https://github.com/cran/spatstat
Tip revision: d606122dc24b56ecf537d55eda38f4bf5ac4de1f authored by Adrian Baddeley on 25 October 2010, 10:40:51 UTC
version 1.20-5
version 1.20-5
Tip revision: d606122
rStrauss.Rd
\name{rStrauss}
\alias{rStrauss}
\title{Perfect Simulation of the Strauss Process}
\description{
Generate a random pattern of points, a simulated realisation
of the Strauss process, using a perfect simulation algorithm.
}
\usage{
rStrauss(beta, gamma = 1, R = 0, W = owin())
}
\arguments{
\item{beta}{
intensity parameter (a positive number).
}
\item{gamma}{
interaction parameter (a number between 0 and 1, inclusive).
}
\item{R}{
interaction radius (a non-negative number).
}
\item{W}{
window (object of class \code{"owin"}) in which to
generate the random pattern. Currently this must be a rectangular
window.
}
}
\details{
This function generates a realisation of the
Strauss point process in the window \code{W}
using a \sQuote{perfect simulation} algorithm.
The Strauss process (Strauss, 1975; Kelly and Ripley, 1976)
is a model for spatial inhibition, ranging from
a strong `hard core' inhibition to a completely random pattern
according to the value of \code{gamma}.
The Strauss process with interaction radius \eqn{R} and
parameters \eqn{\beta}{beta} and \eqn{\gamma}{gamma}
is the pairwise interaction point process
with probability density
\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.
Intuitively, each point of the pattern
contributes a factor \eqn{\beta}{beta} to the
probability density, and each pair of points
closer than \eqn{r} units apart contributes a factor
\eqn{\gamma}{gamma} to the density.
The interaction parameter \eqn{\gamma}{gamma} must be less than
or equal to \eqn{1} in order that the process be well-defined
(Kelly and Ripley, 1976).
This model describes an ``ordered'' or ``inhibitive'' pattern.
If \eqn{\gamma=1}{gamma=1} it reduces to a Poisson process
(complete spatial randomness) with intensity \eqn{\beta}{beta}.
If \eqn{\gamma=0}{gamma=0} it is called a ``hard core process''
with hard core radius \eqn{R/2}, since no pair of points is permitted
to lie closer than \eqn{R} units apart.
The simulation algorithm used to generate the point pattern
is \sQuote{dominated coupling from the past}
as implemented by Berthelsen and Moller (2002, 2003).
This is a \sQuote{perfect simulation} or \sQuote{exact simulation}
algorithm, so called because the output of the algorithm is guaranteed
to have the correct probability distribution exactly (unlike the
Metropolis-Hastings algorithm used in \code{\link{rmh}}, whose output
is only approximately correct).
The implementation is currently \bold{experimental}.
There is a tiny chance that the algorithm will
run out of space before it has terminated. If this occurs, an error
message will be generated.
}
\value{
A point pattern (object of class \code{"ppp"}).
}
\references{
Berthelsen, K.K. and Moller, J. (2002)
A primer on perfect simulation for spatial point processes.
\emph{Bulletin of the Brazilian Mathematical Society} 33, 351-367.
Berthelsen, K.K. and Moller, J. (2003)
Likelihood and non-parametric Bayesian MCMC inference
for spatial point processes based on perfect simulation and
path sampling.
\emph{Scandinavian Journal of Statistics} 30, 549-564.
Kelly, F.P. and Ripley, B.D. (1976)
On Strauss's model for clustering.
\emph{Biometrika} \bold{63}, 357--360.
Moller, J. and Waagepetersen, R. (2003).
\emph{Statistical Inference and Simulation for Spatial Point Processes.}
Chapman and Hall/CRC.
Strauss, D.J. (1975)
A model for clustering.
\emph{Biometrika} \bold{63}, 467--475.
}
\author{
Kasper Klitgaard Berthelsen
\email{k.berthelsen@lancaster.ac.uk},
adapted for \pkg{spatstat} by Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
}
\examples{
X <- rStrauss(0.05,0.2,1.5,square(141.4))
Z <- rStrauss(100,0.7,0.05)
}
\seealso{
\code{\link{rmh}}
}
\keyword{spatial}
\keyword{datagen}