R: Fit Cluster or Cox Point Process Model

kppm {spatstat} | R Documentation

## Fit Cluster or Cox Point Process Model

### Description

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

### Usage

      kppm(X, ...)

      ## S3 method for class 'formula'
    kppm(X,
                    clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"),
                    ...,
                    data=NULL)

      ## S3 method for class 'ppp'
    kppm(X,
           trend = ~1,
           clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"),
           data = NULL,
           ...,
           covariates=data,
           subset,
           method = c("mincon", "clik2", "palm"),
           improve.type = c("none", "clik1", "wclik1", "quasi"),
           improve.args = list(),
           weightfun=NULL,
           control=list(),
           algorithm="Nelder-Mead",
           statistic="K",
           statargs=list(),
           rmax = NULL,
           covfunargs=NULL,
           use.gam=FALSE,
           nd=NULL, eps=NULL)

    ## S3 method for class 'quad'
    kppm(X,
           trend = ~1,
           clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"),
           data = NULL,
           ...,
           covariates=data,
           subset,
           method = c("mincon", "clik2", "palm"),
           improve.type = c("none", "clik1", "wclik1", "quasi"),
           improve.args = list(),
           weightfun=NULL,
           control=list(),
           algorithm="Nelder-Mead",
           statistic="K",
           statargs=list(),
           rmax = NULL,
           covfunargs=NULL,
           use.gam=FALSE,
           nd=NULL, eps=NULL)

### Arguments

`X` | A point pattern dataset (object of class `"ppp"` or `"quad"`) to which the model should be fitted, or a `formula` in the R language defining the model. See Details.
`trend` | An R formula, with no left hand side, specifying the form of the log intensity.
`clusters` | Character string determining the cluster model. Partially matched. Options are `"Thomas"`, `"MatClust"`, `"Cauchy"`, `"VarGamma"` and `"LGCP"`.
`data,covariates` | The values of spatial covariates (other than the Cartesian coordinates) required by the model. A named list of pixel images, functions, windows, tessellations or numeric constants.
`...` | Additional arguments. See Details.
`subset` | Optional. A subset of the spatial domain, to which the model-fitting should be restricted. A window (object of class `"owin"`) or a logical-valued pixel image (object of class `"im"`), or an expression (possibly involving the names of entries in `data`) which can be evaluated to yield a window or pixel image.
`method` | The fitting method. Either `"mincon"` for minimum contrast, `"clik2"` for second order composite likelihood, or `"palm"` for Palm likelihood. Partially matched.
`improve.type` | Method for updating the initial estimate of the trend. Initially the trend is estimated as if the process is an inhomogeneous Poisson process. The default, `improve.type = "none"`, is to use this initial estimate. Otherwise, the trend estimate is updated by `improve.kppm`, using information about the pair correlation function. Options are `"clik1"` (first order composite likelihood, essentially equivalent to `"none"`), `"wclik1"` (weighted first order composite likelihood) and `"quasi"` (quasi likelihood).
`improve.args` | Additional arguments passed to `improve.kppm` when `improve.type != "none"`. See Details.
`weightfun` | Optional weighting function w in the composite likelihood or Palm likelihood. A `function` in the R language. See Details.
`control` | List of control parameters passed to the optimization function `optim`.
`algorithm` | Character string determining the mathematical optimisation algorithm to be used by `optim`. This argument is passed to `optim` as the argument `method`.
`statistic` | Name of the summary statistic to be used for minimum contrast estimation: either `"K"` or `"pcf"`.
`statargs` | Optional list of arguments to be used when calculating the `statistic`. See Details.
`rmax` | Maximum value of interpoint distance to use in the composite likelihood.
`covfunargs,use.gam,nd,eps` | Arguments passed to `ppm` when fitting the intensity.

### 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.

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`.

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`.

See also `ppm`

### Examples

      # method for point patterns
      kppm(redwood, ~1, "Thomas")
      # method for formulas
      kppm(redwood ~ 1, "Thomas")

      # different models for clustering
      kppm(redwood ~ x, "MatClust")
      kppm(redwood ~ x, "MatClust", statistic="pcf", statargs=list(stoyan=0.2))
      kppm(redwood ~ x, cluster="Cauchy", statistic="K")
      kppm(redwood, cluster="VarGamma", nu = 0.5, statistic="pcf")

      # log-Gaussian Cox process (LGCP) models
      kppm(redwood ~ 1, "LGCP", statistic="pcf")
      if(require("RandomFields")) {
        # Random Fields package is needed for non-default choice of covariance model
        kppm(redwood ~ x, "LGCP", statistic="pcf",
                                  model="matern", nu=0.3,
                                  control=list(maxit=10))
      }

      # Different fitting techniques
      kppm(redwood ~ 1, "Thomas", method="c")
      kppm(redwood ~ 1, "Thomas", method="p")
      # composite likelihood method
      kppm(redwood ~ x, "VarGamma", method="clik2", nu.ker=-3/8)
      # quasi-likelihood method
      kppm(redwood ~ x, "Thomas", improve.type = "quasi")

[Package spatstat version 1.63-3 Index]