VS3 - Spatial Point Patterns

Elvan Ceyhan

2023-12-18

First we load the pcds package:

library(pcds)

1 Generation of Points from CSR, Segregation, and Association Patterns

We provide the functions for random point generation from three patterns, namely, complete spatial randomness (CSR) which is usually the benchmark pattern to test or compare other patterns, segregation and association patterns (of different types).

2 Generation of Points from CSR

We consider generation of points from CSR in one triangle, multiple triangles, and also in one tetrahedron. The 1D point generation from CSR can easily be handled with the usual runif function in base R in both the one interval and multiple interval cases, hence no specialized functions are provided for the CSR in the 1D setting.

2.1 Generation of Uniform Points in One Triangle

We choose the triangle \(T=T(A,B,C)\) with vertices \(A=(1,1)\), \(B=(2,0)\), and \(C=(1.5,2)\) as the support of uniform points. The same triangle \(T\) was used in other pcds Vignette Sections as well.

A<-c(1,1); B<-c(2,0); C<-c(1.5,2);
Tr<-rbind(A,B,C)
n<-5  #try also n<-10, 20, 50 or 100

We generate \(n=\) 5 \(\mathcal{X}\) points inside the triangle \(T\) using the function runif.tri, and provide the summary of these uniform points and the corresponding scatterplot (together with the triangle). The function runif.tri is an object of class Uniform and takes arguments k,tri where

Its call (with Xdt in the below script) just returns the type of the pattern. Its summary returns a description of the pattern, the study window for the points, the vertices of the support of the uniform distribution (i.e., vertices of \(T\)), first 6 or fewer of the generated points, and the number of points. The plot function (or plot.Uniform) returns the plot of the triangle together with the generated points.

set.seed(123)
Xdt<-runif.tri(n,Tr)
Xdt
#> Call:
#> runif.tri(n = n, tri = Tr)
#> 
#> Type:
#> [1] "Uniform Distribution in the Triangle with Vertices (1,1), (2,0) and (1.5,2)"
summary(Xdt)
#> Call:
#> runif.tri(n = n, tri = Tr)
#> 
#> Type of the Pattern : [1] "Uniform Distribution in the Triangle with Vertices (1,1), (2,0) and (1.5,2)"
#> 
#> Study Window
#> range in x-coordinate = 1 2 
#> range in y-coordinate = 0 2 
#> 
#>  Vertices of the Support of the Uniform Distribution
#>   [,1] [,2]
#> A  1.0    1
#> B  2.0    0
#> C  1.5    2
#> 
#>  5 uniform points in the triangle with vertices (1,1), (2,0) and (1.5,2)  
#>  (first 6 or fewer are printed) 
#>          [,1]      [,2]
#> [1,] 1.408977 1.7660348
#> [2,] 1.940467 0.0911130
#> [3,] 1.528105 1.7848381
#> [4,] 1.551435 0.9132295
#> [5,] 1.677571 1.1452668
#> 
#> Number of points
#> nx ny 
#>  5  3 
#> nx : the number of uniform points
#> ny : the number of vertices of the support region
plot(Xdt)

There is also uniform point generation capability in a standard equilateral triangle, or in the standard basic triangle, with the functions runif.std.tri and runif.basic.tri, with arguments n and n,c1,c2, respectively, see the corresponding help pages for the details. These functions are mostly for simulation purposes, as PE- and CS-PCDs are geometry invariant, and AS-PCD is scale invariant for uniform data in one triangle.

2.2 Generation of Uniform Points in Multiple Triangles

We can generate uniform points in the union of multiple Delaunay triangles, that is, in the convex hull of non-target points, where \(n_x\) is number of \(\mathcal{X}\) (i.e. target) points and \(n_y\) is number of \(\mathcal{Y}\) (nontarget) points. \(\mathcal{Y}\) points constitute the vertices of the Delaunay triangles, and when uniform points are generated in the convex hull of nontarget points, then points in each triangle will also be uniformly distributed (restricted to the triangle, i.e. scaled by the ratio of the area of the convex hull to the area of the particular triangle).

nx<-10; ny<-5  #try also nx<-40; ny<-5 or nx<-100;  #try also nx<-1000;  ny<-10;
set.seed(1)
Yp<-cbind(runif(ny,0,10),runif(ny,0,10))

We first generate 5 \(\mathcal{Y}\) points uniformly in the unit square \((0,1) \times (0,1)\) (which will form the vertices of the Delaunay triangles).

