Geyer.Rd
\name{Geyer}
\alias{Geyer}
\title{Geyer's Saturation Point Process Model}
\description{
Creates an instance of Geyer's saturation point process model
which can then be fitted to point pattern data.
}
\usage{
Geyer(r,sat)
}
\arguments{
\item{r}{Interaction radius. A positive real number.}
\item{sat}{Saturation threshold. A non-negative real number.}
}
\value{
An object of class \code{"interact"}
describing the interpoint interaction
structure of Geyer's saturation point process
with interaction radius \eqn{r} and saturation threshold \code{sat}.
}
\details{
Geyer (1999) introduced the \dQuote{saturation process},
a modification of the Strauss process (see \code{\link{Strauss}})
in which the total contribution
to the potential from each point (from its pairwise interaction with all
other points) is trimmed to a maximum value \eqn{s}.
The interaction structure of this
model is implemented in the function \code{\link{Geyer}()}.
The saturation point process with interaction radius \eqn{r},
saturation threshold \eqn{s}, and
parameters \eqn{\beta}{beta} and \eqn{\gamma}{gamma},
is the point process
in which each point
\eqn{x_i}{x[i]} in the pattern \eqn{X}
contributes a factor
\deqn{
\beta \gamma^{\min(s, t(x_i, X))}
}{
beta gamma^min(s, t(x[i],X))
}
to the probability density of the point pattern,
where \eqn{t(x_i, X)}{t(x[i],X)} denotes the
number of \sQuote{close neighbours} of \eqn{x_i}{x[i]} in the pattern
\eqn{X}. A close neighbour of \eqn{x_i}{x[i]} is a point
\eqn{x_j}{x[j]} with \eqn{j \neq i}{j != i}
such that the distance between
\eqn{x_i}{x[i]} and \eqn{x_j}{x[j]} is less than or equal to \eqn{r}.
If the saturation threshold \eqn{s} is set to infinity,
this model reduces to the Strauss process (see \code{\link{Strauss}})
with interaction parameter \eqn{\gamma^2}{gamma^2}.
If \eqn{s = 0}, the model reduces to the Poisson point process.
If \eqn{s} is a finite positive number, then the interaction parameter
\eqn{\gamma}{gamma} may take any positive value (unlike the case
of the Strauss process), with
values \eqn{\gamma < 1}{gamma < 1}
describing an \sQuote{ordered} or \sQuote{inhibitive} pattern,
and
values \eqn{\gamma > 1}{gamma > 1}
describing a \sQuote{clustered} or \sQuote{attractive} pattern.
The nonstationary saturation process is similar except that
the value \eqn{\beta}{beta}
is replaced by a function \eqn{\beta(x_i)}{beta(x[i])}
of location.
The function \code{\link{ppm}()}, which fits point process models to
point pattern data, requires an argument
of class \code{"interact"} describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the saturation process interaction is
yielded by \code{Geyer(r, sat)} where the
arguments \code{r} and \code{sat} specify
the Strauss interaction radius \eqn{r} and the saturation threshold
\eqn{s}, respectively. See the examples below.
Note the only arguments are the interaction radius \code{r}
and the saturation threshold \code{sat}.
When \code{r} and \code{sat} are fixed,
the model becomes an exponential family.
The canonical parameters \eqn{\log(\beta)}{log(beta)}
and \eqn{\log(\gamma)}{log(gamma)}
are estimated by \code{\link{ppm}()}, not fixed in
\code{Geyer()}.
}
\section{Zero saturation}{
The value \code{sat=0} is permitted by \code{Geyer},
but this is not very useful.
For technical reasons, when \code{\link{ppm}} fits a
Geyer model with \code{sat=0}, the default behaviour is to return
an \dQuote{invalid} fitted model in which the estimate of
\eqn{\gamma}{gamma} is \code{NA}. In order to get a Poisson
process model returned when \code{sat=0},
you would need to set \code{emend=TRUE} in
the call to \code{\link{ppm}}.
}
\seealso{
\code{\link{ppm}},
\code{\link{pairwise.family}},
\code{\link{ppm.object}},
\code{\link{Strauss}},
\code{\link{SatPiece}}
}
\references{
Geyer, C.J. (1999)
Likelihood Inference for Spatial Point Processes.
Chapter 3 in
O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. Van Lieshout (eds)
\emph{Stochastic Geometry: Likelihood and Computation},
Chapman and Hall / CRC,
Monographs on Statistics and Applied Probability, number 80.
Pages 79--140.
}
\examples{
ppm(cells, ~1, Geyer(r=0.07, sat=2))
# fit the stationary saturation process to `cells'
}
\author{\adrian
and \rolf
}
\keyword{spatial}
\keyword{models}