lgcp.estK.Rd
\name{lgcp.estK}
\alias{lgcp.estK}
\title{Fit a Log-Gaussian Cox Point Process by Minimum Contrast}
\description{
Fits a log-Gaussian Cox point process model
(with exponential covariance function)
to a point pattern dataset by the Method of Minimum Contrast.
}
\usage{
lgcp.estK(X, startpar=list(sigma2=1,alpha=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL)
}
\arguments{
\item{X}{
Data to which the model will be fitted.
Either a point pattern or a summary statistic.
See Details.
}
\item{startpar}{
Vector of starting values for the parameters of the
log-Gaussian Cox process model.
}
\item{lambda}{
Optional. An estimate of the intensity of the point process.
}
\item{q,p}{
Optional. Exponents for the contrast criterion.
}
\item{rmin, rmax}{
Optional. The interval of \eqn{r} values for the contrast criterion.
}
}
\details{
This algorithm fits a log-Gaussian Cox point process model
to a point pattern dataset
by the Method of Minimum Contrast, using the K function.
The argument \code{X} can be either
\describe{
\item{a point pattern:}{An object of class \code{"ppp"}
representing a point pattern dataset.
The \eqn{K} function of the point pattern will be computed
using \code{\link{Kest}}, and the method of minimum contrast
will be applied to this.
}
\item{a summary statistic:}{An object of class \code{"fv"} containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the \eqn{K} function,
and this object should have been obtained by a call to
\code{\link{Kest}} or one of its relatives.
}
}
The algorithm fits a log-Gaussian Cox point process (LGCP)
model to \code{X}, by finding the parameters of the LGCP model
which give the closest match between the
theoretical \eqn{K} function of the LGCP model
and the observed \eqn{K} function.
For a more detailed explanation of the Method of Minimum Contrast,
see \code{\link{mincontrast}}.
The model fitted is a stationary, isotropic log-Gaussian Cox process
with exponential covariance
(Moller and Waagepetersen, 2003, pp. 72-76).
To define this process we start with
a stationary Gaussian random field \eqn{Z} in the two-dimensional plane,
with constant mean \eqn{\mu}{mu} and covariance function
\deqn{
c(r) = \sigma^2 e^{-r/\alpha}
}{
c(r) = sigma^2 * exp(-r/alpha)
}
where \eqn{\sigma^2}{sigma^2} and \eqn{\alpha}{alpha} are parameters.
Given \eqn{Z}, we generate a Poisson point process \eqn{Y} with intensity
function \eqn{\lambda(u) = \exp(Z(u))}{lambda(u) = exp(Z(u))} at
location \eqn{u}. Then \eqn{Y} is a log-Gaussian Cox process.
The theoretical \eqn{K}-function of the LGCP is
\deqn{
K(r) = \int_0^r 2\pi s \exp(\sigma^2 \exp(-s/\alpha)) \, {\rm d}s.
}{
K(r) = integral from 0 to r of
(2 * pi * s * exp(sigma^2 * exp(-s/alpha))) ds.
}
The theoretical intensity of the LGCP is
\deqn{
\lambda = \exp(\mu + \frac{\sigma^2}{2}).
}{
lambda= exp(mu + sigma^2/2).
}
In this algorithm, the Method of Minimum Contrast is first used to find
optimal values of the parameters \eqn{\sigma^2}{sigma^2}
and \eqn{\alpha}{alpha^2}. Then the remaining parameter
\eqn{\mu}{mu} is inferred from the estimated intensity
\eqn{\lambda}{lambda}.
If the argument \code{lambda} is provided, then this is used
as the value of \eqn{\lambda}{lambda}. Otherwise, if \code{X} is a
point pattern, then \eqn{\lambda}{lambda}
will be estimated from \code{X}.
If \code{X} is a summary statistic and \code{lambda} is missing,
then the intensity \eqn{\lambda}{lambda} cannot be estimated, and
the parameter \eqn{\mu}{mu} will be returned as \code{NA}.
The remaining arguments \code{rmin,rmax,q,p} control the
method of minimum contrast; see \code{\link{mincontrast}}.
}
\value{
An object of class \code{"minconfit"}. There are methods for printing
and plotting this object. It contains the following main components:
\item{par }{Vector of fitted parameter values.}
\item{fit }{Function value table (object of class \code{"fv"})
containing the observed values of the summary statistic
(\code{observed}) and the theoretical values of the summary
statistic computed from the fitted model parameters.
}
}
\references{
Moller, J. and Waagepetersen, R. (2003).
Statistical Inference and Simulation for Spatial Point Processes.
Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2006).
An estimation function approach to inference for inhomogeneous
Neyman-Scott processes. Submitted.
}
\author{Rasmus Waagepetersen
\email{rw@math.auc.dk}.
Adapted for \pkg{spatstat} by Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
}
\seealso{
\code{\link{lgcp.estK}},
\code{\link{matclust.estK}},
\code{\link{mincontrast}},
\code{\link{Kest}}
}
\examples{
data(redwood)
u <- lgcp.estK(redwood, c(sigma2=0.1, alpha=1))
u
plot(u)
}
\keyword{spatial}
\keyword{models}