPEint

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)

n<-3
Xp<-runif.std.tetra(n)$g
r<-1.5
NPEstd.tetra(Xp[1,],r)

#or try
RV<-rel.vert.tetraCC(Xp[1,],tetra)$rv
NPEstd.tetra(Xp[1,],r,rv=RV)

NPEstd.tetra(c(-1,-1,-1),r,rv=NULL)

The vertices of the Proportional Edge (PE) Proximity Region in a tetrahedron

Description

Returns the vertices of the PE proximity region (which is itself a tetrahedron) for a point in the tetrahedron th.

PE proximity region is defined with respect to the tetrahedron th with expansion parameter r \ge 1 and vertex regions based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM".

Vertex regions are labeled as 1,2,3,4 rowwise for the vertices of the tetrahedron th. rv is the index of the vertex region p resides, with default=NULL. If p is outside of th, it returns NULL for the proximity region.

See also (Ceyhan (2005, 2010)).

Usage

NPEtetra(p, th, r, M = "CM", rv = NULL)

Arguments

Value

Vertices of the tetrahedron which constitutes the PE proximity region with expansion parameter r and circumcenter (or center of mass) for a point p in the tetrahedron

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.

See Also

NPEstd.tetra, NPEtri and NPEint

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
set.seed(1)
tetra<-rbind(A,B,C,D)+matrix(runif(12,-.25,.25),ncol=3)
n<-3  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.5

NPEtetra(Xp[1,],tetra,r)  #uses the default M="CM"
NPEtetra(Xp[1,],tetra,r,M="CC")

#or try
RV<-rel.vert.tetraCM(Xp[1,],tetra)$rv
NPEtetra(Xp[1,],tetra,r,M,rv=RV)

P1<-c(.1,.1,.1)
NPEtetra(P1,tetra,r,M)

The vertices of the Proportional Edge (PE) Proximity Region in a general triangle

Description

Returns the vertices of the PE proximity region (which is itself a triangle) for a point in the triangle tri=T(A,B,C)=(rv=1,rv=2,rv=3).

PE proximity region is defined with respect to the triangle tri with expansion parameter r \ge 1 and vertex regions 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.

Vertex regions are labeled as 1,2,3 rowwise for the vertices of the triangle tri. rv is the index of the vertex region p resides, with default=NULL. If p is outside of tri, it returns NULL for the proximity region.

See also (Ceyhan (2005); Ceyhan et al. (2006); Ceyhan (2011)).

Usage

NPEtri(p, tri, r, M = c(1, 1, 1), rv = NULL)

Arguments

Value

Vertices of the triangular region which constitutes the PE proximity region with expansion parameter r and center M for a point p

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 (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

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

NPEbasic.tri, NAStri, NCStri, and IarcPEtri

Examples

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

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.0)

r<-1.5

n<-3
set.seed(1)
Xp<-runif.tri(n,Tr)$g

NPEtri(Xp[3,],Tr,r,M)

P1<-as.numeric(runif.tri(1,Tr)$g)  #try also P1<-c(.4,.2)
NPEtri(P1,Tr,r,M)

M<-c(1.3,1.3)
r<-2

P1<-c(1.4,1.2)
P2<-c(1.5,1.26)
NPEtri(P1,Tr,r,M)
NPEtri(P2,Tr,r,M)

#or try
Rv<-rel.vert.tri(P1,Tr,M)$rv
NPEtri(P1,Tr,r,M,Rv)

A test of segregation/association based on arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of Xp points in the convex hull of Yp points against the alternatives of segregation (where Xp points cluster away from Yp points) and association (where Xp points cluster around Yp points) based on the normal approximation of the arc density of the PE-PCD for uniform 2D data.

The function yields the test statistic, p-value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is the arc density), and method and name of the data set used.

Under the null hypothesis of uniformity of Xp points in the convex hull of Yp points, arc density of PE-PCD whose vertices are Xp points equals to its expected value under the uniform distribution and alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association) or right-sided (i.e., data is accumulated around the centers of the triangles, or segregation).

