Softcore.Rd
\name{Softcore}
\alias{Softcore}
\title{The Soft Core Point Process Model}
\description{
Creates an instance of the Soft Core point process model
which can then be fitted to point pattern data.
}
\usage{
Softcore(kappa)
}
\arguments{
\item{kappa}{The exponent \eqn{\kappa}{kappa} of the Soft Core interaction}
}
\value{
An object of class \code{"interact"}
describing the interpoint interaction
structure of the Soft Core process with exponent \eqn{\kappa}{kappa}.
}
\details{
The (stationary)
Soft Core point process with parameters \eqn{\beta}{beta} and
\eqn{\sigma}{sigma} and exponent \eqn{\kappa}{kappa}
is the pairwise interaction point process in which
each point contributes a factor \eqn{\beta}{beta} to the
probability density of the point pattern, and each pair of points
contributes a factor
\deqn{
\exp \left\{ - \left( \frac{\sigma}{d} \right)^{2/\kappa} \right\}
}{
exp( - (sigma/d)^(2/kappa) )
}
to the density, where \eqn{d} is the distance between the two points.
Thus the process has probability density
\deqn{
f(x_1,\ldots,x_n) =
\alpha \beta^{n(x)}
\exp \left\{ - \sum_{i < j} \left(
\frac{\sigma}{||x_i-x_j||}
\right)^{2/\kappa} \right\}
}{
f(x_1,\ldots,x_n) =
alpha . beta^n(x) exp( - sum (sigma/||x[i]-x[j]||)^(2/kappa))
}
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{\alpha}{alpha} is the normalising constant,
and the sum on the right hand side is
over all unordered pairs of points of the pattern.
This model describes an ``ordered'' or ``inhibitive'' process,
with the interpoint interaction decreasing smoothly with distance.
The strength of interaction is controlled by the
parameter \eqn{\sigma}{sigma}, a positive real number,
with larger values corresponding
to stronger interaction; and by the exponent \eqn{\kappa}{kappa}
in the range \eqn{(0,1)}, with larger values corresponding to
weaker interaction.
If \eqn{\sigma = 0}{sigma = 0}
the model reduces to the Poisson point process.
If \eqn{\sigma > 0}{sigma > 0},
the process is well-defined only for \eqn{\kappa}{kappa} in \eqn{(0,1)}.
The limit of the model as \eqn{\kappa \to 0}{kappa -> 0} is the
hard core process with hard core distance \eqn{h=\sigma}{h=sigma}.
The nonstationary Soft Core process is similar except that
the contribution of each individual point \eqn{x_i}{x[i]}
is a function \eqn{\beta(x_i)}{beta(x[i])}
of location, rather than a constant beta.
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 Soft Core process pairwise interaction is
yielded by the function \code{Softcore()}. See the examples below.
Note the only argument is the exponent \code{kappa}.
When \code{kappa} is fixed, the model becomes an exponential family
with canonical parameters \eqn{\log \beta}{log(beta)}
and \deqn{
\log \gamma = \frac{2}{\kappa} \log\sigma
}{
log(gamma) = (2/kappa) log(sigma)
}
The canonical parameters are estimated by \code{\link{ppm}()}, not fixed in
\code{Softcore()}.
}
\seealso{
\code{\link{ppm}},
\code{\link{pairwise.family}},
\code{\link{ppm.object}}
}
\references{
Ogata, Y, and Tanemura, M. (1981).
Estimation of interaction potentials of spatial point patterns
through the maximum likelihood procedure.
\emph{Annals of the Institute of Statistical Mathematics}, B
\bold{33}, 315--338.
Ogata, Y, and Tanemura, M. (1984).
Likelihood analysis of spatial point patterns.
\emph{Journal of the Royal Statistical Society, series B}
\bold{46}, 496--518.
}
\examples{
data(cells)
ppm(cells, ~1, Softcore(kappa=0.5), correction="isotropic")
# fit the stationary Soft Core process to `cells'
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{models}