R: Fit the Neyman-Scott cluster process with Cauchy kernel

cauchy.estpcf {spatstat.core} | R Documentation

## Fit the Neyman-Scott cluster process with Cauchy kernel

### Description

Fits the Neyman-Scott Cluster point process with Cauchy kernel to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.

### Usage

    cauchy.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 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.
`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.

### Details

This algorithm fits the Neyman-Scott cluster point process model with Cauchy kernel to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.

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 Neyman-Scott cluster point process with Cauchy kernel to `X`, by finding the parameters of the Matern Cluster model which give the closest match between the theoretical pair correlation function of the Matern Cluster process and the observed pair correlation function. For a more detailed explanation of the Method of Minimum Contrast, see `mincontrast`.

The model 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 κ, 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 μ, and the locations of the offspring points of one parent follow a common distribution described in Jalilian et al (2013).

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

For computational reasons, the optimisation procedure internally uses the parameter `eta2`, which is equivalent to `4 * scale^2` where `scale` is the scale parameter for the model as used in `rCauchy`.

Homogeneous or inhomogeneous Neyman-Scott/Cauchy 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 Baddeley Adrian.Baddeley@curtin.edu.au

### References

Ghorbani, M. (2012) Cauchy cluster process. Metrika, to appear.

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`, `cauchy.estK`, `lgcp.estpcf`, `thomas.estpcf`, `vargamma.estpcf`, `mincontrast`, `pcf`, `pcfmodel`.

`rCauchy` to simulate the model.

### Examples

        u <- cauchy.estpcf(redwood)
        u
        plot(u, legendpos="topright")

[Package spatstat.core version 1.65-5 Index]