https://github.com/cran/spatstat
Tip revision: 32c7daeb36b6e48fd0356bdcec9580ae124fee5e authored by Adrian Baddeley on 29 December 2015, 22:08:27 UTC
version 1.44-1
version 1.44-1
Tip revision: 32c7dae
Penttinen.Rd
\name{Penttinen}
\alias{Penttinen}
\title{Penttinen Interaction}
\description{
Creates an instance of the Penttinen pairwise interaction
point process model, which can then be fitted to point pattern data.
}
\usage{
Penttinen(r)
}
\arguments{
\item{r}{circle radius}
}
\value{
An object of class \code{"interact"}
describing the interpoint interaction
structure of a point process.
}
\details{
Penttinen (1984, Example 2.1, page 18), citing Cormack (1979),
described the pairwise interaction point process with interaction factor
\deqn{
h(d) = e^{\theta A(d)} = \gamma^{A(d)}
}{
h(d) = exp(theta * A(d)) = gamma^(A(d))
}
between each pair of points separated by a distance $d$.
Here \eqn{A(d)} is the area of intersection between two discs
of radius \eqn{r} separated by a distance \eqn{d}, normalised so that
\eqn{A(0) = 1}.
The scale of interaction is controlled by the disc radius \eqn{r}:
two points interact if they are closer than \eqn{2 r}{2 * r} apart.
The strength of interaction is controlled by the
canonical parameter \eqn{\theta}{theta}, which
must be less than or equal to zero, or equivalently by the
parameter \eqn{\gamma = e^\theta}{gamma = exp(theta)},
which must lie between 0 and 1.
The potential is inhibitory, i.e.\ this model is only appropriate for
regular point patterns.
For \eqn{\gamma=0}{gamma=0} the model is
a hard core process with hard core diameter \eqn{2 r}{2 * r}.
For \eqn{\gamma=1}{gamma=1} the model is a Poisson process.
The irregular parameter
\eqn{r} must be given in the call to
\code{Penttinen}, while the
regular parameter \eqn{\theta}{theta} will be estimated.
This model can be considered as a pairwise approximation
to the area-interaction model \code{\link{AreaInter}}.
}
\seealso{
\code{\link{ppm}},
\code{\link{ppm.object}},
\code{\link{Pairwise}},
\code{\link{AreaInter}}.
}
\examples{
fit <- ppm(cells ~ 1, Penttinen(0.07))
fit
reach(fit) # interaction range is circle DIAMETER
}
\references{
Cormack, R.M. (1979)
Spatial aspects of competition between individuals.
Pages 151--212 in \emph{Spatial and Temporal Analysis in Ecology},
eds. R.M. Cormack and J.K. Ord, International Co-operative
Publishing House, Fairland, MD, USA.
Penttinen, A. (1984)
\emph{Modelling Interaction in Spatial Point Patterns:
Parameter Estimation by the Maximum Likelihood Method.}
\ifelse{latex}{\out{Jyv\"askyl\"a}}{Jyvaskyla}
Studies in Computer Science, Economics and Statistics \bold{7},
University of \ifelse{latex}{\out{Jyv\"askyl\"a}}{Jyvaskyla}, Finland.
}
\author{
Adrian Baddeley \email{Adrian.Baddeley@curtin.edu.au},
Rolf Turner \email{r.turner@auckland.ac.nz}
and Ege Rubak \email{rubak@math.aau.dk}
}
\keyword{spatial}
\keyword{models}