pcf(X, ..., method="c")
smooth.spline
."a"
, "b"
or "c"
indicating the
method for deriving the pair correlation function from the
K
function.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
"ppp"
,
or in a format recognised byas.ppp()
.Kest
,Kcross
,Kmulti
orKinhom
."fasp"
,
seefasp.object
)
containing several estimates of$K$functions.
This should have been obtained fromalltypes
with 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:
"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.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.
Kest
,
Kinhom
,
Kcross
,
Kdot
,
Kmulti
,
alltypes
,
smooth.spline
,
predict.smooth.spline
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")
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