Next, we generate \(n_x=\) 10 \(\mathcal{X}\) points uniformly in the convex hull of the 5 \(Y\) points (i.e. in the union of the Delaunay triangles based on 5 \(Y\) points) using the function runif.multi.tri, and provide the summary of these uniform points and the corresponding scatterplot (together with the Delaunay triangles). The function runif.multi.tri is an object of class Uniform and takes arguments n,Yp,DTmesh where

Its call and plot is as in runif.tri and its summary is similar, with the addition of the \(x\) and \(y\) coordinates of the nodes on the boundary of the convex hull of \(\mathcal{Y}\) points and their indices (labeled $i). The plot of the object returns the scatterplot of the \(\mathcal{X}\) points together with Delaunay triangulation of \(\mathcal{Y}\) points.

Xdt<-runif.multi.tri(nx,Yp)  #data under CSR in the convex hull of Ypoints
Xdt
#> Call:
#> runif.multi.tri(n = nx, Yp = Yp) 
#> 
#> Type:
#> Uniform distribution in the convex hull of Yp points
summary(Xdt)
#> Call:
#> runif.multi.tri(n = nx, Yp = Yp) 
#> 
#> Type of the Pattern : Uniform distribution in the convex hull of Yp points 
#> 
#> Study Window
#> range in x-coordinate = 2.016819 9.082078 
#> range in y-coordinate = 0.6178627 9.446753 
#> 
#> The x and y coordinates of the nodes on the boundary of the convex hull of Y points and their indices (labeled as i) 
#>       [,1]     [,2]      [,3]     [,4]
#> x 3.721239 2.655087 2.0168193 9.082078
#> y 9.446753 8.983897 0.6178627 6.291140
#> i 2.000000 1.000000 5.0000000 4.000000
#> 
#> Number of points
#> nx ny 
#> 10  5 
#> nx : the number of Uniform points
#> ny : the number of points whose convex hull determines the support
plot(Xdt)
Scatterplot of the uniform $X$ points in the Delaunay triangles based on 5 $Y$ points.

Figure 2.1: Scatterplot of the uniform \(X\) points in the Delaunay triangles based on 5 \(Y\) points.

2.3 Generation of Uniform Points in a Tetrahedron

We first generate four 3D points that will form the arbitrary tetrahedron \(T=T(A,B,C,D)\) as the support of uniform points.

set.seed(11)
A<-sample(1:12,3); B<-sample(1:12,3); C<-sample(1:12,3); D<-sample(1:12,3)
tetra<-rbind(A,B,C,D)/6
n<-5 #try also n<-10, 20, 50, or 100

We generate \(n=\) 5 \(\mathcal{X}\) points uniformly inside the tetrahedron \(T\) using the function runif.tetra, and provide the summary of these uniform points and the corresponding scatterplot (together with the tetrahedron). The function runif.tetra is an object of class Uniform and takes arguments n,th where

Its call, plot, and summary are as in runif.tri. The plot of the object returns the 3D scatterplot of the \(\mathcal{X}\) points together with tetrahedron.

