Pair Correlation Function obtained from K Function
Estimates the pair correlation function of a point pattern, given an estimate of the K function.
## S3 method for class 'fv' pcf(X, ..., method="c")
X |
An estimate of the K function
or one of its variants.
An object of class |
... |
Arguments controlling the smoothing spline
function |
method |
Letter |
The pair correlation function of a stationary point process is
g(r) = 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.
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 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) = K(r)/(2 * pi * r) constraining Y(0) = 0, estimate the derivative of Y, and solve;
"c" apply smoothing to Y(r) = 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 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 r at which the pair correlation function g(r) has been estimated |
pcf |
vector of values of g(r) |
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.
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.
# 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
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