spatstat (version 1.5-9)

pcf: Pair Correlation Function

Description

Estimates the pair correlation function of a point pattern.

Usage

pcf(X, ..., method="c")

Arguments

X

Either the observed data point pattern, or an estimate of its $K$ function, or an array of multitype $K$ functions (see Details).

...

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

• Either a function value table (object of class "fv", see fv.object) representing a pair correlation function, or a function array (object of class "fasp", see fasp.object) representing an array of pair correlation functions.

  If X is an unmarked point pattern, the return value is a function value table (class "fv"). It is 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 $g(r)$
• If code{X} is a function value table (class "fv") representing the estimated $K$ function of a point pattern (obtained from Kest, Kinhom or Kest.fft), then the return value is again a function value table, representing the pair correlation function.

  If X is a multitype point pattern, the return value is a function array (class "fasp"). 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.

  If X is a function array (class "fasp") representing an array of estimated $K$ functions (obtained from alltypes), the return value is another function array, containing the corresponding pair correlation functions.

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 estimate of $K(r)$ or its variants, using smoothing splines to approximate the derivative.

The argument X may be either

• a point pattern for which an estimate of the pair correlation function should be computed. This should be an object of class"ppp", or in a format recognised byas.ppp().
• an estimated$K$function, given as a function value table (object of class"fv", seefv.object). This object should be the value returned byKest,Kcross,KmultiorKinhom.
• a function array (object of class"fasp", seefasp.object) containing several estimates of$K$functions. This should have been obtained fromalltypeswith the argumentfun="K".

If X is a point pattern, the $K$ function is first estimated by Kest. 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.

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

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

  # multitype point pattern
  data(betacells)
  p <- pcf(alltypes(betacells, "K"), spar=0.5, method="b")
  plot(p)
  # short range inhibition between all types
  # strong inhibition between "on" and "off"