Xdt<-runif.tetra(n,tetra)
Xdt
#> Call:
#> runif.tetra(n = n, th = tetra)
#> 
#> Type:
#> [1] "Uniform Distribution in the Tetrahedron with Vertices (1.67,0.33,1.33), (1.5,0.17,0.83), (2,1,0.83) and (1,1.17,0.83)"
summary(Xdt)
#> Call:
#> runif.tetra(n = n, th = tetra)
#> 
#> Type of the Pattern : [1] "Uniform Distribution in the Tetrahedron with Vertices (1.67,0.33,1.33), (1.5,0.17,0.83), (2,1,0.83) and (1,1.17,0.83)"
#> 
#> Study Window
#> range in x-coordinate = 1 2 
#> range in y-coordinate = 0.1666667 1.166667 
#> 
#>  Vertices of the Support of the Uniform Distribution
#>       [,1]      [,2]      [,3]
#> A 1.666667 0.3333333 1.3333333
#> B 1.500000 0.1666667 0.8333333
#> C 2.000000 1.0000000 0.8333333
#> D 1.000000 1.1666667 0.8333333
#> 
#>  5 uniform points in the tetrahedron with vertices (1.67,0.33,1.33), (1.5,0.17,0.83), (2,1,0.83) and (1,1.17,0.83)  
#>  (first 6 or fewer are printed) 
#>          [,1]      [,2]      [,3]
#> [1,] 1.398149 0.6715843 0.9971730
#> [2,] 1.642032 0.4435400 0.8851113
#> [3,] 1.502856 0.7644274 1.0461309
#> [4,] 1.425548 0.5928684 0.9355892
#> [5,] 1.239314 0.9320898 0.9298472
#> 
#> Number of points
#> nx ny 
#>  5  4 
#> nx is the number of Uniform points 
#>  ny is the number of vertices of the support region
plot(Xdt)

Similarly, there is also uniform point generation capability in a standard regular tetrahedron with the function runif.std.tetra with the sole argument n, see the corresponding help page for the details. This function is also mostly for simulation purposes, as PE- and CS-PCDs are geometry invariant for uniform data in a tetrahedron.

3 Generation of Points from the Segregation Pattern

We mainly consider two types of segregation, type I segregation and circular segregation (there is also a type II segregation which is briefly described at the end of Section 3.3, however, it is only implemented in standard equilateral triangle at this point). We provide the description in the 2D setting, the extensions to higher dimensional and 1D settings are straightforward.

See Ceyhan, Priebe, and Wierman (2006), Ceyhan, Priebe, and Marchette (2007), and Ceyhan (2011) for more on the type I segregation pattern.

3.1 Generation of Points Segregated (in a Type I Fashion) from the Vertices of a Triangle

Here, the points are generated in type I segregation, parameterized by delta, which is a positive real number in \((0,1)\) so that \(\delta 100\) % area around vertices of each Delaunay triangle is forbidden for \(\mathcal{X}\) points. The forbidden regions are chosen in such a way that \(\mathcal{X}\) points can not be further from a distance to the opposite edge where this distance depends on delta. Such a construction preserves geometry invariance for arc density and domination number of PE- and CS-PCDs under segregation as well.

A<-c(1,1); B<-c(2,0); C<-c(1.5,7/3);
Tr<-rbind(A,B,C)
del<-.4
n<-10  #try also n<-100 or 1000

The same triangle \(T\) used in the CSR illustration in Section 2.1 is used here as well and the support of the segregated \(\mathcal{X}\) points will be subset of this triangle. We specify delta= 0.4 for the type I segregation pattern.

We generate \(n=\) 10 \(\mathcal{X}\) points uniformly inside the type I segregation support in \(T\) using the function rseg.tri, and provide the summary of these segregated points and the corresponding scatterplot (together with the triangle). The function rseg.tri is an object of class Patterns and has arguments n,tri,delta where

Its call and summary are as in runif.tri with the addition of the segregation parameter. The plot function (or plot.Patterns) returns the plot of the triangle together with the generated points.

