R: Method of Minimum Contrast

mincontrast {spatstat.core} | R Documentation

## Method of Minimum Contrast

### Description

A general low-level algorithm for fitting theoretical point process models to point pattern data by the Method of Minimum Contrast.

### Usage

    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),
              action.bad.values=c("warn", "stop", "silent"),
              pspace=NULL)

### Arguments

`observed` | Summary statistic, computed for the data. An object of class `"fv"`.
`theoretical` | An R language function that calculates the theoretical expected value of the summary statistic, given the model parameters. See Details.
`startpar` | 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`.
`ctrl` | Optional. List of arguments controlling the optimisation. See Details.
`fvlab` | Optional. List containing some labels for the return value. See Details.
`explain` | Optional. List containing strings that give a human-readable description of the model, the data and the summary statistic.
`action.bad.values` | String (partially matched) specifying what to do if values of the summary statistic are `NA`, `NaN` or infinite. See Details.
`pspace` | For internal use by the package only.

### Details

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 (Pfanzagl, 1969; 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) = integral from rmin to rmax of abs(Fhat(r)^q - F(theta,r)^q)^p

where theta is the vector of parameters of the model, Fhat(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, 2007).

The argument `action.bad.values` specifies what to do if some of the values of the summary statistic are `NA`, `NaN` or infinite. If `action.bad.values="stop"`, or if all of the values are bad, then a fatal error occurs. Otherwise, the domain of the summary function is shortened to avoid the bad values. The shortened domain is the longest interval on which the function values are finite (provided this interval is at least half the length of the original domain). A warning is issued if `action.bad.values="warn"` (the default) and no warning is issued if `action.bad.values="silent"`.

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`).

### Value

An object of class `"minconfit"`. There are methods for printing and plotting this object. It contains the following components:

`par` | Vector of fitted parameter values.
`fit` | 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.
`opt` | The return value from the optimizer `optim`.
`crtl` | The control parameters of the algorithm.
`info` | List of explanatory strings.

### Author(s)

Rasmus Waagepetersen rw@math.auc.dk, adapted for spatstat by Adrian Baddeley Adrian.Baddeley@curtin.edu.au.

### References

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.

Pfanzagl, J. (1969). On the measurability and consistency of minimum contrast estimates. Metrika 14, 249–276.

Waagepetersen, R. (2007). An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.

### See Also

`kppm`, `lgcp.estK`, `matclust.estK`, `thomas.estK`,

[Package spatstat.core version 2.3-1 Index]