\name{Linhom}
\alias{Linhom}
\title{L-function}
\description{
Calculates an estimate of the inhomogeneous version of
the \eqn{L}-function (Besag's transformation of Ripley's \eqn{K}-function)
for a spatial point pattern.
}
\usage{
Linhom(...)
}
\arguments{
\item{\dots}{
Arguments passed to \code{\link{Kinhom}}
to estimate the inhomogeneous K-function.
}
}
\details{
This command computes an estimate of the inhomogeneous version of
the \eqn{L}-function for a spatial point pattern
The original \eqn{L}-function is a transformation
(proposed by Besag) of Ripley's \eqn{K}-function,
\deqn{L(r) = \sqrt{\frac{K(r)}{\pi}}}{L(r) = sqrt(K(r)/pi)}
where \eqn{K(r)} is the Ripley \eqn{K}-function of a spatially homogeneous
point pattern, estimated by \code{\link{Kest}}.
The inhomogeneous \eqn{L}-function is the corresponding transformation
of the inhomogeneous \eqn{K}-function, estimated by \code{\link{Kinhom}}.
It is appropriate when the point pattern clearly does not have a
homogeneous intensity of points. It was proposed by
Baddeley, \ifelse{latex}{\out{M\o ller}}{Moller} and Waagepetersen (2000).
The command \code{Linhom} first calls
\code{\link{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 \eqn{L}-function is \eqn{L(r) = r}.
The square root also has the effect of stabilising
the variance of the estimator, so that \eqn{L} is more appropriate
for use in simulation envelopes and hypothesis tests.
}
\value{
An object of class \code{"fv"}, see \code{\link{fv.object}},
which can be plotted directly using \code{\link{plot.fv}}.
Essentially a data frame containing columns
\item{r}{the vector of values of the argument \eqn{r}
at which the function \eqn{L} has been estimated
}
\item{theo}{the theoretical value \eqn{L(r) = r}
for a stationary Poisson process
}
together with columns named
\code{"border"}, \code{"bord.modif"},
\code{"iso"} and/or \code{"trans"},
according to the selected edge corrections. These columns contain
estimates of the function \eqn{L(r)} obtained by the edge corrections
named.
}
\references{
Baddeley, A., \ifelse{latex}{\out{M\o ller}}{Moller}, J. and Waagepetersen, R. (2000)
Non- and semiparametric estimation of interaction in
inhomogeneous point patterns.
\emph{Statistica Neerlandica} \bold{54}, 329--350.
}
\seealso{
\code{\link{Kest}},
\code{\link{Lest}},
\code{\link{Kinhom}},
\code{\link{pcf}}
}
\examples{
data(japanesepines)
X <- japanesepines
L <- Linhom(X, sigma=0.1)
plot(L, main="Inhomogeneous L function for Japanese Pines")
}
\author{Adrian Baddeley \email{Adrian.Baddeley@curtin.edu.au}
and Rolf Turner \email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{nonparametric}