Xdt<-rseg.tri(n,Tr,del)
Xdt
#> Call:
#> rseg.tri(n = n, tri = Tr, delta = del)
#> 
#> Type:
#> [1] "Type I Segregation of 10 points in the triangle with vertices (1,1), (2,0) and (1.5,2.33) and exclusion parameter delta = 0.4"
summary(Xdt)
#> Call:
#> rseg.tri(n = n, tri = Tr, delta = del)
#> 
#> Type of the Pattern
#> [1] "Type I Segregation of 10 points in the triangle with vertices (1,1), (2,0) and (1.5,2.33) and exclusion parameter delta = 0.4"
#> 
#> Parameters of the Pattern
#> exclusion parameter 
#>                 0.4 
#> 
#> Study Window
#> range in x-coordinate = 1 2 
#> range in y-coordinate = 0 2.333333 
#> 
#>  Generated Points from Pattern of Type I Segregation of One Class from Vertices of the Triangle  
#>  (first 6 or fewer are printed) 
#>         [,1]      [,2]
#> pnt 1.587007 0.5205062
#> pnt 1.409048 0.9539932
#> pnt 1.669594 0.7064556
#> pnt 1.523988 0.5721596
#> pnt 1.271635 1.6545486
#> pnt 1.674897 1.3032453
#> 
#> Number of points:
#> nx ny 
#> 10  3 
#> nx = number of generated points according to the pattern 
#> ny = number of reference (i.e. Y) points
plot(Xdt)

3.2 Generation of Points Segregated (in a Type I Fashion) from a Given Set of \(\mathcal{Y}\) Points

This case is equivalent to generation of type I segregation in each of the Delaunay triangles based on the non-target points (i.e., \(\mathcal{Y}\) points). That is, \(\mathcal{X}\) points are segregated in type I fashion from the vertices of each Delaunay triangle at the same level (i.e., parameterized with the same delta). 

ny<-5;
set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
del<-.4
nx<-10;  #try also nx<-100 or 1000;

We first generate 5 \(\mathcal{Y}\) points uniformly in the unit square which will form the vertices of the Delaunay triangles, and specify delta= 0.4 for type I segregation.

We generate \(n=\) 10 \(\mathcal{X}\) points uniformly inside the type I segregation support in the union of the Delaunay triangles (i.e. in the convex hull of the \(\mathcal{Y}\) points) using the function rseg.multi.tri, and provide the summary of these segregated points and the corresponding scatterplot (together with the triangles). The function rseg.multi.tri is an object of class Patterns and takes arguments n,Yp,delta,DTmesh,DTr where

Its call, summary, and plot are as in rseg.tri. The plot of the object is the scatterplot of the \(\mathcal{X}\) points together with Delaunay triangulation of \(\mathcal{Y}\) points.

Xdt<-rseg.multi.tri(nx,Yp,del)
Xdt
#> Call:
#> rseg.multi.tri(n = nx, Yp = Yp, delta = del) 
#> 
#> Type:
#> Type I Segregation of 10 points from 5 Y points with exclusion parameter delta = 0.4
summary(Xdt)
#> Call:
#> rseg.multi.tri(n = nx, Yp = Yp, delta = del) 
#> 
#> Type of the Pattern
#> Type I Segregation of 10 points from 5 Y points with exclusion parameter delta = 0.4 
#> 
#> Parameters of the Pattern
#> exclusion parameter 
#>                 0.4 
#> 
#> Study Window
#> range in x-coordinate = 0.2016819 0.9082078 
#> range in y-coordinate = 0.06178627 0.9446753 
#> 
#> Number of points:
#> nx ny 
#> 10  5 
#> nx = number of generated points according to the pattern 
#> ny = number of reference (i.e. Y) points
plot(Xdt)
Scatterplot of the $X$ points segregated (in a type I fashion) from the $Y$ points.

Figure 3.1: Scatterplot of the \(X\) points segregated (in a type I fashion) from the \(Y\) points.

3.3 Generation of Points Segregated (in a Radial or Circular fashion) from a Given Set of \(\mathcal{Y}\) Points

In this type of segregation, the points are generated with a parameter, e, which is a positive real number representing the radius of the balls centered at \(\mathcal{Y}\) points. These balls are forbidden for the generated points (i.e., \(\mathcal{X}\) points can not reside in the union of these balls). Note that same radius is used for each \(\mathcal{Y}\) point, and the \(\mathcal{X}\) points are not restricted to the convex hull of \(\mathcal{Y}\) points (or the Delaunay triangles). Such a construction, although more intuitive, does not preserve the geometry invariance property for the arc density and domination number of PE- and CS-PCDs under segregation.

nx<-10;  #try also nx<-100 or 1000; 
ny<-5
e<-.15;
#with default bounding box (i.e., unit square)
set.seed(1)
Yp<-cbind(runif(ny),runif(ny))

We first generate 5 \(\mathcal{Y}\) points uniformly in the unit square and specify e= 0.15 for circular segregation.

