Exponential Energy Marks
Given a point process model fitted to a point pattern, compute the Stoyan-Grabarnik diagnostic ``exponential energy marks'' for the data points.
- The fitted point process model. An object of class
- Logical value indicating whether to check the internal format
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
Stoyan and Grabarnik (1991) proposed a diagnostic
tool for point process models fitted to spatial point pattern data.
Each point $x[i]$ of the data pattern $X$
is given a `mark' or `weight'
$$m[i] = 1/lambda-hat(x[i],X)$$
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 $B$ has expected value equal to the
area of $B$.
fit must be a fitted point process model
(object of class
"ppm"). Such objects are produced by the maximum
pseudolikelihood fitting algorithm
This fitted model object contains complete
information about the original data pattern and the model that was
fitted to it.
The value returned by
eem is the vector
of weights $m[i]$ associated with the points $x[i]$
of the original data pattern. The original data pattern
(in corresponding order) can be
produces a set of sensible diagnostic plots based on these weights.
- A vector containing the values of the exponential energy mark for each point in the pattern.
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected with Gibbs point processes. Mathematische Nachrichten, 151:95--100.
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 Exponential energy marks")