spatstat (version 1.23-1)

Linhom: L-function

Description

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

Usage

Linhom(...)

Arguments

...

Arguments passed to Kinhom to estimate the inhomogeneous K-function.

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.

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.

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

Kest, Lest, Kinhom, pcf

Examples

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