# pcf

0th

Percentile

##### Pair Correlation Function

Estimate the pair correlation function.

Keywords
spatial, nonparametric
##### 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 $$g(r) = \frac{K'(r)}{2\pi r}$$ 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.

##### 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.

##### 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

• pcf
##### Examples
# ppp object
data(simdat)
<testonly>simdat <- simdat[seq(1,simdat$n, by=4)]</testonly> p <- pcf(simdat) plot(p) # fv object K <- Kest(simdat) p2 <- pcf(K, spar=0.8, method="b") plot(p2) # multitype pattern; fasp object data(betacells) <testonly>betacells <- betacells[seq(1,betacells$n, by=10)]</testonly>
betaK <- alltypes(betacells, "K")
betap <- pcf(betaK, spar=1, method="b")
plot(betap)
Documentation reproduced from package spatstat, version 1.28-1, License: GPL (>= 2)

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