Triplets.Rd
\name{Triplets}
\alias{Triplets}
\title{The Triplet Point Process Model}
\description{
Creates an instance of Geyer's triplet interaction point process model
which can then be fitted to point pattern data.
}
\usage{
Triplets(r)
}
\arguments{
\item{r}{The interaction radius of the Triplets process}
}
\value{
An object of class \code{"interact"}
describing the interpoint interaction
structure of the Triplets process with interaction radius \eqn{r}.
}
\details{
The (stationary) Geyer triplet process (Geyer, 1999)
with interaction radius \eqn{r} and
parameters \eqn{\beta}{beta} and \eqn{\gamma}{gamma}
is the point process
in which each point contributes a factor \eqn{\beta}{beta} to the
probability density of the point pattern, and each triplet of close points
contributes a factor \eqn{\gamma}{gamma} to the density.
A triplet of close points is a group of 3 points,
each pair of which is closer than \eqn{r} units
apart.
Thus the probability density is
\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 unordered triples of
points that are closer than \eqn{r} units apart,
and \eqn{\alpha}{alpha} is the normalising constant.
The interaction parameter \eqn{\gamma}{gamma} must be less than
or equal to \eqn{1}
so that this model describes an ``ordered'' or ``inhibitive'' pattern.
The nonstationary Triplets 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 Triplets process pairwise interaction is
yielded by the function \code{Triplets()}. See the examples below.
Note the only argument is the interaction radius \code{r}.
When \code{r} is fixed, the model becomes an exponential family.
The canonical parameters \eqn{\log(\beta)}{log(beta)}
and \eqn{\log(\gamma)}{log(gamma)}
are estimated by \code{\link{ppm}()}, not fixed in
\code{Triplets()}.
}
\seealso{
\code{\link{ppm}},
\code{\link{triplet.family}},
\code{\link{ppm.object}}
}
\references{
Geyer, C.J. (1999)
Likelihood Inference for Spatial Point Processes.
Chapter 3 in
O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. Van Lieshout (eds)
\emph{Stochastic Geometry: Likelihood and Computation},
Chapman and Hall / CRC,
Monographs on Statistics and Applied Probability, number 80.
Pages 79--140.
}
\examples{
Triplets(r=0.1)
# prints a sensible description of itself
\dontrun{
ppm(cells, ~1, Triplets(r=0.1))
# fit the stationary Triplets process to `cells'
}
ppm(cells, ~polynom(x,y,3), Triplets(r=0.1))
# fit a nonstationary Triplets process with log-cubic polynomial trend
}
\author{Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{models}