PE proximity region is constructed with the expansion parameter r \ge 1 and CM-vertex regions (i.e., the test is not available for a general center M at this version of the function).

**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are assumed to be fixed (i.e., the test is conditional on Yp points). Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points, say at least 5 times more. This test is more appropriate when supports of Xp and Yp have a substantial overlap. Currently, the Xp points outside the convex hull of Yp points are handled with a convex hull correction factor, ch.cor, which is derived under the assumption of uniformity of Xp and Yp points in the study window, (see the description below and the function code.) However, in the special case of no Xp points in the convex hull of Yp points, arc density is taken to be 1, as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is an ongoing topic of research of the author of the package.

ch.cor is for convex hull correction (default is "no convex hull correction", i.e., ch.cor=FALSE) which is recommended when both Xp and Yp have the same rectangular support.

See also (Ceyhan (2005); Ceyhan et al. (2006)) for more on the test based on the arc density of PE-PCDs.

Usage

PEarc.dens.test(
  Xp,
  Yp,
  r,
  ch.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

Value

A list with the elements

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

CSarc.dens.test and PEarc.dens.test1D

Examples

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

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))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

plotDelaunay.tri(Xp,Yp,xlab="",ylab="")

PEarc.dens.test(Xp,Yp,r=1.25)
PEarc.dens.test(Xp,Yp,r=1.25,ch=TRUE)
#since Y points are not uniform, convex hull correction is invalid here

A test of uniformity of 1D data in a given interval based on Proportional Edge Proximity Catch Digraph (PE-PCD)

Description

An object of class "htest". This is an "htest" (i.e., hypothesis test) function which performs a hypothesis test of uniformity of 1D data in one interval based on the normal approximation of the arc density of the PE-PCD with expansion parameter r \ge 1 and centrality parameter c \in (0,1).

The function yields the test statistic, p-value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is the arc density), and method and name of the data set used.

The null hypothesis is that data is uniform in a finite interval (i.e., arc density of PE-PCD equals to its expected value under uniform distribution) and alternative could be two-sided, or left-sided (i.e., data is accumulated around the end points) or right-sided (i.e., data is accumulated around the mid point or center M_c).

See also (Ceyhan (2012, 2016)).

Usage

PEarc.dens.test.int(
  Xp,
  int,
  r,
  c = 0.5,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

Ceyhan E (2016). “Density of a Random Interval Catch Digraph Family and its Use for Testing Uniformity.” REVSTAT, 14(4), 349-394.

See Also

CSarc.dens.test.int

Examples

c<-.4
r<-2
a<-0; b<-10; int<-c(a,b)

n<-100  #try also n<-20, 1000
Xp<-runif(n,a,b)

PEarc.dens.test.int(Xp,int,r,c)
PEarc.dens.test.int(Xp,int,r,c,alt="g")
PEarc.dens.test.int(Xp,int,r,c,alt="l")

A test of segregation/association based on arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 1D data

Description

An object of class "htest" (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of Xp points in the range (i.e., range) of Yp points against the alternatives of segregation (where Xp points cluster away from Yp points) and association (where Xp points cluster around Yp points) based on the normal approximation of the arc density of the PE-PCD for uniform 1D data.

The function yields the test statistic, p-value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is the arc density), and method and name of the data set used.

Under the null hypothesis of uniformity of Xp points in the range of Yp points, arc density of PE-PCD whose vertices are Xp points equals to its expected value under the uniform distribution and alternative could be two-sided, or left-sided (i.e., data is accumulated around the Yp points, or association) or right-sided (i.e., data is accumulated around the centers of the intervals, or segregation).

PE proximity region is constructed with the expansion parameter r \ge 1 and centrality parameter c which yields M-vertex regions. More precisely, for a middle interval (y_{(i)},y_{(i+1)}), the center is M=y_{(i)}+c(y_{(i+1)}-y_{(i)}) for the centrality parameter c \in (0,1). If there are duplicates of Yp points, only one point is retained for each duplicate value, and a warning message is printed.

