rself.ref: Generation of Points from Self Correspondence Pattern

In nnspat: Nearest Neighbor Methods for Spatial Patterns

Generation of Points from Self Correspondence Pattern

Description

An object of class "SpatPatterns".

Generates n_1 2D points from class 1 and n_2 (denoted as n2 as an argument) 2D points from class 2 in such a way that self-reflexive pairs are more frequent than expected under CSR independence.

If distribution="uniform", the points from class 1, say X_i are generated as follows: X_i \stackrel{iid}{\sim} Uniform(S_1) for S_1=(c1r[1],c1r[2])^2 for i=1,2,…,n_{1h} where n_{1h}=\lfloor n_1/2 \rfloor, and for k=n_{1h},+1,…,n_1, X_k=X_{k-n_{1h}}+r (\cos(T_k), \sin(T_k)) where r \sim Uniform(0,r_0) and T_k are iid \sim Uniform(0,2 π). Similarly, the points from class 2, say Y_j are generated as follows: Y_j \stackrel{iid}{\sim} Uniform(S_2) for S_2=(c2r[1],c2r[2])^2 for j=1,2,…,n_{2h} where n_{2h}=\lfloor n_2/2\rfloor), and for l=n_{2h},+1,…,n_2, Y_l=Y_{l-n_{2h}}+r (\cos(T_l), \sin(T_l)) where r \sim Uniform(0,r_0) and T_l \stackrel{iid}{\sim} Uniform(0,2 π). This version is the case IV in the article (\insertCiteceyhan:NNCorrespond2018;textualnnspat).

If distribution="bvnormal", the points from class 1, say X_i are generated as follows: X_i \stackrel{iid}{\sim} BVN(CM(S_1),I_{2x}) where CM(S_1) is the center of mass of S_1 and I_2x is a 2 \times 2 matrix with diagonals equal to s_1^2 with s_1=(c1r[2]-c1r[1])/3 and off-diagonals are 0 for i=1,2,…,n_{1h} where n_{1h}=\lfloor{n_1/2\rfloor}, and for k=n_{1h}+1,…,n_1, X_k = Z_k+r (\cos(T_k), \sin(T_k)) where Z_k \sim BVN(X_{k-n_{1h}}, I_2(r_0)) with I_2(r_0) being the 2 \times 2 matrix with diagonals r_0/3 and 0 off-diagonals, r \sim Uniform(0,r_0) and T_k are iid \sim Uniform(0,2 π). Similarly, the points from class 2, say Y_j are generated as follows: Y_j \stackrel{iid}{\sim} BVN(CM(S_2),I_{2y}) where CM(S_1) is the center of mass of S_1 and I_2y is a 2 \times 2 matrix with diagonals equal to s_2^2 with s_2=(c2r[2]-c2r[1])/3 and off-diagonals are 0 for j=1,2,…,n_{2h} where n_{2h}=\lfloor n_2/2\rfloor), and for l=n_{2h},+1,…,n_2, Y_l = W_k+r (\cos(T_l), \sin(T_l)) where W_l \sim BVN(Y_{l-n_{2h}}, I_2(r_0)) with I_2(r_0) being the 2 \times 2 matrix with diagonals r_0/3 and 0 off-diagonals, r \sim Uniform(0,r_0) and T_l \stackrel{iid}{\sim} Uniform(0,2 π).

Notice that the classes will be segregated if the supports S_1 and S_2 are separated, with more separation implying stronger segregation. Furthermore, r_0 (denoted as r0 as an argument) determines the level of self-reflexivity or self correspondence, i.e. smaller r_0 implies a higher level of self correspondence and vice versa for higher r_0 .

See also (\insertCiteceyhan:NNCorrespond2018;textualnnspat) and the references therein.

Usage

rself.ref(n1, n2, c1r, c2r, r0, distribution = c("uniform", "bvnormal"))

Arguments

n1, n2	Positive integers representing the numbers of points to be generated from the two classes
c1r, c2r	Ranges of the squares which constitute the supports of the two classes
r0	The radius of attraction which determines the level of self-reflexivity (or self correspondence) in both the uniform and bvnormal distributions for the two classes
distribution	The argument determining the distribution of each class. Takes on values "uniform" and "bvnormal" (see the description for the details).

Value

A list with the elements

pat.type	"2c" for the 2-class pattern of self-correspondence of the two classes
type	The type of the spatial pattern
parameters	The radius of attraction r_0 which determines the level of self-correspondence.
lab	The class labels of the generated points, it is 1 class 1 or X1 points and 2 for class 2 or X_2 points
init.cases	The initial points for class 1 and class 2, one initial point for each class, marked with a cross in the plot.
gen.points	The output set of generated points from the self-correspondence pattern.
ref.points	The input set of reference points, it is NULL for this function.
desc.pat	Description of the species correspondence pattern
mtitle	The "main" title for the plot of the point pattern
num.points	The number of generated points.
xlimit,ylimit	The possible ranges of the x- and y-coordinates of the generated points

Author(s)

Elvan Ceyhan

See Also

Zself.ref and Xsq.spec.cor

Examples

n1<-50;  #try also n1<-50; n1<-1000;
n2<-50;  #try also n2<-50; n2<-1000;

c1r<-c(0,1) #try also c(0,5/6), C(0,3/4), c(0,2/3)
c2r<-c(0,1) #try also c(1/6,1), c(1/4,1), c(1/3,1)
r0<-1/9 #try also 1/7, 1/8

#data generation
Xdat<-rself.ref(n1,n2,c1r,c2r,r0)
Xdat

summary(Xdat)
plot(Xdat,asp=1)
plot(Xdat)

#data generation (bvnormal)
Xdat<-rself.ref(n1,n2,c1r,c2r,r0,distr="bvnormal")
Xdat

summary(Xdat)
plot(Xdat,asp=1)
plot(Xdat)

nnspat documentation built on Aug. 30, 2022, 9:06 a.m.