Calculates an estimate of the inhomogeneous version of the \(L\)-function (Besag's transformation of Ripley's \(K\)-function) for a spatial point pattern.

`Linhom(X, ..., correction)`

An object of class `"fv"`

, see `fv.object`

,
which can be plotted directly using `plot.fv`

.

Essentially a data frame containing columns

- r
the vector of values of the argument \(r\) at which the function \(L\) has been estimated

- theo
the theoretical value \(L(r) = r\) for a stationary Poisson process

together with columns named

`"border"`

, `"bord.modif"`

,

`"iso"`

and/or `"trans"`

,
according to the selected edge corrections. These columns contain
estimates of the function \(L(r)\) obtained by the edge corrections
named.

- X
The observed point pattern, from which an estimate of \(L(r)\) will be computed. An object of class

`"ppp"`

, or data in any format acceptable to`as.ppp()`

.- correction,...
Other arguments passed to

`Kinhom`

to control the estimation procedure.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz

This command computes an estimate of the inhomogeneous version of the \(L\)-function for a spatial point pattern.

The original \(L\)-function is a transformation
(proposed by Besag) of Ripley's \(K\)-function,
$$L(r) = \sqrt{\frac{K(r)}{\pi}}$$
where \(K(r)\) is the Ripley \(K\)-function of a spatially homogeneous
point pattern, estimated by `Kest`

.

The inhomogeneous \(L\)-function is the corresponding transformation
of the inhomogeneous \(K\)-function, estimated by `Kinhom`

.
It is appropriate when the point pattern clearly does not have a
homogeneous intensity of points. It was proposed by
Baddeley, Moller and Waagepetersen (2000).

The command `Linhom`

first calls
`Kinhom`

to compute the estimate of the inhomogeneous K-function,
and then applies the square root transformation.

For a Poisson point pattern (homogeneous or inhomogeneous), the theoretical value of the inhomogeneous \(L\)-function is \(L(r) = r\). The square root also has the effect of stabilising the variance of the estimator, so that \(L\) is more appropriate for use in simulation envelopes and hypothesis tests.

Baddeley, A., Moller, J. and Waagepetersen, R. (2000)
Non- and semiparametric estimation of interaction in
inhomogeneous point patterns.
*Statistica Neerlandica* **54**, 329--350.

`Kest`

,
`Lest`

,
`Kinhom`

,
`pcf`

```
X <- japanesepines
L <- Linhom(X, sigma=0.1)
plot(L, main="Inhomogeneous L function for Japanese Pines")
```

Run the code above in your browser using DataLab