## S3 method for class 'fv':
pcf(X, \dots, method="c")
"fv"
.smooth.spline
."a"
, "b"
, "c"
or "d"
indicating the
method for deriving the pair correlation function from the
K
function."fv"
, see fv.object
)
representing a pair correlation function.Essentially a data frame containing (at least) the variables
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.
It is a method for the generic function pcf
for the class "fv"
.
The argument X
should be an estimated $K$ function,
given as a function value table (object of class "fv"
,
see fv.object
).
This object should be the value returned by
Kest
, Kcross
, Kmulti
or Kinhom
.
The smoothing spline operations are performed by
smooth.spline
and predict.smooth.spline
from the modreg
library.
Four 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.
pcf
,
pcf.ppp
,
Kest
,
Kinhom
,
Kcross
,
Kdot
,
Kmulti
,
alltypes
,
smooth.spline
,
predict.smooth.spline
# univariate point pattern
data(simdat)
<testonly>simdat <- simdat[seq(1,simdat$n, by=4)]</testonly>
K <- Kest(simdat)
p <- pcf.fv(K, spar=0.5, method="b")
plot(p, main="pair correlation function for simdat")
# indicates inhibition at distances r < 0.3
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