\name{rThomas}
\alias{rThomas}
\title{Simulate Thomas Process}
\description{
Generate a random point pattern, a realisation of the
Thomas cluster process.
}
\usage{
rThomas(kappa, sigma, mu, win = owin(c(0,1),c(0,1)))
}
\arguments{
\item{kappa}{
Intensity of the Poisson process of cluster centres.
A single positive number.
}
\item{sigma}{
Standard deviation of displacement of a point from its cluster centre.
}
\item{mu}{
Expected number of points per cluster.
}
\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 the
Thomas process, a special case of the Neyman-Scott process.
The algorithm
generates a uniform 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
per cluster being Poisson (\code{mu}) distributed, and their
positions being isotropic Gaussian displacements from the
cluster parent location.
This classical model can be fitted to data by the method of minimum contrast,
using \code{\link{thomas.estK}}.
The algorithm can also generate spatially inhomogeneous versions of
the Thomas 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 \code{mu(x,y) * f(x,y)}
where \code{f} is the Gaussian density centred at the parent point.
}
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{thomas.estK}} applied to the inhomogeneous
\eqn{K} function.
}
\seealso{
\code{\link{rpoispp}},
\code{\link{rMatClust}},
\code{\link{rGaussPoisson}},
\code{\link{rNeymanScott}},
\code{\link{thomas.estK}}
}
\references{
Waagepetersen, R. (2006)
An estimating function approach to inference for inhomogeneous
Neyman-Scott processes.
Submitted for publication.
}
\examples{
#homogeneous
X <- rThomas(10, 0.2, 5)
#inhomogeneous
Z <- as.im(function(x,y){ 5 * exp(2 * x - 1) }, owin())
Y <- rThomas(10, 0.2, Z)
}
\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}