R: Pair Correlation Function

pcf {spatstat.core} | R Documentation

## Pair Correlation Function

### Description

Estimate the pair correlation function.

### Usage

     pcf(X, ...)

### Arguments

`X` | Either the observed data point pattern, or an estimate of its K function, or an array of multitype K functions (see Details).
`...` | Other arguments passed to the appropriate method.

### 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) either directly from a point pattern, or indirectly from an estimate of K(r) or one of its variants.

This function is generic, with methods for the classes `"ppp"`, `"fv"` and `"fasp"`.

If `X` is a point pattern (object of class `"ppp"`) then the pair correlation function is estimated using a traditional kernel smoothing method (Stoyan and Stoyan, 1994). See `pcf.ppp` for details.

If `X` is a function value table (object of class `"fv"`), then it is assumed to contain estimates of the K function or one of its variants (typically obtained from `Kest` or `Kinhom`). This routine computes an estimate of g(r) using smoothing splines to approximate the derivative. See `pcf.fv` for details.

If `X` is a function value array (object of class `"fasp"`), then it is assumed to contain estimates of several K functions (typically obtained from `Kmulti` or `alltypes`). This routine computes an estimate of g(r) for each cell in the array, using smoothing splines to approximate the derivatives. See `pcf.fasp` for details.

### 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.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk

### References

Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

### See Also

`pcf.ppp`, `pcf.fv`, `pcf.fasp`, `Kest`, `Kinhom`, `Kcross`, `Kdot`, `Kmulti`, `alltypes`

### Examples

      # ppp object
      X <- simdat

      p <- pcf(X)
      plot(p)

      # fv object
      K <- Kest(X)
      p2 <- pcf(K, spar=0.8, method="b")
      plot(p2)

      # multitype pattern; fasp object
      amaK <- alltypes(amacrine, "K")
      amap <- pcf(amaK, spar=1, method="b")
      plot(amap)

[Package spatstat.core version 1.65-5 Index]