spatstat (version 1.64-1)

# Lest: L-function

## Description

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

## Usage

Lest(X, ..., correction)

## Arguments

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 Kest to control the estimation procedure.

## 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.

## Details

This command computes an estimate of the $$L$$-function for the spatial point pattern X. 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 $$L(r)$$ is more appropriate for use in simulation envelopes and hypothesis tests.

See Kest for the list of arguments.

## References

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

Kest, pcf
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