Pair Correlation Function obtained from K Function

Description

Estimates the pair correlation function of a point pattern, given an estimate of the K function.

Usage

  ## S3 method for class 'fv':
  pcf(X, ..., method="c")

Arguments

Details

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.

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 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 argument X should be an estimated K function, given as a function value table (object of class "fv", see fv.object). This object should be the value returned by Kest, Kcross, Kmulti or Kinhom.

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.

Value

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

Author(s)

Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf

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

pcf, pcf.ppp, Kest, Kinhom, Kcross, Kdot, Kmulti, alltypes, smooth.spline, predict.smooth.spline

Examples

  # univariate point pattern
  data(simdat)
  
  K <- Kest(simdat)
  p <- pcf.fv(K, spar=0.5, method="b")
  plot(p, main="pair correlation function for simdat")
  # indicates inhibition at distances r < 0.3