**Caveat:** This test is currently a conditional test, where Xp points are assumed to be random, while Yp points are assumed to be fixed (i.e., the test is conditional on Yp points). Furthermore, the test is a large sample test when Xp points are substantially larger than Yp points, say at least 5 times more. This test is more appropriate when supports of Xp and Yp have a substantial overlap. Currently, the Xp points outside the range of Yp points are handled with a range correction (or end-interval correction) factor (see the description below and the function code.) However, in the special case of no Xp points in the range of Yp points, arc density is taken to be 1, as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is an ongoing line of research of the author of the package.

end.int.cor is for end-interval correction, (default is "no end-interval correction", i.e., end.int.cor=FALSE), recommended when both Xp and Yp have the same interval support.

See also (Ceyhan (2012)) for more on the uniformity test based on the arc density of PE-PCDs.

Usage

PEarc.dens.test1D(
  Xp,
  Yp,
  r,
  c = 0.5,
  support.int = NULL,
  end.int.cor = FALSE,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “The Distribution of the Relative Arc Density of a Family of Interval Catch Digraph Based on Uniform Data.” Metrika, 75(6), 761-793.

See Also

PEarc.dens.test, PEdom.num.binom.test1D, and PEarc.dens.test.int

Examples

r<-2
c<-.4
a<-0; b<-10; int=c(a,b)

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-100; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
xf<-(int[2]-int[1])*.1

Xp<-runif(nx,a-xf,b+xf)
Yp<-runif(ny,a,b)

PEarc.dens.test1D(Xp,Yp,r,c,int)
#try also PEarc.dens.test1D(Xp,Yp,r,c,int,alt="l") and PEarc.dens.test1D(Xp,Yp,r,c,int,alt="g")

PEarc.dens.test1D(Xp,Yp,r,c,int,end.int.cor = TRUE)

Arc density of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one tetrahedron case

Description

Returns the arc density of PE-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the tetrahedron th.

PE proximity region is constructed with respect to the tetrahedron th and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM". For the number of arcs, loops are not allowed so arcs are only possible for points inside the tetrahedron th for this function.

th.cor is a logical argument for tetrahedron correction (default is TRUE), if TRUE, only the points inside the tetrahedron are considered (i.e., digraph induced by these vertices are considered) in computing the arc density, otherwise all points are considered (for the number of vertices in the denominator of arc density).

See also (Ceyhan (2005, 2010)).

Usage

PEarc.dens.tetra(Xp, th, r, M = "CM", th.cor = FALSE)

Arguments

Value

Arc density of PE-PCD whose vertices are the 2D numerical data set, Xp; PE proximity regions are defined with respect to the tetrahedron th 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.

See Also

PEarc.dens.tri and num.arcsPEtetra

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-10  #try also n<-20

set.seed(1)
Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.5

num.arcsPEtetra(Xp,tetra,r,M)
PEarc.dens.tetra(Xp,tetra,r,M)
PEarc.dens.tetra(Xp,tetra,r,M,th.cor = FALSE)

Arc density of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case

Description

Returns the arc density of PE-PCD whose vertex set is the given 2D numerical data set, Xp, (some of its members are) in the triangle tri.

PE proximity regions is defined with respect to tri with expansion parameter r \ge 1 and 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 circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri. The function also provides arc density standardized by the mean and asymptotic variance of the arc density of PE-PCD for uniform data in the triangle tri only when M is the center of mass. For the number of arcs, loops are not allowed.

in.tri.only is a logical argument (default is FALSE) for considering only the points inside the triangle or all the points as the vertices of the digraph. if in.tri.only=TRUE, arc density is computed only for the points inside the triangle (i.e., arc density of the subdigraph induced by the vertices in the triangle is computed), otherwise arc density of the entire digraph (i.e., digraph with all the vertices) is computed.