# Lest

##### L-function

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

- Keywords
- spatial, nonparametric

##### Usage

`Lest(...)`

##### Arguments

- ...
- Arguments passed to
`Kest`

to estimate the $K$-function.

##### Details

This command computes an estimate of the $L$-function for a spatial point pattern. The $L$-function is a transformation of Ripley's $K$-function, $$L(r) = \sqrt{\frac{K(r)}{\pi}}$$ where $K(r)$ is the $K$-function.

See `Kest`

for information
about Ripley's $K$-function. The transformation to $L$ was
proposed by Besag (1977).

The command `Lest`

first calls
`Kest`

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

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

See `Kest`

for the list of arguments.

##### 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.

##### Variance approximations

If the argument `var.approx=TRUE`

is given, the return value
includes columns `rip`

and `ls`

containing approximations
to the variance of $\hat L(r)$ under CSR.
These are obtained by the delta method from the variance
approximations described in `Kest`

.

##### References

Besag, J. (1977)
Discussion of Dr Ripley's paper.
*Journal of the Royal Statistical Society, Series B*,
**39**, 193--195.

##### See Also

##### Examples

```
data(cells)
L <- Lest(cells)
plot(L, main="L function for cells")
```

*Documentation reproduced from package spatstat, version 1.24-1, License: GPL (>= 2)*