mincontrast(observed, theoretical, startpar, ...,
ctrl=list(q = 1/4, p = 2, rmin=NULL, rmax=NULL),
fvlab=list(label=NULL, desc="minimum contrast fit"),
explain=list(dataname=NULL, modelname=NULL, fname=NULL))"fv".theoretical).theoretical
and to the optimisation algorithm optim."minconfit". There are methods for printing
and plotting this object. It contains the following components:"fv")
containing the observed values of the summary statistic
(observed) and the theoretical values of the summary
statistic computed from the fitted model parameters.optim.thomas.estK.
If you want to fit a log-Gaussian Cox process, see
lgcp.estK. If you want to fit the Matern cluster
process, see matclust.estK.The Method of Minimum Contrast (Diggle and Gratton, 1984) is a general technique for fitting a point process model to point pattern data. First a summary function (typically the $K$ function) is computed from the data point pattern. Second, the theoretical expected value of this summary statistic under the point process model is derived (if possible, as an algebraic expression involving the parameters of the model) or estimated from simulations of the model. Then the model is fitted by finding the optimal parameter values for the model to give the closest match between the theoretical and empirical curves.
The argument observed should be an object of class "fv"
(see fv.object) containing the values of a summary
statistic computed from the data point pattern. Usually this is the
function $K(r)$ computed by Kest or one of its relatives.
The argument theoretical should be a user-supplied function
that computes the theoretical expected value of the summary statistic.
It must have an argument named par that will be the vector
of parameter values for the model (the length and format of this
vector are determined by the starting values in startpar).
The function theoretical should also expect a second argument
(the first argument other than par)
containing values of the distance $r$ for which the theoretical
value of the summary statistic $K(r)$ should be computed.
The value returned by theoretical should be a vector of the
same length as the given vector of $r$ values.
The argument ctrl determines the contrast criterion
(the objective function that will be minimised).
The algorithm minimises the criterion
$$D(\theta)=
\int_{r_{\mbox{\scriptsize min}}}^{r_{\mbox{\scriptsize max}}}
|\hat F(r)^q - F_\theta(r)^q|^p \, {\rm d}r$$
where $\theta$ is the vector of parameters of the model,
$\hat F(r)$ is the observed value of the summary statistic
computed from the data, $F_\theta(r)$ is the
theoretical expected value of the summary statistic,
and $p,q$ are two exponents. The default is q = 1/4,
p=2 so that the contrast criterion is the integrated squared
difference between the fourth roots of the two functions
(Waagepetersen, 2006).
The other arguments just make things print nicely.
The argument fvlab contains labels for the component
fit of the return value.
The argument explain contains human-readable strings
describing the data, the model and the summary statistic.
The "..." argument of mincontrast can be used to
pass extra arguments to the function theoretical
and/or to the optimisation function optim.
In this case, the function theoretical
should also have a "..." argument and should ignore it
(so that it ignores arguments intended for optim).
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2006). An estimation function approach to inference for inhomogeneous Neyman-Scott processes. Submitted.
kppm,
lgcp.estK,
matclust.estK,
thomas.estK,