# Linhom

##### L-function

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

- Keywords
- spatial, nonparametric

##### Usage

`Linhom(...)`

##### Arguments

- ...
- Arguments passed to
`Kinhom`

to estimate the inhomogeneous K-function.

##### Details

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.

##### Value

- 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.

##### References

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

##### See Also

##### Examples

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

*Documentation reproduced from package spatstat, version 1.23-2, License: GPL (>= 2)*