spatstat (version 1.48-0)

pcf.fasp: Pair Correlation Function obtained from array of K functions

Description

Estimates the (bivariate) pair correlation functions of a point pattern, given an array of (bivariate) K functions.

Usage

"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 the K function.

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.

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 $Y(r) = K(r)/(2 * pi * r)$ 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.

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
  KK <- alltypes(amacrine, "K")
  p <- pcf.fasp(KK, spar=0.5, method="b")
  plot(p)
  # strong inhibition between points of the same type