We generate \(n=\) 10 \(\mathcal{X}\) points uniformly inside the circular segregation support using the function rseg.circular, and provide the summary of these segregated points and the corresponding scatterplot (together with the \(\mathcal{Y}\) points). The function rseg.circular is an object of class Patterns and takes arguments n,Yp,e,a1,a2,b1,b2) where

Its call, summary, and plot are as in rseg.multi.tri. The plot of the object is the scatterplot of the \(\mathcal{X}\) points together with \(\mathcal{Y}\) points (here we use asp=1 so that the prohibited regions are actually depicted as circles).

Xdt<-rseg.circular(nx,Yp,e)
Xdt
#> Call:
#> rseg.circular(n = nx, Yp = Yp, e = e)
#> 
#> Type:
#> [1] "Segregation of 10 X points from 5 Y points with circular exclusion parameter e = 0.15"
summary(Xdt)
#> Call:
#> rseg.circular(n = nx, Yp = Yp, e = e)
#> 
#> Type of the Pattern
#> [1] "Segregation of 10 X points from 5 Y points with circular exclusion parameter e = 0.15"
#> 
#> Parameters of the Pattern
#> exclusion parameter 
#>                0.15 
#> 
#> Study Window
#> range in x-coordinate = 0.05168193 1.058208 
#> range in y-coordinate = -0.08821373 1.094675 
#> 
#>  Generated Points from Pattern of Segregation of Class X from Class Y  
#>  (first 6 or fewer are printed) 
#>            [,1]       [,2]
#> [1,] 0.82654723 0.50050923
#> [2,] 0.77398352 1.08510108
#> [3,] 0.99248692 0.16272732
#> [4,] 0.70760843 0.06030401
#> [5,] 0.32064644 0.36851638
#> [6,] 0.06515965 0.36410878
#> 
#> Number of points:
#> nx ny 
#> 10  5 
#> nx = number of generated points according to the pattern 
#> ny = number of reference (i.e. Y) points
plot(Xdt,asp=1)

There is also point generation capability from segregation pattern in a standard equilateral triangle with two types of segregation, type I segregation is the above described one and type II segregation (described below), with the relevant functions rseg.std.tri and rsegII.std.tri, with arguments n,eps for both, see the corresponding help pages for the details. These functions are mostly for simulation purposes, as arc density and domination number of PE- and CS-PCDs are geometry invariant. Type II segregation is the pattern in which a region at a certain distance from the boundary of the triangle is forbidden. That is, there is a parameter, eps, which is a positive real number representing the distance from the interior triangle points to the boundary of \(T_e\), (i.e., an annular region in the interior of the triangle around the edges) which is forbidden under this type of segregation.

4 Generation of Points from the Association Pattern

We mainly consider two types of association, type I association and circular association (there is also a type II association which is briefly described at the end of Section 4.4, however, it is only implemented in standard equilateral triangle at this point). We provide the description in the 2D setting, the extensions to higher dimensional and 1D settings are straightforward.

See Ceyhan, Priebe, and Wierman (2006), Ceyhan, Priebe, and Marchette (2007), and Ceyhan (2011) for more on the type I association pattern.

4.1 Generation of Points Associated (in a Type I Fashion) with the Vertices of a Triangle

Here, the points are generated in type I association, parameterized by delta, which is a positive real number in \((0,1)\) so that \(\delta 100\) % area around vertices of each Delaunay triangle is the only region allowed for \(\mathcal{X}\) points. The allowed regions are chosen in such a way that \(\mathcal{X}\) points must be further from a distance to the opposite edge where this distance depends on delta. Such a construction preserves geometry invariance for arc density and domination number of PE- and CS-PCDs under association as well.

