Revision c9b2c621c3bff55aaa77646dc1ba7316765cd7e4 authored by Adrian Baddeley on 25 April 2013, 00:00:00 UTC, committed by Gabor Csardi on 25 April 2013, 00:00:00 UTC
1 parent f86606a
Lest.Rd
\name{Lest}
\alias{Lest}
\title{L-function}
\description{
Calculates an estimate of the \eqn{L}-function (Besag's
transformation of Ripley's \eqn{K}-function)
for a spatial point pattern.
}
\usage{
Lest(X, ...)
}
\arguments{
\item{X}{
The observed point pattern,
from which an estimate of \eqn{L(r)} will be computed.
An object of class \code{"ppp"}, or data
in any format acceptable to \code{\link{as.ppp}()}.
}
\item{\dots}{
Other arguments passed to \code{\link{Kest}}
to control the estimation procedure.
}
}
\details{
This command computes an estimate of the \eqn{L}-function
for the spatial point pattern \code{X}.
The \eqn{L}-function is a transformation 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 \eqn{K}-function.
See \code{\link{Kest}} for information
about Ripley's \eqn{K}-function. The transformation to \eqn{L} was
proposed by Besag (1977).
The command \code{Lest} first calls
\code{\link{Kest}} to compute the estimate of the \eqn{K}-function,
and then applies the square root transformation.
For a completely random (uniform Poisson) point pattern,
the theoretical value of the \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{K} is more appropriate
for use in simulation envelopes and hypothesis tests.
See \code{\link{Kest}} for the list of arguments.
}
\section{Variance approximations}{
If the argument \code{var.approx=TRUE} is given, the return value
includes columns \code{rip} and \code{ls} containing approximations
to the variance of \eqn{\hat L(r)}{Lest(r)} under CSR.
These are obtained by the delta method from the variance
approximations described in \code{\link{Kest}}.
}
\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{
Besag, J. (1977)
Discussion of Dr Ripley's paper.
\emph{Journal of the Royal Statistical Society, Series B},
\bold{39}, 193--195.
}
\seealso{
\code{\link{Kest}},
\code{\link{pcf}}
}
\examples{
data(cells)
L <- Lest(cells)
plot(L, main="L function for cells")
}
\author{Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{nonparametric}
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