vargamma.estpcf.Rd
\name{vargamma.estpcf}
\alias{vargamma.estpcf}
\title{Fit the Neyman-Scott Cluster Point Process with Variance Gamma kernel}
\description{
Fits the Neyman-Scott cluster point process, with Variance Gamma
kernel, to a point pattern dataset by the Method of Minimum Contrast,
using the pair correlation function.
}
\usage{
vargamma.estpcf(X, startpar=c(kappa=1,eta=1), nu.ker = -1/4, lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs = list())
}
\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 model.
}
\item{nu.ker}{
Numerical value controlling the shape of the tail of the clusters.
A number greater than \code{-1/2}.
}
\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 Neyman-Scott Cluster point process model
with Variance Gamma kernel (Jalilian et al, 2011)
to a point pattern dataset
by the Method of Minimum Contrast, using the pair correlation 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 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 Neyman-Scott Cluster point process
with Variance Gamma kernel to \code{X},
by finding the parameters of the model
which give the closest match between the
theoretical pair correlation function of the model
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see \code{\link{mincontrast}}.
The Neyman-Scott cluster point process with Variance Gamma
kernel is described in Jalilian et al (2011).
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
have a common distribution described in Jalilian et al (2011).
If the argument \code{lambda} is provided, then this is used
as the value of the point process intensity \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 corresponding model can be simulated using \code{\link{rVarGamma}}.
The parameter \code{eta} appearing in \code{startpar} is equivalent to the
scale parameter \code{omega} used in \code{\link{rVarGamma}}.
Homogeneous or inhomogeneous Neyman-Scott/VarGamma models can also be
fitted using the function \code{\link{kppm}} and the fitted models
can be simulated using \code{\link{simulate.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{
Jalilian, A., Guan, Y. and Waagepetersen, R. (2011)
Decomposition of variance for spatial Cox processes.
Manuscript submitted for publication.
Waagepetersen, R. (2007)
An estimating function approach to inference for
inhomogeneous Neyman-Scott processes.
\emph{Biometrics} \bold{63}, 252--258.
}
\author{Abdollah Jalilian and Rasmus Waagepetersen.
Adapted for \pkg{spatstat} by Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
}
\seealso{
\code{\link{kppm}},
\code{\link{vargamma.estK}},
\code{\link{lgcp.estpcf}},
\code{\link{thomas.estpcf}},
\code{\link{cauchy.estpcf}},
\code{\link{mincontrast}},
\code{\link{pcf}},
\code{\link{pcfmodel}}.
\code{\link{rVarGamma}} to simulate the model.
}
\examples{
u <- vargamma.estpcf(redwood)
u
plot(u, legendpos="topright")
}
\keyword{spatial}
\keyword{models}