spatstat (version 1.55-0)

thomas.estpcf: 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 pair correlation function.

Usage

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

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.

pcfargs

Optional list containing arguments passed to pcf.ppp to control the smoothing in the estimation of the pair correlation function.

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.

Details

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

The argument X can be either

a point pattern:

An object of class "ppp" representing a point pattern dataset. The pair correlation function of the point pattern will be computed using pcf, 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 pair correlation function, and this object should have been obtained by a call to pcf 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 pair correlation function of the Thomas process and the observed pair correlation 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 pair correlation function of the Thomas process is $$ g(r) = 1 + \frac 1 {4\pi \kappa \sigma^2} \exp(-\frac{r^2}{4\sigma^2})). $$ 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.

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

thomas.estK mincontrast, pcf, rThomas to simulate the fitted model.

Examples

# NOT RUN {
    data(redwood)
    u <- thomas.estpcf(redwood, c(kappa=10, scale=0.1))
    u
    plot(u, legendpos="topright")
    u2 <- thomas.estpcf(redwood, c(kappa=10, scale=0.1),
          pcfargs=list(stoyan=0.12))
# }