pcf.fasp {spatstat.explore} | R Documentation |
Estimates the (bivariate) pair correlation functions of a point pattern, given an array of (bivariate) K functions.
## S3 method for class 'fasp'
pcf(X, ..., method="c")
X |
An array of multitype |
... |
Arguments controlling the smoothing spline
function |
method |
Letter |
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 array of estimates of K(r)
or its variants,
using smoothing splines to approximate the derivatives.
It is a method for the generic function pcf
.
The argument X
should be
a function array (object of class "fasp"
,
see fasp.object
)
containing several estimates of K
functions.
This should have been obtained from alltypes
with the argument fun="K"
.
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 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.
"d" apply smoothing to
V(r) = \sqrt{K(r)}
,
estimate its derivative, and solve.
Method "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.
A function array (object of class "fasp"
,
see fasp.object
)
representing an array of pair correlation functions.
This can be thought of as a matrix Y
each of whose entries
Y[i,j]
is a function value table (class "fv"
)
representing the pair correlation function between
points of type i
and points of type j
.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
and Rolf Turner rolfturner@posteo.net
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.
Kest
,
Kinhom
,
Kcross
,
Kdot
,
Kmulti
,
alltypes
,
smooth.spline
,
predict.smooth.spline
# multitype point pattern
KK <- alltypes(amacrine, "K")
p <- pcf.fasp(KK, spar=0.5, method="b")
plot(p)
# strong inhibition between points of the same type