A<-c(1,1); B<-c(2,0); C<-c(1.5,7/3);
Tr<-rbind(A,B,C)
del<-.4
n<-5  #try also n<-100 or 1000

The same triangle \(T\) used in the CSR illustration in Section 2.1 is used here as well and the support of the segregated \(\mathcal{X}\) points will be subset of this triangle. We specify delta= 0.4 for the type I association pattern.

We generate \(n=\) 5 \(\mathcal{X}\) points uniformly inside the type I association support in \(T\) using the function rassoc.tri, and provide the summary of these associated points and the corresponding scatterplot (together with the triangle). The function racs.tri is an object of class Patterns and has the arguments as rseg.tri. Its call, summary, and plot are also as in rseg.tri.

Xdt<-rassoc.tri(n,Tr,del)
Xdt
#> Call:
#> rassoc.tri(n = n, tri = Tr, delta = del)
#> 
#> Type:
#> [1] "Type I Association of 5 points in the triangle with vertices (1,1), (2,0) and (1.5,2.33) with attraction parameter delta = 0.4"
summary(Xdt)
#> Call:
#> rassoc.tri(n = n, tri = Tr, delta = del)
#> 
#> Type of the Pattern
#> [1] "Type I Association of 5 points in the triangle with vertices (1,1), (2,0) and (1.5,2.33) with attraction parameter delta = 0.4"
#> 
#> Parameters of the Pattern
#> attraction parameter 
#>                  0.4 
#> 
#> Study Window
#> range in x-coordinate = 1 2 
#> range in y-coordinate = 0 2.333333 
#> 
#>  Generated Points from Pattern of Type I Association of One Class with Vertices of the Triangle  
#>  (first 6 or fewer are printed) 
#>         [,1]      [,2]
#> pnt 1.529720 1.8418312
#> pnt 1.477620 2.0094888
#> pnt 1.478545 1.7880582
#> pnt 1.476351 2.0817961
#> pnt 1.757087 0.4729486
#> 
#> Number of points:
#> nx ny 
#>  5  3 
#> nx = number of generated points according to the pattern 
#> ny = number of reference (i.e. Y) points
plot(Xdt)

4.2 Generation of Points Associated (in a Type I Fashion) with a Given Set of \(\mathcal{Y}\) Points

This case is equivalent to generation of type I associated points in each of the Delaunay triangles based on the non-target points (i.e., \(\mathcal{Y}\) points). That is, \(\mathcal{X}\) points are associated in type I fashion with the vertices of each Delaunay triangle at the same level (i.e., parameterized with the same delta).

ny<-5;
set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
del<-.4
nx<-10;  #try also nx<-100 or 1000;

We first generate 5 \(\mathcal{Y}\) points uniformly in the unit square which will form the vertices of the Delaunay triangles, and specify delta= 0.4 for type I association.

We generate \(n=\) 10 \(\mathcal{X}\) points uniformly inside the type I association support in the union of the Delaunay triangles (or in the convex hull of the \(\mathcal{Y}\) points) using the function rassoc.multi.tri, and provide the summary of these associated points and the corresponding scatterplot (together with the triangles). The function rassoc.multi.tri is an object of class Patterns and its arguments, call, summary, and plot are as in rseg.multi.tri.

Xdt<-rassoc.multi.tri(nx,Yp,del)
Xdt
#> Call:
#> rassoc.multi.tri(n = nx, Yp = Yp, delta = del) 
#> 
#> Type:
#> Type I Association of 10 points with 5 Y points with attraction parameter delta = 0.4
summary(Xdt)
#> Call:
#> rassoc.multi.tri(n = nx, Yp = Yp, delta = del) 
#> 
#> Type of the Pattern
#> Type I Association of 10 points with 5 Y points with attraction parameter delta = 0.4 
#> 
#> Parameters of the Pattern
#> attraction parameter 
#>                  0.4 
#> 
#> Study Window
#> range in x-coordinate = 0.2016819 0.9082078 
#> range in y-coordinate = 0.06178627 0.9446753 
#> 
#> Number of points:
#> nx ny 
#> 10  5 
#> nx = number of generated points according to the pattern 
#> ny = number of reference (i.e. Y) points
plot(Xdt)
Scatterplot of the $X$ points associated (in a type I fashion) with the $Y$ points.

