spatstat (version 1.64-1)

# pcf: 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.

## 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.

## 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 $$g(r)$$ 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.

## References

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

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

## Examples

Run this code
# NOT RUN {
# ppp object
X <- simdat

# }
# NOT RUN {
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)
# }


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