\name{bw.relrisk}
\alias{bw.relrisk}
\title{
Cross Validated Bandwidth Selection for Relative Risk Estimation
}
\description{
Uses cross-validation to select a smoothing bandwidth
for the estimation of relative risk.
}
\usage{
bw.relrisk(X, method = "likelihood", nh = spatstat.options("n.bandwidth"),
hmin=NULL, hmax=NULL, warn=TRUE)
}
\arguments{
\item{X}{
A multitype point pattern (object of class \code{"ppp"}
which has factor valued marks).
}
\item{method}{
Character string determining the cross-validation method.
Current options are \code{"likelihood"},
\code{"leastsquares"} or
\code{"weightedleastsquares"}.
}
\item{nh}{
Number of trial values of smoothing bandwith \code{sigma}
to consider. The default is 32.
}
\item{hmin, hmax}{
Optional. Numeric values.
Range of trial values of smoothing bandwith \code{sigma}
to consider. There is a sensible default.
}
\item{warn}{
Logical. If \code{TRUE}, issue a warning if the minimum of
the cross-validation criterion occurs at one of the ends of the
search interval.
}
}
\details{
This function selects an appropriate bandwidth for the nonparametric
estimation of relative risk using \code{\link{relrisk}}.
Consider the indicators \eqn{y_{ij}}{y[i,j]} which equal \eqn{1} when
data point \eqn{x_i}{x[i]} belongs to type \eqn{j}, and equal \eqn{0}
otherwise.
For a particular value of smoothing bandwidth,
let \eqn{\hat p_j(u)}{p*[j](u)} be the estimated
probabilities that a point at location \eqn{u} will belong to
type \eqn{j}.
Then the bandwidth is chosen to minimise either the likelihood,
the squared error, or the approximately standardised squared error, of the
indicators \eqn{y_{ij}}{y[i,j]} relative to the fitted
values \eqn{\hat p_j(x_i)}{p*[j](x[i])}. See Diggle (2003).
The result is a numerical value giving the selected bandwidth \code{sigma}.
The result also belongs to the class \code{"bw.optim"}
allowing it to be printed and plotted. The plot shows the cross-validation
criterion as a function of bandwidth.
The range of values for the smoothing bandwidth \code{sigma}
is set by the arguments \code{hmin, hmax}. There is a sensible default,
based on multiples of Stoyan's rule of thumb \code{\link{bw.stoyan}}.
If the optimal bandwidth is achieved at an endpoint of the
interval \code{[hmin, hmax]}, the algorithm will issue a warning
(unless \code{warn=FALSE}). If this occurs, then it is probably advisable
to expand the interval by changing the arguments \code{hmin, hmax}.
Computation time depends on the number \code{nh} of trial values
considered, and also on the range \code{[hmin, hmax]} of values
considered, because larger values of \code{sigma} require
calculations involving more pairs of data points.
}
\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{relrisk}},
\code{\link{bw.stoyan}}
}
\examples{
data(urkiola)
\testonly{op <- spatstat.options(n.bandwidth=8)}
b <- bw.relrisk(urkiola)
b
plot(b)
b <- bw.relrisk(urkiola, hmax=20)
plot(b)
\testonly{spatstat.options(op)}
}
\references{
Diggle, P.J. (2003)
\emph{Statistical analysis of spatial point patterns},
Second edition. Arnold.
Kelsall, J.E. and Diggle, P.J. (1995)
Kernel estimation of relative risk.
\emph{Bernoulli} \bold{1}, 3--16.
}
\author{Adrian Baddeley \email{Adrian.Baddeley@curtin.edu.au}
and Rolf Turner \email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{methods}
\keyword{smooth}