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 \(i=1,2,\ldots,n_{1h}\)
where \(n_{1h}=\lfloor n_1/2 \rfloor\),
and for \(k=n_{1h},+1,\ldots,n_1\), \(X_k=X_{k-n_{1h}}+r (\cos(T_k), \sin(T_k))\) where \(r \sim Uniform(0,r_0)\)
and \(T_k\) are iid \(\sim Uniform(0,2 \pi)\).
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 \(j=1,2,\ldots,n_{2h}\) where \(n_{2h}=\lfloor n_2/2\rfloor)\),
and for \(l=n_{2h},+1,\ldots,n_2\), \(Y_l=Y_{l-n_{2h}}+r (\cos(T_l), \sin(T_l))\) where \(r \sim Uniform(0,r_0)\) and
\(T_l \stackrel{iid}{\sim} Uniform(0,2 \pi)\).
This version is the case IV in the article (ceyhan: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 \(i=1,2,\ldots,n_{1h}\)
where \(n_{1h}=\lfloor{n_1/2\rfloor}\),
and for \(k=n_{1h}+1,\ldots,n_1\), \(X_k = Z_k+r (\cos(T_k), \sin(T_k))\)
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 \(\sim Uniform(0,2 \pi)\).
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 \(j=1,2,\ldots,n_{2h}\)
where \(n_{2h}=\lfloor n_2/2\rfloor)\),
and for \(l=n_{2h},+1,\ldots,n_2\), \(Y_l = W_k+r (\cos(T_l), \sin(T_l))\)
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
\(T_l \stackrel{iid}{\sim} Uniform(0,2 \pi)\).
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 (ceyhan:NNCorrespond2018;textualnnspat) and the references therein.
rself.ref(n1, n2, c1r, c2r, r0, distribution = c("uniform", "bvnormal"))A list with the elements
"2c" for the 2-class pattern of self-correspondence of the two classes
The type of the spatial pattern
The radius of attraction \(r_0\) which determines the level of self-correspondence.
The class labels of the generated points, it is 1 class 1 or X1 points and 2 for class 2 or \(X_2\) points
The initial points for class 1 and class 2, one initial point for each class, marked with a cross in the plot.
The output set of generated points from the self-correspondence pattern.
The input set of reference points, it is NULL for this function.
Description of the species correspondence pattern
The "main" title for the plot of the point pattern
The number of generated points.
The possible ranges of the \(x\)- and \(y\)-coordinates of the generated points
Positive integers representing the numbers of points to be generated from the two classes
Ranges of the squares which constitute the supports of the two classes
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
The argument determining the distribution of each class. Takes on values "uniform" and
"bvnormal" (see the description for the details).
Elvan Ceyhan
Zself.ref and Xsq.spec.cor
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)
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