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 C fashion with a radius of association \(r_0\) (denoted as r0 as
an argument of the function) which is a positive real number.
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 \(n_2\) \(X_2\) 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 within a
circle with radius equal to \(r_0\) (uniform in the polar coordinates).
That is, \(x_{2new} = x_{1ref}+r_u c(\cos(t_u),\sin(t_u))\)
where \(r_u \sim U(0,r_0)\) and \(t_u \sim U(0, 2\pi)\).
Note that, the level of association increases as \(r_0\) decreases, and the association vanishes when \(r_0\) is
sufficiently large.
For type C association, it is recommended to take \(r_0 \le 0.25\) 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 \(r_0 \le 0.25\) 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 C association is closely related to Type U association, see the function rassocC
and the other association types.
In the type U association pattern
the new point from the class 2, \(x_{2new}\), is generated uniformly within a circle
centered at \(x_{1ref}\) with radius equal to \(r_0\).
In type G association, \(x_{2new}\) is generated from the bivariate normal distribution centered at \(x_{1ref}\) with covariance
\(\sigma I_2\) where \(I_2\) is \(2 \times 2\) identity matrix.
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.