pcf.fasp
Pair Correlation Function obtained from array of K functions
Estimates the (bivariate) pair correlation functions of a point pattern, given an array of (bivariate) K functions.
- Keywords
- spatial, nonparametric
Usage
# S3 method for fasp
pcf(X, …, method="c")
Arguments
- X
An array of multitype \(K\) functions (object of class
"fasp"
).- …
Arguments controlling the smoothing spline function
smooth.spline
.- method
Letter
"a"
,"b"
,"c"
or"d"
indicating the method for deriving the pair correlation function from theK
function.
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 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.
Value
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
.
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
# NOT RUN {
# 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
# }