https://github.com/cran/spatstat
Tip revision: 330e89946e924decc594678849a81f2a76e93730 authored by Adrian Baddeley on 05 February 2012, 07:53:32 UTC
version 1.25-3
version 1.25-3
Tip revision: 330e899
psst.Rd
\name{psst}
\alias{psst}
\title{
Pseudoscore Diagnostic For Fitted Model against General Alternative
}
\description{
Given a point process model fitted to a point pattern dataset,
and any choice of functional summary statistic,
this function computes the pseudoscore test statistic
of goodness-of-fit for the model.
}
\usage{
psst(object, fun, r = NULL, breaks = NULL, ...,
trend = ~1, interaction = Poisson(), rbord = reach(interaction),
truecoef=NULL, hi.res=NULL, funargs = list(correction="best"))
}
\arguments{
\item{object}{
Object to be analysed.
Either a fitted point process model (object of class \code{"ppm"})
or a point pattern (object of class \code{"ppp"})
or quadrature scheme (object of class \code{"quad"}).
}
\item{fun}{
Summary function to be applied to each point pattern.
}
\item{r}{
Optional.
Vector of values of the argument \eqn{r} at which the
function \eqn{S(r)} should be computed.
This argument is usually not specified. There is a sensible default.
}
\item{breaks}{
Optional alternative to \code{r} for advanced use.
}
\item{\dots}{
Ignored.
}
\item{trend,interaction,rbord}{
Optional. Arguments passed to \code{\link{ppm}}
to fit a point process model to the data,
if \code{object} is a point pattern.
See \code{\link{ppm}} for details.
}
\item{truecoef}{
Optional. Numeric vector. If present, this will be treated as
if it were the true coefficient vector of the point process model,
in calculating the diagnostic. Incompatible with \code{hi.res}.
}
\item{hi.res}{
Optional. List of parameters passed to \code{\link{quadscheme}}.
If this argument is present, the model will be
re-fitted at high resolution as specified by these parameters.
The coefficients
of the resulting fitted model will be taken as the true coefficients.
Then the diagnostic will be computed for the default
quadrature scheme, but using the high resolution coefficients.
}
\item{funargs}{
Optional.
List of additional arguments to be passed to \code{fun}.
}
}
\details{
Let \eqn{x} be a point pattern dataset consisting of points
\eqn{x_1,\ldots,x_n}{x[1],...,x[n]} in a window \eqn{W}.
Consider a point process model fitted to \eqn{x}, with
conditional intensity
\eqn{\lambda(u,x)}{lambda(u,x)} at location \eqn{u}.
For the purpose of testing goodness-of-fit, we regard the fitted model
as the null hypothesis. Given a functional summary statistic \eqn{S},
consider a family of alternative models obtained by exponential
tilting of the null model by \eqn{S}.
The pseudoscore for the null model is
\deqn{
V(r) = \sum_i \Delta S(x_i, x, r) - \int_W \Delta S(u,x, r) \lambda(u,x)
{\rm d} u
}{
V(r) = sum( Delta S(x[i], x, r)) - integral( Delta S(u,x, r) lambda(u,x) du)
}
where the \eqn{\Delta}{Delta} operator is
\deqn{
\Delta S(u,x, r) = S(x\cup\{u\}, r) - S(x\setminus u, r)
}{
Delta S(u,x, r) = S(x union u, r) - S(x setminus u, r)
}
the difference between the values of \eqn{S} for the
point pattern with and without the point \eqn{u}.
According to the Georgii-Nguyen-Zessin formula, \eqn{V(r)} should have
mean zero if the model is correct (ignoring the fact that the
parameters of the model have been estimated). Hence \eqn{V(r)} can be
used as a diagnostic for goodness-of-fit.
This algorithm computes \eqn{V(r)} by direct evaluation of the sum and
integral. It is computationally intensive, but it is available for
any summary statistic \eqn{S(r)}.
The diagnostic \eqn{V(r)} is also called
the \bold{pseudoresidual} of \eqn{S}. On the right
hand side of the equation for \eqn{V(r)} given above,
the sum over points of \eqn{x} is called the
\bold{pseudosum} and the integral is called the \bold{pseudocompensator}.
}
\value{
A function value table (object of class \code{"fv"}),
essentially a data frame of function values.
Columns in this data frame include \code{dat} for the pseudosum,
\code{com} for the compensator and \code{res} for the
pseudoresidual.
There is a plot method for this class. See \code{\link{fv.object}}.
}
\references{
Baddeley, A., Rubak, E. and Moller, J. (2011)
Score, pseudo-score and residual
diagnostics for spatial point process models.
To appear in \emph{Statistical Science}.
}
\author{
Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
Ege Rubak and Jesper Moller.
}
\seealso{
Special cases:
\code{\link{psstA}},
\code{\link{psstG}}.
Alternative functions:
\code{\link{Kres}},
\code{\link{Gres}}.
}
\examples{
data(cells)
fit0 <- ppm(cells, ~1) # uniform Poisson
\testonly{fit0 <- ppm(cells, ~1, nd=8)}
G0 <- psst(fit0, Gest)
G0
if(interactive()) plot(G0)
}
\keyword{spatial}
\keyword{models}