Fiksel.Rd
\name{Fiksel}
\alias{Fiksel}
\title{The Fiksel Interaction}
\description{
Creates an instance of Fiksel's double exponential
pairwise interaction point process model,
which can then be fitted to point pattern data.
}
\usage{
Fiksel(r, hc, kappa)
}
\arguments{
\item{r}{The interaction radius of the Fiksel model}
\item{hc}{The hard core distance}
\item{kappa}{The rate parameter}
}
\value{
An object of class \code{"interact"}
describing the interpoint interaction
structure of the Fiksel
process with interaction radius \eqn{r},
hard core distance \code{hc} and
rate parameter \code{kappa}.
}
\details{
Fiksel (1984) introduced a pairwise interaction point process
with the following interaction function \eqn{c}.
For two points \eqn{u} and \eqn{v} separated by a distance
\eqn{d=||u-v||}, the interaction
\eqn{c(u,v)} is equal to \eqn{0} if \eqn{d < h},
equal to \eqn{1} if \eqn{d > r}, and
equal to
\deqn{ \exp(a \exp(-\kappa d))}{exp(a * exp(-kappa * d))}
if \eqn{h \le d \le r}{h <= d <= r}, where
\eqn{h,r,\kappa,a}{h,r,kappa,a} are parameters.
A graph of this interaction function is shown in the Examples.
The interpretation of the parameters is as follows.
\itemize{
\item \eqn{h} is the hard core distance: distinct points are
not permitted to come closer than a distance \eqn{h} apart.
\item \eqn{r} is the interaction range: points further than
this distance do not interact.
\item \eqn{\kappa}{kappa} is the rate or slope parameter,
controlling the decay of the interaction as distance increases.
\item \eqn{a} is the interaction strength parameter,
controlling the strength and type of interaction.
If \eqn{a} is zero, the process is Poisson. If \code{a} is positive,
the process is clustered. If \code{a} is negative, the process is
inhibited (regular).
}
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 Fiksel
pairwise interaction is
yielded by the function \code{Fiksel()}. See the examples below.
The parameters \eqn{h}, \eqn{r} and \eqn{\kappa}{kappa} must be
fixed and given in the call to \code{Fiksel}, while the canonical
parameter \eqn{a} is estimated by \code{\link{ppm}()}.
To estimate \eqn{h}, \eqn{r} and\eqn{\kappa}{kappa}
it is possible to use \code{\link{profilepl}}. The maximum likelihood
estimator of\eqn{h} is the minimum interpoint distance.
See also Stoyan, Kendall and Mecke (1987) page 161.
}
\seealso{
\code{\link{ppm}},
\code{\link{pairwise.family}},
\code{\link{ppm.object}},
\code{\link{StraussHard}}
}
\references{
Baddeley, A. and Turner, R. (2000)
Practical maximum pseudolikelihood for spatial point patterns.
\emph{Australian and New Zealand Journal of Statistics}
\bold{42}, 283--322.
Fiksel, T. (1984)
Estimation of parameterized pair potentials
of marked and non-marked Gibbsian point processes.
\emph{Electronische Informationsverabeitung und Kybernetika}
\bold{20}, 270--278.
Stoyan, D, Kendall, W.S. and Mecke, J. (1987)
\emph{Stochastic geometry and its applications}. Wiley.
}
\examples{
Fiksel(r=1,hc=0.02, kappa=2)
# prints a sensible description of itself
data(spruces)
X <- unmark(spruces)
fit <- ppm(X, ~1, Fiksel(r=3.5, hc=1, kappa=1))
plot(fitin(fit))
}
\author{Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{models}