ippm.Rd
\name{ippm}
\alias{ippm}
\title{
Fit Point Process Model Involving Irregular Trend Parameters
}
\description{
Experimental extension to \code{ppm}
which finds optimal values of the irregular trend parameters in a
point process model.
}
\usage{
ippm(Q, \dots,
iScore=NULL,
start=list(),
covfunargs=start,
nlm.args=list(),
silent=FALSE,
warn.unused=TRUE)
}
\arguments{
\item{Q,\dots}{
Arguments passed to \code{\link{ppm}}
to fit the point process model.
}
\item{iScore}{
Optional. A named list of \R functions that compute the partial derivatives
of the logarithm of the trend, with respect to each irregular parameter.
See Details.
}
\item{start}{
Named list containing initial values of the
irregular parameters over which to optimise.
}
\item{covfunargs}{
Argument passed to \code{\link{ppm}}.
A named list containing values for \emph{all} irregular parameters
required by the covariates in the model.
Must include all the parameters named in \code{start}.
}
\item{nlm.args}{
Optional list of arguments passed to \code{\link[stats]{nlm}}
to control the optimization algorithm.
}
\item{silent}{
Logical. Whether to print warnings if the optimization algorithm
fails to converge.
}
\item{warn.unused}{
Logical. Whether to print a warning if some of the parameters
in \code{start} are not used in the model.
}
}
\details{
This function is an experimental extension to the
point process model fitting command \code{\link{ppm}}.
The extension allows the trend of the model to include irregular parameters,
which will be maximised by a Newton-type iterative
method, using \code{\link[stats]{nlm}}.
For the sake of explanation,
consider a Poisson point process with intensity function
\eqn{\lambda(u)}{lambda(u)} at location \eqn{u}. Assume that
\deqn{
\lambda(u) = \exp(\alpha + \beta Z(u)) \, f(u, \gamma)
}{
lambda(u) = \exp(alpha + beta * Z(u)) * f(u, gamma)
}
where \eqn{\alpha,\beta,\gamma}{alpha,beta,gamma} are
parameters to be estimated, \eqn{Z(u)} is a spatial covariate
function, and \eqn{f} is some known function.
Then the parameters
\eqn{\alpha,\beta}{alpha,beta} are called \emph{regular} because they
appear in a loglinear form; the parameter
\eqn{\gamma}{gamma} is called \emph{irregular}.
To fit this model using \code{ippm}, we specify the
intensity using the \code{trend} formula
in the same way as usual for \code{\link{ppm}}.
The trend formula is a representation of the log intensity.
In the above example the log intensity is
\deqn{
\log\lambda(u) = \alpha + \beta Z(u) + \log f(u, \gamma)
}{
log(lambda(u)) = alpha + beta * Z(u) + log(f(u, gamma))
}
So the model above would be encoded with the trend formula
\code{~Z + offset(log(f))}. Note that the irregular part of the model
is an \emph{offset} term, which means that it is included in the log trend
as it is, without being multiplied by another regular parameter.
The optimisation runs faster if we specify the derivative
of \eqn{\log f(u,\gamma)}{log(f(u,gamma))} with
respect to \eqn{\gamma}{gamma}. We call this the
\emph{irregular score}. To specify this, the user must write an \R function
that computes the irregular score for any value of
\eqn{\gamma}{gamma} at any location \code{(x,y)}.
Thus, to code such a problem,
\enumerate{
\item The argument \code{trend} should define the
log intensity, with the irregular part as an offset;
\item The argument \code{start} should be a list
containing initial values of each of the irregular parameters;
\item The argument \code{iScore}, if provided,
must be a list (with one entry
for each entry of \code{start}) of functions
with arguments \code{x,y,\dots}, that evaluate the partial derivatives
of \eqn{\log f(u,\gamma)}{log(f(u,gamma))} with
respect to each irregular parameter.
}
The coded example below illustrates the model with two irregular
parameters \eqn{\gamma,\delta}{gamma,delta} and irregular term
\deqn{
f((x,y), (\gamma, \delta)) = 1 + \exp(\gamma - \delta x^3)
}{
f((x,y), (gamma, delta)) = 1 + \exp(gamma - delta * x^3)
}
Arguments \code{\dots} passed to \code{\link{ppm}} may
also include \code{interaction}. In this case the model is not
a Poisson point process but a more general Gibbs point process;
the trend formula \code{trend}
determines the first-order trend
of the model (the first order component of the conditional intensity),
not the intensity.
}
\value{
A fitted point process model (object of class \code{"ppm"}).
}
\author{\adrian
and \rolf
}
\seealso{
\code{\link{ppm}},
\code{\link{profilepl}}
}
\examples{
nd <- 32
\testonly{nd <- 10}
gamma0 <- 3
delta0 <- 5
POW <- 3
# Terms in intensity
Z <- function(x,y) { -2*y }
f <- function(x,y,gamma,delta) { 1 + exp(gamma - delta * x^POW) }
# True intensity
lamb <- function(x,y,gamma,delta) { 200 * exp(Z(x,y)) * f(x,y,gamma,delta) }
# Simulate realisation
lmax <- max(lamb(0,0,gamma0,delta0), lamb(1,1,gamma0,delta0))
set.seed(42)
X <- rpoispp(lamb, lmax=lmax, win=owin(), gamma=gamma0, delta=delta0)
# Partial derivatives of log f
DlogfDgamma <- function(x,y, gamma, delta) {
topbit <- exp(gamma - delta * x^POW)
topbit/(1 + topbit)
}
DlogfDdelta <- function(x,y, gamma, delta) {
topbit <- exp(gamma - delta * x^POW)
- (x^POW) * topbit/(1 + topbit)
}
# irregular score
Dlogf <- list(gamma=DlogfDgamma, delta=DlogfDdelta)
# fit model
ippm(X, ~Z + offset(log(f)),
covariates=list(Z=Z, f=f),
iScore=Dlogf,
start=list(gamma=1, delta=1),
nd=nd)
}
\keyword{spatial}
\keyword{models}