StraussHard.Rd
\name{StraussHard}
\alias{StraussHard}
\title{The Strauss / Hard Core Point Process Model}
\description{
Creates an instance of the ``Strauss/ hard core'' point process model
which can then be fitted to point pattern data.
}
\usage{
StraussHard(r, hc=NA)
}
\arguments{
\item{r}{The interaction radius of the Strauss interaction}
\item{hc}{The hard core distance. Optional.}
}
\value{
An object of class \code{"interact"}
describing the interpoint interaction
structure of the ``Strauss/hard core''
process with Strauss interaction radius \eqn{r}
and hard core distance \code{hc}.
}
\details{
A Strauss/hard core process with interaction radius \eqn{r},
hard core distance \eqn{h < r}, and
parameters \eqn{\beta}{beta} and \eqn{\gamma}{gamma},
is a pairwise interaction point process
in which
\itemize{
\item distinct points are not allowed to come closer
than a distance \eqn{h} apart
\item each pair of points closer than \eqn{r} units apart
contributes a factor \eqn{\gamma}{gamma} to the probability density.
}
This is a hybrid of the Strauss process and the hard core process.
The probability density is zero if any pair of points
is closer than \eqn{h} units apart, and otherwise equals
\deqn{
f(x_1,\ldots,x_n) =
\alpha \beta^{n(x)} \gamma^{s(x)}
}{
f(x_1,\ldots,x_n) =
alpha . beta^n(x) gamma^s(x)
}
where \eqn{x_1,\ldots,x_n}{x[1],\ldots,x[n]} represent the
points of the pattern, \eqn{n(x)} is the number of points in the
pattern, \eqn{s(x)} is the number of distinct unordered pairs of
points that are closer than \eqn{r} units apart,
and \eqn{\alpha}{alpha} is the normalising constant.
The interaction parameter \eqn{\gamma}{gamma} may take any
positive value (unlike the case for the Strauss process).
If \eqn{\gamma < 1}{gamma < 1},
the model describes an ``ordered'' or ``inhibitive'' pattern.
If \eqn{\gamma > 1}{gamma > 1},
the model is ``ordered'' or ``inhibitive'' up to the distance
\eqn{h}, but has an ``attraction'' between points lying at
distances in the range between \eqn{h} and \eqn{r}.
If \eqn{\gamma = 1}{gamma = 1}, the process reduces to a classical
hard core process with hard core distance \eqn{h}.
If \eqn{\gamma = 0}{gamma = 0}, the process reduces to a classical
hard core process with hard core distance \eqn{r}.
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 Strauss/hard core process
pairwise interaction is
yielded by the function \code{StraussHard()}. See the examples below.
The canonical parameter \eqn{\log(\gamma)}{log(gamma)}
is estimated by \code{\link{ppm}()}, not fixed in
\code{StraussHard()}.
If the hard core distance argument \code{hc} is missing or \code{NA},
it will be estimated from the data when \code{\link{ppm}} is called.
The estimated value of \code{hc} 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{pairwise.family}},
\code{\link{ppm.object}}
}
\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.
Ripley, B.D. (1981)
\emph{Spatial statistics}.
John Wiley and Sons.
Strauss, D.J. (1975)
A model for clustering.
\emph{Biometrika} \bold{62}, 467--475.
}
\examples{
StraussHard(r=1,hc=0.02)
# prints a sensible description of itself
data(cells)
\dontrun{
ppm(cells, ~1, StraussHard(r=0.1, hc=0.05))
# fit the stationary Strauss/hard core process to `cells'
}
ppm(cells, ~ polynom(x,y,3), StraussHard(r=0.1, hc=0.05))
# fit a nonstationary Strauss/hard core process
# with log-cubic polynomial trend
}
\author{\adrian
and \rolf
}
\keyword{spatial}
\keyword{models}