Description

Fit a homogeneous or inhomogeneous cluster process or Cox point process model to a point pattern.

Usage

Arguments

Details

This function fits a clustered point process model to the point pattern dataset X.

The model may be either a Neyman-Scott cluster process or another Cox process. The type of model is determined by the argument clusters. Currently the options are clusters="Thomas" for the Thomas process, clusters="MatClust" for the Matern cluster process, clusters="Cauchy" for the Neyman-Scott cluster process with Cauchy kernel, clusters="VarGamma" for the Neyman-Scott cluster process with Variance Gamma kernel (requires an additional argument nu to be passed through the dots; see rVarGamma for details), and clusters="LGCP" for the log-Gaussian Cox process (may require additional arguments passed through ...; see rLGCP for details on argument names). The first four models are Neyman-Scott cluster processes.

The algorithm first estimates the intensity function of the point process using ppm. The argument X may be a point pattern (object of class "ppp") or a quadrature scheme (object of class "quad"). The intensity is specified by the trend argument. If the trend formula is ~1 (the default) then the model is homogeneous. The algorithm begins by estimating the intensity as the number of points divided by the area of the window. Otherwise, the model is inhomogeneous. The algorithm begins by fitting a Poisson process with log intensity of the form specified by the formula trend. (See ppm for further explanation).

The argument X may also be a formula in the R language. The right hand side of the formula gives the trend as described above. The left hand side of the formula gives the point pattern dataset to which the model should be fitted.

If improve.type="none" this is the final estimate of the intensity. Otherwise, the intensity estimate is updated, as explained in improve.kppm. Additional arguments to improve.kppm are passed as a named list in improve.args.

The clustering parameters of the model are then fitted either by minimum contrast estimation, or by maximising a composite likelihood.

Minimum contrast:

If method = "mincon" (the default) clustering parameters of the model will be fitted by minimum contrast estimation, that is, by matching the theoretical K-function of the model to the empirical K-function of the data, as explained in mincontrast.

For a homogeneous model ( trend = ~1 ) the empirical K-function of the data is computed using Kest, and the parameters of the cluster model are estimated by the method of minimum contrast.

For an inhomogeneous model, the inhomogeneous K function is estimated by Kinhom using the fitted intensity. Then the parameters of the cluster model are estimated by the method of minimum contrast using the inhomogeneous K function. This two-step estimation procedure is due to Waagepetersen (2007).

If statistic="pcf" then instead of using the K-function, the algorithm will use the pair correlation function pcf for homogeneous models and the inhomogeneous pair correlation function pcfinhom for inhomogeneous models. In this case, the smoothing parameters of the pair correlation can be controlled using the argument statargs, as shown in the Examples.

Additional arguments ... will be passed to clusterfit to control the minimum contrast fitting algorithm.

Composite likelihood:

If method = "clik2" the clustering parameters of the model will be fitted by maximising the second-order composite likelihood (Guan, 2006). The log composite likelihood is

sum[i,j] w(d[i,j]) log(rho(d[i,j]; theta)) - (sum[i,j] w(d[i,j])) log(integral[D,D] w(||u-v||) rho(||u-v||; theta) du dv)

where the sums are taken over all pairs of data points x[i], x[j] separated by a distance d[i,j] = ||x[i] - x[j]|| less than rmax, and the double integral is taken over all pairs of locations u,v in the spatial window of the data. Here rho(d;theta) is the pair correlation function of the model with cluster parameters theta.

The function w in the composite likelihood is a weighting function and may be chosen arbitrarily. It is specified by the argument weightfun. If this is missing or NULL then the default is a threshold weight function, w(d) = 1(d <= R), where R is rmax/2.

Palm likelihood:

If method = "palm" the clustering parameters of the model will be fitted by maximising the Palm loglikelihood (Tanaka et al, 2008)

sum[i,j] w(x[i], x[j]) log(lambdaP(x[j] | x[i]; theta) - integral[D] w(x[i], u) lambdaP(u | x[i]; theta) du

with the same notation as above. Here lambdaP(u|v;theta) is the Palm intensity of the model at location u given there is a point at v.

In all three methods, the optimisation is performed by the generic optimisation algorithm optim. The behaviour of this algorithm can be modified using the arguments control and algorithm. Useful control arguments include trace, maxit and abstol (documented in the help for optim).

Fitting the LGCP model requires the RandomFields package, except in the default case where the exponential covariance is assumed.

Value

An object of class "kppm" representing the fitted model. There are methods for printing, plotting, predicting, simulating and updating objects of this class.

Log-Gaussian Cox Models

To fit a log-Gaussian Cox model with non-exponential covariance, specify clusters="LGCP" and use additional arguments to specify the covariance structure. These additional arguments can be given individually in the call to kppm, or they can be collected together in a list called covmodel.

For example a Matern model with parameter ν=0.5 could be specified either by kppm(X, clusters="LGCP", model="matern", nu=0.5) or by kppm(X, clusters="LGCP", covmodel=list(model="matern", nu=0.5)).

The argument model specifies the type of covariance model: the default is model="exp" for an exponential covariance. Alternatives include "matern", "cauchy" and "spheric". Model names correspond to functions beginning with RM in the RandomFields package: for example model="matern" corresponds to the function RMmatern in the RandomFields package.

Additional arguments are passed to the relevant function in the RandomFields package: for example if model="matern" then the additional argument nu is required, and is passed to the function RMmatern in the RandomFields package.

Note that it is not possible to use anisotropic covariance models because the kppm technique assumes the pair correlation function is isotropic.

Error and warning messages

See ppm.ppp for a list of common error messages and warnings originating from the first stage of model-fitting.

Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk, with contributions from Abdollah Jalilian and Rasmus Waagepetersen.

References

Guan, Y. (2006) A composite likelihood approach in fitting spatial point process models. Journal of the American Statistical Association 101, 1502–1512.

Guan, Y., Jalilian, A. and Waagepetersen, R. (2015) Quasi-likelihood for spatial point processes. Journal of the Royal Statistical Society, Series B 77, 677-697.

Jalilian, A., Guan, Y. and Waagepetersen, R. (2012) Decomposition of variance for spatial Cox processes. Scandinavian Journal of Statistics 40, 119–137.

Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott point processes. Biometrical Journal 50, 43–57.

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

See Also

Methods for kppm objects: plot.kppm, fitted.kppm, predict.kppm, simulate.kppm, update.kppm, vcov.kppm, methods.kppm, as.ppm.kppm, as.fv.kppm, Kmodel.kppm, pcfmodel.kppm.

See also improve.kppm for improving the fit of a kppm object.

Minimum contrast fitting algorithm: higher level interface clusterfit; low-level algorithm mincontrast.

Alternative fitting algorithms: thomas.estK, matclust.estK, lgcp.estK, cauchy.estK, vargamma.estK, thomas.estpcf, matclust.estpcf, lgcp.estpcf, cauchy.estpcf, vargamma.estpcf.

Summary statistics: Kest, Kinhom, pcf, pcfinhom.

For fitting Poisson or Gibbs point process models, see ppm.

Examples