pcf.fv {spatstat.explore} | R Documentation |
Pair Correlation Function obtained from K Function
Description
Estimates the pair correlation function of a point pattern, given an estimate of the K function.
Usage
## S3 method for class 'fv'
pcf(X, ..., method="c")
Arguments
X |
An estimate of the |
... |
Arguments controlling the smoothing spline
function |
method |
Letter |
Details
The pair correlation function of a stationary point process is
where is the derivative of
, the
reduced second moment function (aka “Ripley's
function”)
of the point process. See
Kest
for information
about . For a stationary Poisson process, the
pair correlation function is identically equal to 1. Values
suggest inhibition between points;
values greater than 1 suggest clustering.
We also apply the same definition to
other variants of the classical function,
such as the multitype
functions
(see
Kcross
, Kdot
) and the
inhomogeneous function (see
Kinhom
).
For all these variants, the benchmark value of
corresponds to
.
This routine computes an estimate of
from an estimate of
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 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:
-
"a" apply smoothing to
, estimate its derivative, and plug in to the formula above;
-
"b" apply smoothing to
constraining
, estimate the derivative of
, and solve;
-
"c" apply smoothing to
constraining
, estimate its derivative, and solve.
-
"d" apply smoothing to
, estimate its derivative, and solve.
Method "c"
seems to be the best at
suppressing variability for small values of .
However it effectively constrains
.
If the point pattern seems to have inhibition at small distances,
you may wish to experiment with method
"b"
which effectively
constrains . 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 function value table
(object of class "fv"
, see fv.object
)
representing a pair correlation function.
Essentially a data frame containing (at least) the variables
r |
the vector of values of the argument |
pcf |
vector of values of |
Author(s)
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner rolfturner@posteo.net and Ege Rubak rubak@math.aau.dk.
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
pcf
,
pcf.ppp
,
Kest
,
Kinhom
,
Kcross
,
Kdot
,
Kmulti
,
alltypes
,
smooth.spline
,
predict.smooth.spline
Examples
# univariate point pattern
X <- simdat
K <- Kest(X)
p <- pcf.fv(K, spar=0.5, method="b")
plot(p, main="pair correlation function for simdat")
# indicates inhibition at distances r < 0.3