Figure 4.1: Scatterplot of the \(X\) points associated (in a type I fashion) with the \(Y\) points.

4.3 Generation of Points Associated (in a Radial or Circular fashion) with a Given Set of \(\mathcal{Y}\) Points

In this type of association, the points are generated with a parameter, e, which is a positive real number representing the radius of the balls centered at \(\mathcal{Y}\) points. These balls are the only regions allowed for the generated points (i.e., \(\mathcal{X}\) points must reside in the union of these balls). Note that same radius is used for each \(\mathcal{Y}\) point, and the \(\mathcal{X}\) points are not restricted to the convex hull of \(\mathcal{Y}\) points (or the Delaunay triangles). Such a construction, although more intuitive, does not preserve the geometry invariance property for the arc density and domination number of PE- and CS-PCDs, under association.

ny<-5;
e<-.15;
#with default bounding box (i.e., unit square)
set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
nx<-10;  #try also nx<-100 or 1000;

We first generate 5 \(\mathcal{Y}\) points uniformly in the unit square and specify e= 0.15 for circular association.

We generate \(n=\) 10 \(\mathcal{X}\) points uniformly inside the circular association support using the function rassoc.circular, and provide the summary of these associated points and the corresponding scatterplot (together with the \(\mathcal{Y}\) points). The function rassoc.circular is an object of class Patterns and its arguments, call, summary, and plot are as in rseg.circular. We use asp=1 in the plot of the object so that the support of the \(\mathcal{X}\) points (i.e., the allowed regions) are actually depicted as circles.

Xdt<-rassoc.circular(nx,Yp,e)
Xdt
#> Call:
#> rassoc.circular(n = nx, Yp = Yp, e = e)
#> 
#> Type:
#> [1] "Association of 10 points with 5 Y points with circular attraction parameter e = 0.15"
summary(Xdt)
#> Call:
#> rassoc.circular(n = nx, Yp = Yp, e = e)
#> 
#> Type of the Pattern
#> [1] "Association of 10 points with 5 Y points with circular attraction parameter e = 0.15"
#> 
#> Parameters of the Pattern
#> attraction parameter 
#>                 0.15 
#> 
#> Study Window
#> range in x-coordinate = 0.05168193 1.058208 
#> range in y-coordinate = -0.08821373 1.094675 
#> 
#>  Generated Points from Pattern of Association of one Class with Class Y  
#>  (first 6 or fewer are printed) 
#>           [,1]      [,2]
#> [1,] 0.4341972 0.8314177
#> [2,] 0.5369080 0.6210061
#> [3,] 0.8844546 0.7025082
#> [4,] 0.8779856 0.6771867
#> [5,] 0.8397240 0.5659668
#> [6,] 0.1228222 0.0294437
#> 
#> Number of points:
#> nx ny 
#> 10  5 
#> nx = number of generated points according to the pattern 
#> ny = number of reference (i.e. Y) points
plot(Xdt,asp=1)

4.4 Generation of Points Associated (in a Matérn-like fashion) with a Given Set of \(\mathcal{Y}\) Points

In this case, the points generated uniformly in \(\cup_{i=1}^{n_y} B(y_i,e)\) where \(\mathcal{Y}_{n_y}=\{\mathsf{y}_1,\mathsf{y}_2,\ldots,\mathsf{y}_{n_y}\}\) for various values of e under the association pattern where \(n_y\) is the number of \(\mathcal{Y}\) points and \(B(\mathsf{y}_i,e)\) is the ball centered at \(\mathsf{y}_i\) with radius e.

The pattern resembles the Matérn cluster pattern (see rMatClust function in spatstat.random package for further information (Baddeley and Turner (2005)). rMatClust(kappa, scale, mu, win) in the simplest case generates a uniform Poisson point process of “parent” points with intensity kappa. Then each parent point is replaced by a random cluster of “offspring” points, the number of points per cluster being Poisson(mu) distributed, and their positions being placed and uniformly inside a disc of radius scale centered on the parent point. The resulting point pattern is a realization of the classical “stationary Matérn cluster process” generated inside the window win.

