pcf
Pair Correlation Function
Estimate the pair correlation function.
- Keywords
- spatial, nonparametric
Usage
pcf(X, ...)Arguments
- X
- Either the observed data point pattern, or an estimate of its $K$ function, or an array of multitype $K$ functions (see Details).
- ...
- Other arguments passed to the appropriate method.
Details
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.
Value
- Either a function value table
(object of class
"fv", seefv.object) representing a pair correlation function, or a function array (object of class"fasp", seefasp.object) representing an array of pair correlation functions.
References
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.
See Also
pcf.ppp,
pcf.fv,
pcf.fasp,
Kest,
Kinhom,
Kcross,
Kdot,
Kmulti,
alltypes
Examples
# ppp object
data(simdat)
<testonly>simdat <- simdat[seq(1,npoints(simdat), by=4)]</testonly>
p <- pcf(simdat)
plot(p)
# fv object
K <- Kest(simdat)
p2 <- pcf(K, spar=0.8, method="b")
plot(p2)
# multitype pattern; fasp object
amaK <- alltypes(amacrine, "K")
amap <- pcf(amaK, spar=1, method="b")
plot(amap)