R: Fit the Neyman-Scott Cluster Point Process with Variance...

vargamma.estpcf {spatstat.core} | R Documentation

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

### Description

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

### Usage

    vargamma.estpcf(X, startpar=c(kappa=1,scale=1), nu = -1/4, 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.
`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.
`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 Variance Gamma kernel (Jalilian et al, 2013) 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 Variance Gamma kernel to `X`, by finding the parameters of the model which give the closest match between the theoretical pair correlation function of the model and the observed pair correlation 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 Baddeley Adrian.Baddeley@curtin.edu.au

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

`rVarGamma` to simulate the model.

### Examples

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

[Package spatstat.core version 1.65-5 Index]