\name{eem}
\alias{eem}
\title{
Exponential Energy Marks
}
\description{
Given a point process model fitted to a point pattern,
compute the Stoyan-Grabarnik diagnostic ``exponential energy marks''
for the data points.
}
\usage{
eem(fit, check=TRUE)
}
\arguments{
\item{fit}{
The fitted point process model. An object of class \code{"ppm"}.
}
\item{check}{
Logical value indicating whether to check the internal format
of \code{fit}. If there is any possibility that this object
has been restored from a dump file, or has otherwise lost track of
the environment where it was originally computed, set
\code{check=TRUE}.
}
}
\value{
A vector containing the values of the exponential energy mark
for each point in the pattern.
}
\details{
Stoyan and Grabarnik (1991) proposed a diagnostic
tool for point process models fitted to spatial point pattern data.
Each point \eqn{x[i]}{x_i} of the data pattern \eqn{X}
is given a `mark' or `weight'
\deqn{m[i] = 1/lambda-hat(x[i],X)}{m_i = \frac 1 {\hat\lambda(x_i,X)}}
where \eqn{lambda-hat(x[i],X)}{\hat\lambda(x_i,X)}
is the conditional intensity of the fitted model.
If the fitted model is correct, then the sum of these marks
for all points in a region \eqn{B} has expected value equal to the
area of \eqn{B}.
The argument \code{fit} must be a fitted point process model
(object of class \code{"ppm"}). Such objects are produced by the maximum
pseudolikelihood fitting algorithm \code{\link{ppm}}).
This fitted model object contains complete
information about the original data pattern and the model that was
fitted to it.
The value returned by \code{eem} is the vector
of weights \eqn{m[i]}{m_i} associated with the points \eqn{x[i]}{x_i}
of the original data pattern. The original data pattern
(in corresponding order) can be
extracted from \code{fit} using \code{\link{data.ppm}}.
The function \code{\link{diagnose.ppm}}
produces a set of sensible diagnostic plots based on these weights.
}
\references{
Stoyan, D. and Grabarnik, P. (1991)
Second-order characteristics for stochastic structures connected with
Gibbs point processes.
\emph{Mathematische Nachrichten}, 151:95--100.
}
\seealso{
\code{\link{diagnose.ppm}},
\code{\link{ppm.object}},
\code{\link{data.ppm}},
\code{\link{residuals.ppm}},
\code{\link{ppm}}
}
\examples{
data(cells)
fit <- ppm(cells, ~x, Strauss(r=0.15), rbord=0.15)
ee <- eem(fit)
sum(ee)/area.owin(cells$window) # should be about 1 if model is correct
Y <- setmarks(cells, ee)
plot(Y, main="Cells data\n Exponential energy marks")
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{rolf@math.unb.ca}
\url{http://www.math.unb.ca/~rolf}
}
\keyword{spatial}
\keyword{models}