DiggleGratton.Rd
\name{DiggleGratton}
\alias{DiggleGratton}
\title{Diggle-Gratton model}
\description{
Creates an instance of the Diggle-Gratton pairwise interaction
point process model, which can then be fitted to point pattern data.
}
\usage{
DiggleGratton(delta=NA, rho)
}
\arguments{
\item{delta}{lower threshold \eqn{\delta}{\delta}}
\item{rho}{upper threshold \eqn{\rho}{\rho}}
}
\value{
An object of class \code{"interact"}
describing the interpoint interaction
structure of a point process.
}
\details{
Diggle and Gratton (1984, pages 208-210)
introduced the pairwise interaction point
process with pair potential \eqn{h(t)} of the form
\deqn{
h(t) = \left( \frac{t-\delta}{\rho-\delta} \right)^\kappa
\quad\quad \mbox{ if } \delta \le t \le \rho
}{
h(t) = ((t - \delta)/(\rho - \delta))^\kappa, { } \delta \le t \le \rho
}
with \eqn{h(t) = 0} for \eqn{t < \delta}{t < \delta}
and \eqn{h(t) = 1} for \eqn{t > \rho}{t > \rho}.
Here \eqn{\delta}{\delta}, \eqn{\rho}{\rho} and \eqn{\kappa}{\kappa}
are parameters.
Note that we use the symbol \eqn{\kappa}{\kappa}
where Diggle and Gratton (1984) and Diggle, Gates and Stibbard (1987)
use \eqn{\beta}{\beta}, since in \pkg{spatstat} we reserve the symbol
\eqn{\beta}{\beta} for an intensity parameter.
The parameters must all be nonnegative,
and must satisfy \eqn{\delta \le \rho}{\delta \le \rho}.
The potential is inhibitory, i.e.\ this model is only appropriate for
regular point patterns. The strength of inhibition increases with
\eqn{\kappa}{\kappa}. For \eqn{\kappa=0}{\kappa=0} the model is
a hard core process with hard core radius \eqn{\delta}{\delta}.
For \eqn{\kappa=\infty}{\kappa=Inf} the model is a hard core
process with hard core radius \eqn{\rho}{\rho}.
The irregular parameters
\eqn{\delta, \rho}{\delta, \rho} must be given in the call to
\code{DiggleGratton}, while the
regular parameter \eqn{\kappa}{\kappa} will be estimated.
If the lower threshold \code{delta} is missing or \code{NA},
it will be estimated from the data when \code{\link{ppm}} is called.
The estimated value of \code{delta} is the minimum nearest neighbour distance
multiplied by \eqn{n/(n+1)}, where \eqn{n} is the
number of data points.
}
\seealso{
\code{\link{ppm}},
\code{\link{ppm.object}},
\code{\link{Pairwise}}
}
\examples{
ppm(cells ~1, DiggleGratton(0.05, 0.1))
}
\references{
Diggle, P.J., Gates, D.J. and Stibbard, A. (1987)
A nonparametric estimator for pairwise-interaction point processes.
\emph{Biometrika} \bold{74}, 763 -- 770.
Diggle, P.J. and Gratton, R.J. (1984)
Monte Carlo methods of inference for implicit statistical models.
\emph{Journal of the Royal Statistical Society, series B}
\bold{46}, 193 -- 212.
}
\author{
\spatstatAuthors
}
\keyword{spatial}
\keyword{models}