The pair correlation function of a stationary point process is
$$
g(r) = \frac{K'(r)}{2\pi r}
$$
where \(K'(r)\) is the derivative of \(K(r)\), the
reduced second moment function (aka ``Ripley's \(K\) function'')
of the point process. See Kest for information
about \(K(r)\). For a stationary Poisson process, the
pair correlation function is identically equal to 1. Values
\(g(r) < 1\) suggest inhibition between points;
values greater than 1 suggest clustering.
We also apply the same definition to
other variants of the classical \(K\) function,
such as the multitype \(K\) functions
(see Kcross, Kdot) and the
inhomogeneous \(K\) function (see Kinhom).
For all these variants, the benchmark value of
\(K(r) = \pi r^2\) corresponds to
\(g(r) = 1\).
This routine computes an estimate of \(g(r)\)
either directly from a point pattern,
or indirectly from an estimate of \(K(r)\) or one of its variants.
This function is generic, with methods for
the classes "ppp", "fv" and "fasp".
If X is a point pattern (object of class "ppp")
then the pair correlation function is estimated using
a traditional kernel smoothing method (Stoyan and Stoyan, 1994).
See pcf.ppp for details.
If X is a function value table (object of class "fv"),
then it is assumed to contain estimates of the \(K\) function
or one of its variants (typically obtained from Kest or
Kinhom).
This routine computes an estimate of \(g(r)\)
using smoothing splines to approximate the derivative.
See pcf.fv for details.
If X is a function value array (object of class "fasp"),
then it is assumed to contain estimates of several \(K\) functions
(typically obtained from Kmulti or
alltypes). This routine computes
an estimate of \(g(r)\) for each cell in the array,
using smoothing splines to approximate the derivatives.
See pcf.fasp for details.