# pcf: Pair Correlation Function In spatstat.core: Core Functionality of the 'spatstat' Family

 pcf 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.

\spatstatAuthors

### 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

```  # 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)
```

spatstat.core documentation built on May 18, 2022, 9:05 a.m.