An object of class "SpatPatterns".
Generates n_2 2D points associated with the given set of points (i.e., reference points) \(X_1\) in the
type G fashion with the parameter sigma which is a positive real number representing the variance of the
Gaussian marginals.
The generated points are intended to be from a different class, say class 2 (or \(X_2\) points) than the reference
(i.e., \(X_1\) points, say class 1 points, denoted as X1 as an argument of the function), say class 1 points).
To generate \(n_2\) (denoted as n2 as an argument of the function)\(X_2\) points, \(n_2\) of \(X_1\) points are randomly selected (possibly with replacement) and
for a selected X1 point, say \(x_{1ref}\),
a new point from the class 2, say \(x_{2new}\), is generated from a bivariate normal distribution centered at \(x_{1ref}\)
where the covariance matrix of the bivariate normal is a diagonal matrix with sigma in the diagonals.
That is, \(x_{2new} = x_{1ref}+V\) where \(V \sim BVN((0,0),\sigma I_2)\) with \(I_2\) being the \(2 \times 2\) identity matrix.
Note that, the level of association increases as sigma decreases, and the association vanishes when sigma
goes to infinity.
For type G association, it is recommended to take \(\sigma \le 0.10\) times length of the shorter
edge of a rectangular study region, or take \(r_0 = 1/(k \sqrt{\hat \rho})\) with the appropriate choice of \(k\)
to get an association pattern more robust to differences in relative abundances
(i.e., the choice of \(k\) implies \(\sigma \le 0.10\) times length of the shorter edge to have alternative patterns more
robust to differences in sample sizes).
Here \(\hat \rho\) is the
estimated intensity of points in the study region (i.e., # of points divided by the area of the region).
Type G association is closely related to Types C and U association,
see the functions rassocC and rassocU and
the other association types.
In the type C association pattern
the new point from the class 2, \(x_{2new}\), is generated (uniform in the polar coordinates) within a circle
centered at \(x_{1ref}\) with radius equal to \(r_0\),
in type U association pattern \(x_{2new}\) is generated similarly except it is uniform in the circle.
In type I association, first a \(Uniform(0,1)\) number, \(U\), is generated.
If \(U \le p\), \(x_{2new}\) is generated (uniform in the polar coordinates) within a
circle with radius equal to the distance to the closest \(X_1\) point,
else it is generated uniformly within the smallest bounding box containing \(X_1\) points.
See ceyhan:serra-2014;textualnnspat for more detail.