Fit the Neyman-Scott Cluster Point Process with Variance Gamma kernel

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Description

Fits the Neyman-Scott cluster point process, with Variance Gamma kernel, to a point pattern dataset by the Method of Minimum Contrast.

Usage

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

Arguments

X

Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details.

startpar

Vector of starting values for the parameters of the model.

nu

Numerical value controlling the shape of the tail of the clusters. A number greater than -1/2.

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 Neyman-Scott Cluster point process model with Variance Gamma kernel (Jalilian et al, 2013) 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 Neyman-Scott Cluster point process with Variance Gamma kernel to X, by finding the parameters of the model which give the closest match between the theoretical K function of the model and the observed K function. For a more detailed explanation of the Method of Minimum Contrast, see mincontrast.

The Neyman-Scott cluster point process with Variance Gamma kernel is described in Jalilian et al (2013). 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 have a common distribution described in Jalilian et al (2013).

The shape of the kernel is determined by the dimensionless index nu. This is the parameter nu' = alpha/2 - 1 appearing in equation (12) on page 126 of Jalilian et al (2013). In previous versions of spatstat instead of specifying nu (called nu.ker at that time) the user could specify nu.pcf which is the parameter nu = alpha-1 appearing in equation (13), page 127 of Jalilian et al (2013). These are related by nu.pcf = 2 * nu.ker + 1 and nu.ker = (nu.pcf - 1)/2. This syntax is still supported but not recommended for consistency across the package. In that case exactly one of nu.ker or nu.pcf must be specified.

If the argument lambda is provided, then this is used as the value of the point process intensity 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 corresponding model can be simulated using rVarGamma.

The parameter eta appearing in startpar is equivalent to the scale parameter omega used in rVarGamma.

Homogeneous or inhomogeneous Neyman-Scott/VarGamma models can also be fitted using the function kppm and the fitted models can be simulated using simulate.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)

Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by \adrian

References

Jalilian, A., Guan, Y. and Waagepetersen, R. (2013) Decomposition of variance for spatial Cox processes. Scandinavian Journal of Statistics 40, 119-137.

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

See Also

kppm, vargamma.estpcf, lgcp.estK, thomas.estK, cauchy.estK, mincontrast, Kest, Kmodel.

rVarGamma to simulate the model.

Examples

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