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 class 'fasp':
pcf(X, \dots, 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.
"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"
, seefasp.object
) representing an array of pair correlation functions. This can be thought of as a matrixY
each of whose entriesY[i,j]
is a function value table (class"fv"
) representing the pair correlation function between points of typei
and points of typej
.
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
# multitype point pattern
data(betacells)
<testonly>betacells <- betacells[seq(1,betacells$n, by=10)]</testonly>
KK <- alltypes(betacells, "K")
p <- pcf.fasp(KK, spar=0.5, method="b")
plot(p)
# short range inhibition between all types
# strong inhibition between "on" and "off"