The main difference of rassoc.matern and rMatClust in spatstat.random package is that in rassoc.matern the “parent” points are \(\mathcal{Y}\) points which are given beforehand and are not discarded at the end and the offspring points are the points associated with the reference points, \(\mathcal{Y}\). Notice that in rassoc.matern, the parent and offspring points belong to different classes. The argument e must be positive and very large values of e provide patterns closer to CSR pattern.

This function is also similar to rassoc.circular, where rassoc.circular needs the study window to be specified, while rassoc.matern does not.

ny<-5;
e<-.15;
#with default bounding box (i.e., unit square)
set.seed(1)
Yp<-cbind(runif(ny),runif(ny))
nx<-10;  #try also nx<-100 or 1000;

We first generate 5 \(\mathcal{Y}\) points and specify e= 0.15 for the Matérn-like association.

We generate \(n=\) 10 \(\mathcal{X}\) points from the Matérn-like association using the function rassoc.matern, and provide the summary of these associated points and the corresponding scatterplot (together with the \(\mathcal{Y}\) points). The function rassoc.matern is an object of class Patterns and its arguments, call, summary, and plot are as in rassoc.circular. We use asp=1 in the plot of the object so that the support of the \(\mathcal{X}\) points (i.e., the allowed regions) are actually depicted as circles.

Xdt<-rassoc.matern(nx,Yp,e)
Xdt
#> Call:
#> rassoc.matern(n = nx, Yp = Yp, e = e)
#> 
#> Type:
#> [1] "Matern-like Association of 10 points with 5 Y points with circular attraction parameter e = 0.15"
summary(Xdt)
#> Call:
#> rassoc.matern(n = nx, Yp = Yp, e = e)
#> 
#> Type of the Pattern
#> [1] "Matern-like Association of 10 points with 5 Y points with circular attraction parameter e = 0.15"
#> 
#> Parameters of the Pattern
#> attraction parameter 
#>                 0.15 
#> 
#> Study Window
#> range in x-coordinate = 0.05168193 1.058208 
#> range in y-coordinate = -0.08821373 1.094675 
#> 
#>  Generated Points from Pattern of Matern-like Association of one Class with Class Y  
#>  (first 6 or fewer are printed) 
#>            [,1]       [,2]
#> [1,] 0.56278796 0.75345984
#> [2,] 0.66528156 0.66859254
#> [3,] 0.32528878 0.83448201
#> [4,] 0.08626992 0.07483616
#> [5,] 0.13700582 0.06441234
#> [6,] 0.42942439 0.83624485
#> 
#> Number of points:
#> nx ny 
#> 10  5 
#> nx = number of generated points according to the pattern 
#> ny = number of reference (i.e. Y) points
plot(Xdt,asp=1)

There is also point generation capability from association pattern in a standard equilateral triangle with two types of association. Type I is the above described one and Type II association (described below), with the relevant functions rassoc.std.tri and rassocII.std.tri, with arguments n,eps for both, see the corresponding help pages for the details. These functions are mostly for simulation purposes, as arc density and domination number of PE- and CS-PCDs are geometry invariant. Type II association is the pattern in which only a region of points inside the triangle closer to the boundary of the triangle than a certain distance is allowed only. That is, there is a parameter, eps, which is a positive real number representing the distance from the interior triangle points to the boundary of \(T_e\), i.e., an annular region in the interior of the triangle around the edges which is the only allowed region under this type of association.

References

Baddeley, A. J., and R. Turner. 2005. “Spatstat: An R Package for Analyzing Spatial Point Patterns.” Journal of Statistical Software 12(6): 1–42.
Ceyhan, E. 2011. “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods 40(8): 1363–95.
Ceyhan, E., C. E. Priebe, and D. J. Marchette. 2007. “A New Family of Random Graphs for Testing Spatial Segregation.” Canadian Journal of Statistics 35(1): 27–50.
Ceyhan, E., C. E. Priebe, and J. C. Wierman. 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): 1925–64.