Estimates the pair correlation function of a point pattern, given an estimate of the K function.

```
# S3 method for fv
pcf(X, ..., method="c")
```

A function value table
(object of class `"fv"`

, see `fv.object`

)
representing a pair correlation function.

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 \(g(r)\)

- X
An estimate of the \(K\) function or one of its variants. An object of class

`"fv"`

.- ...
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.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner rolfturner@posteo.net and Ege Rubak rubak@math.aau.dk.

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.
It is a method for the generic function `pcf`

for the class `"fv"`

.

The argument `X`

should be an estimated \(K\) function,
given as a function value table (object of class `"fv"`

,
see `fv.object`

).
This object should be the value returned by
`Kest`

, `Kcross`

, `Kmulti`

or `Kinhom`

.

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.

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.

`pcf`

,
`pcf.ppp`

,
`Kest`

,
`Kinhom`

,
`Kcross`

,
`Kdot`

,
`Kmulti`

,
`alltypes`

,
`smooth.spline`

,
`predict.smooth.spline`

```
# univariate point pattern
X <- simdat
# \testonly{
X <- X[seq(1,npoints(X), by=4)]
# }
K <- Kest(X)
p <- pcf.fv(K, spar=0.5, method="b")
plot(p, main="pair correlation function for simdat")
# indicates inhibition at distances r < 0.3
```

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