thomas.estpcf.Rd
\name{thomas.estpcf}
\alias{thomas.estpcf}
\title{Fit the Thomas Point Process by Minimum Contrast}
\description{
Fits the Thomas point process to a point pattern dataset by the Method of
Minimum Contrast using the pair correlation function.
}
\usage{
thomas.estpcf(X, startpar=c(kappa=1,sigma2=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
}
\arguments{
\item{X}{
Data to which the Thomas 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
Thomas process.
}
\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.
}
\item{\dots}{
Optional arguments passed to \code{\link[stats]{optim}}
to control the optimisation algorithm. See Details.
}
\item{pcfargs}{
Optional list containing arguments passed to \code{\link{pcf.ppp}}
to control the smoothing in the estimation of the
pair correlation function.
}
}
\details{
This algorithm fits the Thomas point process model to a point pattern dataset
by the Method of Minimum Contrast, using the pair correlation function
\code{\link{pcf}}.
The argument \code{X} can be either
\describe{
\item{a point pattern:}{An object of class \code{"ppp"}
representing a point pattern dataset.
The pair correlation function of the point pattern will be computed
using \code{\link{pcf}}, 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 pair correlation function,
and this object should have been obtained by a call to
\code{\link{pcf}} or one of its relatives.
}
}
The algorithm fits the Thomas point process to \code{X},
by finding the parameters of the Thomas model
which give the closest match between the
theoretical pair correlation function of the Thomas process
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see \code{\link{mincontrast}}.
The Thomas point process is described in Moller and Waagepetersen
(2003, pp. 61--62). It is a cluster process formed by taking a
pattern of parent points, generated according to a Poisson process
with intensity \eqn{\kappa}{kappa}, and around each parent point,
generating a random number of offspring points, such that the
number of offspring of each parent is a Poisson random variable with mean
\eqn{\mu}{mu}, and the locations of the offspring points of one parent
are independent and isotropically Normally distributed around the parent
point with standard deviation \eqn{\sigma}{sigma}.
The theoretical pair correlation function of the Thomas process is
\deqn{
g(r) = 1 + \frac 1 {4\pi \kappa \sigma^2} \exp(-\frac{r^2}{4\sigma^2})).
}{
g(r) = 1 + exp(-r^2/(4 * sigma^2)))/(4 * pi * kappa * sigma^2).
}
The theoretical intensity
of the Thomas process is \eqn{\lambda = \kappa \mu}{lambda=kappa* mu}.
In this algorithm, the Method of Minimum Contrast is first used to find
optimal values of the parameters \eqn{\kappa}{kappa}
and \eqn{\sigma^2}{sigma^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}}.
The Thomas process can be simulated, using \code{\link{rThomas}}.
Homogeneous or inhomogeneous Thomas process models can also
be fitted using the function \code{\link{kppm}}.
The optimisation algorithm can be controlled through the
additional arguments \code{"..."} which are passed to the
optimisation function \code{\link[stats]{optim}}. For example,
to constrain the parameter values to a certain range,
use the argument \code{method="L-BFGS-B"} to select an optimisation
algorithm that respects box constraints, and use the arguments
\code{lower} and \code{upper} to specify (vectors of) minimum and
maximum values for each parameter.
}
\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. (2007)
An estimating function approach to inference for
inhomogeneous Neyman-Scott processes.
\emph{Biometrics} \bold{63}, 252--258.
}
\author{
Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
}
\seealso{
\code{\link{thomas.estK}}
\code{\link{mincontrast}},
\code{\link{pcf}},
\code{\link{rThomas}} to simulate the fitted model.
}
\examples{
data(redwood)
u <- thomas.estpcf(redwood, c(kappa=10, sigma2=0.1))
u
plot(u)
u2 <- thomas.estpcf(redwood, c(kappa=10, sigma2=0.1),
pcfargs=list(stoyan=0.12))
}
\keyword{spatial}
\keyword{models}