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.
## S3 method for class 'fasp' pcf(X, ..., method="c")
X |
An array of multitype K functions
(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 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) = 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 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 r.turner@auckland.ac.nz
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.
# 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
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.