https://github.com/cran/spatstat
Tip revision: 95fd631dcecc6e90c663947d755f3243af3ee213 authored by Adrian Baddeley on 22 July 2008, 00:00:00 UTC
version 1.14-1
version 1.14-1
Tip revision: 95fd631
PairPiece.Rd
\name{PairPiece}
\alias{PairPiece}
\title{The Piecewise Constant Pairwise Interaction Point Process Model}
\description{
Creates an instance of a pairwise interaction point process model
with piecewise constant potential function. The model
can then be fitted to point pattern data.
}
\usage{
PairPiece(r)
}
\arguments{
\item{r}{vector of jump points for the potential function}
}
\value{
An object of class \code{"interact"}
describing the interpoint interaction
structure of a point process. The process is a pairwise interaction process,
whose interaction potential is piecewise constant, with jumps
at the distances given in the vector \eqn{r}.
}
\details{
A pairwise interaction point process in a bounded region
is a stochastic point process with probability density of the form
\deqn{
f(x_1,\ldots,x_n) =
\alpha \prod_i b(x_i) \prod_{i < j} h(x_i, x_j)
}{
f(x_1,\ldots,x_n) =
alpha . product { b(x[i]) } product { h(x_i, x_j) }
}
where \eqn{x_1,\ldots,x_n}{x[1],\ldots,x[n]} represent the
points of the pattern. The first product on the right hand side is
over all points of the pattern; the second product is over all
unordered pairs of points of the pattern.
Thus each point \eqn{x_i}{x[i]} of the pattern contributes a factor
\eqn{b(x_i)}{b(x[i])} to the probability density, and each pair of
points \eqn{x_i, x_j}{x[i], x[j]} contributes a factor
\eqn{h(x_i,x_j)}{h(x[i], x[j])} to the density.
The pairwise interaction term \eqn{h(u, v)} is called piecewise constant
if it depends only on the distance between \eqn{u} and \eqn{v},
say \eqn{h(u,v) = H(||u-v||)}, and \eqn{H} is a piecewise constant
function (a function which is constant except for jumps at a finite
number of places).
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 pairwise
interaction is yielded by the function \code{PairPiece()}.
See the examples below.
The entries of \code{r} must be strictly increasing, positive numbers.
They are interpreted as the points of discontinuity of \eqn{H}.
It is assumed that \eqn{H(s) =1} for all \eqn{s > r_{max}}{s > rmax}
where \eqn{r_{max}}{rmax} is the maximum value in \code{r}. Thus the
model has as many regular parameters (see \code{\link{ppm}})
as there are entries in \code{r}. The \eqn{i}-th regular parameter
\eqn{\theta_i}{theta[i]} is the logarithm of the value of the
interaction function \eqn{H} on the interval
\eqn{[r_{i-1},r_i)}{[r[i-1],r[i])}.
If \code{r} is a single number, this model is similar to the
Strauss process, see \code{\link{Strauss}}. The difference is that
in \code{PairPiece} the interaction function is continuous on the
right, while in \code{\link{Strauss}} it is continuous on the left.
The analogue of this model for multitype point processes
has not yet been implemented.
}
\seealso{
\code{\link{ppm}},
\code{\link{pairwise.family}},
\code{\link{ppm.object}},
\code{\link{Strauss}}
\code{\link{rmh.ppm}}
}
\examples{
PairPiece(c(0.1,0.2))
# prints a sensible description of itself
data(cells)
ppm(cells, ~1, PairPiece(r = c(0.05, 0.1, 0.2)))
# fit a stationary piecewise constant pairwise interaction process
ppm(cells, ~polynom(x,y,3), PairPiece(c(0.05, 0.1)))
# 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}
}
\keyword{spatial}
\keyword{models}