R: Fit the Thomas Point Process by Minimum Contrast

thomas.estK {spatstat.core} | R Documentation

## Fit the Thomas Point Process by Minimum Contrast

### Description

Fits the Thomas point process to a point pattern dataset by the Method of Minimum Contrast using the K function.

### Usage

    thomas.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL,
                q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)

### Arguments

`X` | Data to which the Thomas model will be fitted. Either a point pattern or a summary statistic. See Details.
`startpar` | Vector of starting values for the parameters of the Thomas process.
`lambda` | Optional. An estimate of the intensity of the point process.
`q,p` | Optional. Exponents for the contrast criterion.
`rmin, rmax` | Optional. The interval of r values for the contrast criterion.
`...` | Optional arguments passed to `optim` to control the optimisation algorithm. See Details.

### Details

This algorithm fits the Thomas point process model to a point pattern dataset by the Method of Minimum Contrast, using the K function.

The argument `X` can be either

a point pattern:

  An object of class `"ppp"` representing a point pattern dataset. The K function of the point pattern will be computed using `Kest`, and the method of minimum contrast will be applied to this.

a summary statistic:

  An object of class `"fv"` containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the K function, and this object should have been obtained by a call to `Kest` or one of its relatives.

The algorithm fits the Thomas point process to `X`, by finding the parameters of the Thomas model which give the closest match between the theoretical K function of the Thomas process and the observed K function. For a more detailed explanation of the Method of Minimum Contrast, see `mincontrast`.

The Thomas point process is described in Moller and Waagepetersen (2003, pp. 61–62). It is a cluster process formed by taking a pattern of parent points, generated according to a Poisson process with intensity kappa, and around each parent point, generating a random number of offspring points, such that the number of offspring of each parent is a Poisson random variable with mean mu, and the locations of the offspring points of one parent are independent and isotropically Normally distributed around the parent point with standard deviation sigma which is equal to the parameter `scale`. The named vector of stating values can use either `sigma2` (sigma^2) or `scale` as the name of the second component, but the latter is recommended for consistency with other cluster models.

The theoretical K-function of the Thomas process is

K(r) = pi r^2 + (1 - exp(-r^2/(4 sigma^2)))/kappa.

The theoretical intensity of the Thomas process is lambda=kappa* mu.

In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters kappa and sigma^2. Then the remaining parameter mu is inferred from the estimated intensity lambda.

If the argument `lambda` is provided, then this is used as the value of lambda. Otherwise, if `X` is a point pattern, then lambda will be estimated from `X`. If `X` is a summary statistic and `lambda` is missing, then the intensity lambda cannot be estimated, and the parameter mu will be returned as `NA`.

The remaining arguments `rmin,rmax,q,p` control the method of minimum contrast; see `mincontrast`.

The Thomas process can be simulated, using `rThomas`.

Homogeneous or inhomogeneous Thomas process models can also be fitted using the function `kppm`.

The optimisation algorithm can be controlled through the additional arguments `"..."` which are passed to the optimisation function `optim`. For example, to constrain the parameter values to a certain range, use the argument `method="L-BFGS-B"` to select an optimisation algorithm that respects box constraints, and use the arguments `lower` and `upper` to specify (vectors of) minimum and maximum values for each parameter.

### Value

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

### Author(s)

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

### References

Diggle, P. J., Besag, J. and Gleaves, J. T. (1976) Statistical analysis of spatial point patterns by means of distance methods. Biometrics 32 659–667.

Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.

Thomas, M. (1949) A generalisation of Poisson's binomial limit for use in ecology. Biometrika 36, 18–25.

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`, `mincontrast`, `Kest`, `rThomas` to simulate the fitted model.

### Examples

        data(redwood)
        u <- thomas.estK(redwood, c(kappa=10, scale=0.1))
        u
        plot(u)

[Package spatstat.core version 2.3-0 Index]