pcf
Pair Correlation Function
Estimates the pair correlation function of a point pattern.
- Keywords
- spatial
Usage
pcf(X, ..., method="c")Arguments
- X
- Either the observed data point pattern, or an estimate of its $K$ function, or an array of multitype $K$ functions (see Details).
- ...
- Arguments controlling the smoothing spline
function
smooth.spline. - method
- Letter
"a","b"or"c"indicating the method for deriving the pair correlation function from theKfunction.
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 $g(r)$ from an estimate of $K(r)$ or its variants, using smoothing splines to approximate the derivative.
The argument X may be either
- a point pattern for which an estimate of the pair correlation function
should be computed. This should be an object of class
"ppp", or in a format recognised byas.ppp(). - a data frame containing an estimate of a$K$function.
This data frame should be the value returned by
Kest,Kcross,KmultiorKinhom. - a function array (object of class
"fasp", seefasp.object) containing several estimates of$K$functions. This should have been obtained fromalltypeswith the argumentfun="K".
X is a point pattern, the $K$ function is
first estimated by Kest.
The smoothing spline operations are performed by
smooth.spline and predict.smooth.spline
from the modreg library.
Three numerical methods are available:
- "a"apply smoothing to$K(r)$, estimate its derivative, and plug in to the formula above;
- "b"apply smoothing to$Y(r) = \frac{K(r)}{2 \pi r}$constraining$Y(0) = 0$, estimate the derivative of$Y$, and solve;
- "c"apply smoothing to$Z(r) = \frac{K(r)}{\pi r^2}$constraining$Z(0)=1$, estimate its derivative, and solve.
"c" seems to be the best at
suppressing variability for small values of $r$.
However it effectively constrains $g(0) = 1$.
If the point pattern seems to have inhibition at small distances,
you may wish to experiment with method "b" which effectively
constrains $g(0)=0$. Method "a" seems
comparatively unreliable. Useful arguments to control the splines
include the smoothing tradeoff parameter spar
and the degrees of freedom df. See smooth.spline
for details.
Value
- A data frame containing (at least) the variables
r the vector of values of the argument $r$ at which the pair correlation function $g(r)$ has been estimated pcf vector of values of $g(r)$
References
Stoyan, D, Kendall, W.S. and Mecke, J. (1995) Stochastic geometry and its applications. 2nd edition. Springer Verlag.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.
See Also
Kest,
Kinhom,
Kcross,
Kdot,
Kmulti,
alltypes,
smooth.spline,
predict.smooth.spline
Examples
library(spatstat)
data(simdat)
p <- pcf(simdat)
plot(p$r, p$pcf, type="l", xlab="r", ylab="g(r)",
main="pair correlation")
abline(h=1, lty=1)
# multitype point pattern
data(ganglia)
p <- pcf(alltypes(ganglia, "K"), spar=0.5, method="b")
conspire(p, cbind(pcf,1) ~ r, subset="r <= 0.2",
title="Pair correlation functions for ganglia")