SatPiece.Rd
\name{SatPiece}
\alias{SatPiece}
\title{Piecewise Constant Saturated Pairwise Interaction Point Process Model}
\description{
Creates an instance of a saturated pairwise interaction point process model
with piecewise constant potential function. The model
can then be fitted to point pattern data.
}
\usage{
SatPiece(r, sat)
}
\arguments{
\item{r}{vector of jump points for the potential function}
\item{sat}{
vector of saturation values,
or a single saturation value
}
}
\value{
An object of class \code{"interact"}
describing the interpoint interaction
structure of a point process.
}
\details{
This is a generalisation of the Geyer saturation point process model,
described in \code{\link{Geyer}}, to the case of multiple interaction
distances. It can also be described as the saturated analogue of a
pairwise interaction process with piecewise-constant pair potential,
described in \code{\link{PairPiece}}.
The saturated point process with interaction radii
\eqn{r_1,\ldots,r_k}{r[1], ..., r[k]},
saturation thresholds \eqn{s_1,\ldots,s_k}{s[1],...,s[k]},
intensity parameter \eqn{\beta}{beta} and
interaction parameters
\eqn{\gamma_1,\ldots,gamma_k}{gamma[1], ..., gamma[k]},
is the point process
in which each point
\eqn{x_i}{x[i]} in the pattern \eqn{X}
contributes a factor
\deqn{
\beta \gamma_1^{v_1(x_i, X)} \ldots gamma_k^{v_k(x_i,X)}
}{
beta gamma[1]^v(1, x_i, X) ... gamma[k]^v(k, x_i, X)
}
to the probability density of the point pattern,
where
\deqn{
v_j(x_i, X) = \min( s_j, t_j(x_i,X) )
}{
v(j, x_i, X) = min(s[j], t(j, x_i, X))
}
where \eqn{t_j(x_i, X)}{t(j,x[i],X)} denotes the
number of points in the pattern \eqn{X} which lie
at a distance between \eqn{r_{j-1}}{r[j-1]} and \eqn{r_j}{r[j]}
from the point \eqn{x_i}{x[i]}. We take \eqn{r_0 = 0}{r[0] = 0}
so that \eqn{t_1(x_i,X)}{t(1, x[i], X)} is the number of points of
\eqn{X} that lie within a distance \eqn{r_1}{r[1]} of the point
\eqn{x_i}{x[i]}.
\code{SatPiece} is used to fit this model to data.
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 piecewise constant Saturated pairwise
interaction is yielded by the function \code{SatPiece()}.
See the examples below.
Simulation of this point process model is not yet implemented.
This model is not locally stable (the conditional intensity is
unbounded).
The argument \code{r} specifies the vector of interaction distances.
The entries of \code{r} must be strictly increasing, positive numbers.
The argument \code{sat} specifies the vector of saturation parameters.
It should be a vector of the same length as \code{r}, and its entries
should be nonnegative numbers. Thus \code{sat[1]} corresponds to the
distance range from \code{0} to \code{r[1]}, and \code{sat[2]} to the
distance range from \code{r[1]} to \code{r[2]}, etc.
Alternatively \code{sat} may be a single number, and this saturation
value will be applied to every distance range.
Infinite values of the
saturation parameters are also permitted; in this case
\eqn{v_j(x_i,X) = t_j(x_i,X)}{v(j, x_i, X) = t(j, x_i, X)}
and there is effectively no `saturation' for the distance range in
question. If all the saturation parameters are set to \code{Inf} then
the model is effectively a pairwise interaction process, equivalent to
\code{\link{PairPiece}} (however the interaction parameters
\eqn{\gamma}{gamma} obtained from \code{\link{SatPiece}} are the
square roots of the parameters \eqn{\gamma}{gamma}
obtained from \code{\link{PairPiece}}).
If \code{r} is a single number, this model is virtually equivalent to the
Geyer process, see \code{\link{Geyer}}.
}
\seealso{
\code{\link{ppm}},
\code{\link{pairsat.family}},
\code{\link{Geyer}},
\code{\link{PairPiece}},
\code{\link{BadGey}}.
}
\examples{
SatPiece(c(0.1,0.2), c(1,1))
# prints a sensible description of itself
SatPiece(c(0.1,0.2), 1)
data(cells)
ppm(cells, ~1, SatPiece(c(0.07, 0.1, 0.13), 2))
# fit a stationary piecewise constant Saturated pairwise interaction process
\dontrun{
ppm(cells, ~polynom(x,y,3), SatPiece(c(0.07, 0.1, 0.13), 2))
# nonstationary process with log-cubic polynomial trend
}
}
\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}
in collaboration with Hao Wang and Jeff Picka
}
\keyword{spatial}
\keyword{models}