A general low-level algorithm for fitting theoretical point process models to point pattern data by the Method of Minimum Contrast.
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),
adjustment=NULL)Summary statistic, computed for the data.
An object of class "fv".
An R language function that calculates the theoretical expected value of the summary statistic, given the model parameters. See Details.
Vector of initial values of the parameters of the
point process model (passed to theoretical).
Additional arguments passed to the function theoretical
and to the optimisation algorithm optim.
Optional. List of arguments controlling the optimisation. See Details.
Optional. List containing some labels for the return value. See Details.
Optional. List containing strings that give a human-readable description of the model, the data and the summary statistic.
Internal use only.
An object of class "minconfit". There are methods for printing
and plotting this object. It contains the following components:
Vector of fitted parameter values.
Function value table (object of class "fv")
containing the observed values of the summary statistic
(observed) and the theoretical values of the summary
statistic computed from the fitted model parameters.
The return value from the optimizer optim.
The control parameters of the algorithm.
List of explanatory strings.
This function is a general algorithm for fitting point process models
by the Method of Minimum Contrast. If you want to fit the
Thomas process, see 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).
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 -- 212.
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2006). An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63 (2007) 252--258.