bw.diggle.Rd
\name{bw.diggle}
\alias{bw.diggle}
\title{
Cross Validated Bandwidth Selection for Kernel Density
}
\description{
Uses cross-validation to select a smoothing bandwidth
for the kernel estimation of point process intensity.
}
\usage{
bw.diggle(X, ..., correction="good", hmax=NULL, nr=512)
}
\arguments{
\item{X}{
A point pattern (object of class \code{"ppp"}).
}
\item{\dots}{Ignored.}
\item{correction}{
Character string passed to \code{\link{Kest}}
determining the edge correction to be used to
calculate the \eqn{K} function.
}
\item{hmax}{
Numeric. Maximum value of bandwidth that should be considered.
}
\item{nr}{
Integer. Number of steps in the distance value \eqn{r} to use in computing
numerical integrals.
}
}
\details{
This function selects an appropriate bandwidth \code{sigma}
for the kernel estimator of point process intensity
computed by \code{\link{density.ppp}}.
The bandwidth \eqn{\sigma}{\sigma} is chosen to
minimise the mean-square error criterion defined by Diggle (1985).
The algorithm uses the method of Berman and Diggle (1989) to
compute the quantity
\deqn{
M(\sigma) = \frac{\mbox{MSE}(\sigma)}{\lambda^2} - g(0)
}{
M(\sigma) = MSE(\sigma)/\lambda^2 - g(0)
}
as a function of bandwidth \eqn{\sigma}{\sigma},
where \eqn{\mbox{MSE}(\sigma)}{MSE(\sigma)} is the
mean squared error at bandwidth \eqn{\sigma}{\sigma},
while \eqn{\lambda}{\lambda} is the mean intensity,
and \eqn{g} is the pair correlation function.
See Diggle (2003, pages 115-118) for a summary of this method.
The result is a numerical value giving the selected bandwidth.
The result also belongs to the class \code{"bw.optim"}
which can be plotted to show the (rescaled) mean-square error
as a function of \code{sigma}.
}
\section{Definition of bandwidth}{
The smoothing parameter \code{sigma} returned by \code{bw.diggle}
(and displayed on the horizontal axis of the plot)
corresponds to \code{h/2}, where \code{h} is the smoothing
parameter described in Diggle (2003, pages 116-118) and
Berman and Diggle (1989).
In those references, the smoothing kernel
is the uniform density on the disc of radius \code{h}. In
\code{\link{density.ppp}}, the smoothing kernel is the
isotropic Gaussian density with standard deviation \code{sigma}.
When replacing one kernel by another, the usual
practice is to adjust the bandwidths so that the kernels have equal
variance (cf. Diggle 2003, page 118). This implies that \code{sigma = h/2}.
}
\value{
A numerical value giving the selected bandwidth.
The result also belongs to the class \code{"bw.optim"}
which can be plotted.
}
\seealso{
\code{\link{density.ppp}},
\code{\link{bw.ppl}},
\code{\link{bw.scott}}
}
\examples{
data(lansing)
attach(split(lansing))
b <- bw.diggle(hickory)
plot(b, ylim=c(-2, 0), main="Cross validation for hickories")
\donttest{
plot(density(hickory, b))
}
}
\references{
Berman, M. and Diggle, P. (1989)
Estimating weighted integrals of the
second-order intensity of a spatial point process.
\emph{Journal of the Royal Statistical Society, series B}
\bold{51}, 81--92.
Diggle, P.J. (1985)
A kernel method for smoothing point process data.
\emph{Applied Statistics} (Journal of the Royal Statistical Society,
Series C) \bold{34} (1985) 138--147.
Diggle, P.J. (2003)
\emph{Statistical analysis of spatial point patterns},
Second edition. Arnold.
}
\author{\adrian
and \rolf
}
\keyword{spatial}
\keyword{methods}
\keyword{smooth}