rMatClust.Rd
\name{rMatClust}
\alias{rMatClust}
\title{Simulate Matern Cluster Process}
\description{
Generate a random point pattern, a simulated realisation of the
Mat\'ern Cluster Process.
}
\usage{
rMatClust(kappa, r, mu, win = owin(c(0,1),c(0,1)))
}
\arguments{
\item{kappa}{
Intensity of the Poisson process of cluster centres.
A single positive number, a function, or a pixel image.
}
\item{r}{
Radius parameter of the clusters.
}
\item{mu}{
Mean number of points per cluster (a single positive number)
or reference intensity for the cluster points (a function or
a pixel image).
}
\item{win}{
Window in which to simulate the pattern.
An object of class \code{"owin"}
or something acceptable to \code{\link{as.owin}}.
}
}
\value{
The simulated point pattern (an object of class \code{"ppp"}).
Additionally, some intermediate results of the simulation are
returned as attributes of this point pattern.
See \code{\link{rNeymanScott}}.
}
\details{
This algorithm generates a realisation of Mat\'ern's cluster process
inside the window \code{win}. The process is constructed by first
generating a Poisson point process of ``parent'' points
with intensity \code{kappa}. Then each parent point is
replaced by a random cluster of points, the number of points in each
cluster being random with a Poisson (\code{mu}) distribution,
and the points being placed independently and uniformly inside
a disc of radius \code{r} centred on the parent point.
In this implementation, parent points are not restricted to lie in the
window; the parent process is effectively the uniform
Poisson process on the infinite plane.
This classical model can be fitted to data by the method of minimum contrast,
using \code{\link{matclust.estK}} or \code{\link{kppm}}.
The algorithm can also generate spatially inhomogeneous versions of
the Mat\'ern cluster process:
\itemize{
\item The parent points can be spatially inhomogeneous.
If the argument \code{kappa} is a \code{function(x,y)}
or a pixel image (object of class \code{"im"}), then it is taken
as specifying the intensity function of an inhomogeneous Poisson
process that generates the parent points.
\item The offspring points can be inhomogeneous. If the
argument \code{mu} is a \code{function(x,y)}
or a pixel image (object of class \code{"im"}), then it is
interpreted as the reference density for offspring points,
in the sense of Waagepetersen (2006).
For a given parent point, the offspring constitute a Poisson process
with intensity function equal to the \emph{average} value of
\code{mu} inside the disc of radius \code{r} centred on the parent
point, and zero intensity outside this disc.
}
When the parents are homogeneous (\code{kappa} is a single number)
and the offspring are inhomogeneous (\code{mu} is a
function or pixel image), the model can be fitted to data
using \code{\link{kppm}}, or using \code{\link{matclust.estK}}
applied to the inhomogeneous \eqn{K} function.
}
\seealso{
\code{\link{rpoispp}},
\code{\link{rThomas}},
\code{\link{rGaussPoisson}},
\code{\link{rNeymanScott}},
\code{\link{matclust.estK}},
\code{\link{kppm}}.
}
\examples{
# homogeneous
X <- rMatClust(10, 0.05, 4)
# inhomogeneous
Z <- as.im(function(x,y){ 4 * exp(2 * x - 1) }, owin())
Y <- rMatClust(10, 0.05, Z)
}
\references{
Mat\'ern, B. (1960)
\emph{Spatial Variation}.
Meddelanden fraan Statens Skogsforskningsinstitut,
volume 59, number 5.
Statens Skogsforskningsinstitut, Sweden.
Mat\'ern, B. (1986)
\emph{Spatial Variation}.
Lecture Notes in Statistics 36, Springer-Verlag, New York.
Waagepetersen, R. (2006)
An estimating function approach to inference for inhomogeneous
Neyman-Scott processes.
Submitted for publication.
}
\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{datagen}