R: Log Likelihood and AIC for Point Process Model

logLik.ppm {spatstat.core} | R Documentation

## Log Likelihood and AIC for Point Process Model

### Description

Extracts the log likelihood, deviance, and AIC of a fitted Poisson point process model, or analogous quantities based on the pseudolikelihood or logistic likelihood for a fitted Gibbs point process model.

### Usage

    ## S3 method for class 'ppm'
    logLik(object, ..., new.coef=NULL, warn=TRUE, absolute=FALSE)

    ## S3 method for class 'ppm'
    deviance(object, ...)

    ## S3 method for class 'ppm'
    AIC(object, ..., k=2, takeuchi=TRUE)

    ## S3 method for class 'ppm'
    extractAIC(fit, scale=0, k=2, ..., takeuchi=TRUE)

    ## S3 method for class 'ppm'
    nobs(object, ...)

### Arguments

`object,fit` | Fitted point process model. An object of class `"ppm"`.
`...` | Ignored.
`warn` | If `TRUE`, a warning is given when the pseudolikelihood or logistic likelihood is returned instead of the likelihood.
`absolute` | Logical value indicating whether to include constant terms in the loglikelihood.
`scale` | Ignored.
`k` | Numeric value specifying the weight of the equivalent degrees of freedom in the AIC. See Details.
`new.coef` | New values for the canonical parameters of the model. A numeric vector of the same length as `coef(object)`.
`takeuchi` | Logical value specifying whether to use the Takeuchi penalty (`takeuchi=TRUE`) or the number of fitted parameters (`takeuchi=FALSE`) in calculating AIC.

### Details

These functions are methods for the generic commands `logLik`, `deviance`, `extractAIC` and `nobs` for the class `"ppm"`.

An object of class `"ppm"` represents a fitted Poisson or Gibbs point process model. It is obtained from the model-fitting function `ppm`.

The method `logLik.ppm` computes the maximised value of the log likelihood for the fitted model `object` (as approximated by quadrature using the Berman-Turner approximation) is extracted. If `object` is not a Poisson process, the maximised log pseudolikelihood is returned, with a warning (if `warn=TRUE`).

The Akaike Information Criterion AIC for a fitted model is defined as

AIC = -2 * log(L) + k * penalty

where L is the maximised likelihood of the fitted model, and penalty is a penalty for model complexity, usually equal to the effective degrees of freedom of the model. The method `extractAIC.ppm` returns the analogous quantity AIC* in which L is replaced by L*, the quadrature approximation to the likelihood (if `fit` is a Poisson model) or the pseudolikelihood or logistic likelihood (if `fit` is a Gibbs model).

The penalty term is calculated as follows. If `takeuchi=FALSE` then penalty is the number of fitted parameters. If `takeuchi=TRUE` then penalty = trace(J H^(-1)) where J and H are the estimated variance and hessian, respectively, of the composite score. These two choices are equivalent for a Poisson process.

The method `nobs.ppm` returns the number of points in the original data point pattern to which the model was fitted.

The R function `step` uses these methods.

### Value

`logLik` returns a numerical value, belonging to the class `"logLik"`, with an attribute `"df"` giving the degrees of freedom.

`AIC` returns a numerical value.

`extractAIC` returns a numeric vector of length 2 containing the degrees of freedom and the AIC value.

`nobs` returns an integer value.

### Model comparison

The values of `logLik` and `AIC` returned by these functions are based on the pseudolikelihood of the Gibbs point process model. If the model is a Poisson process, then the pseudolikelihood is the same as the likelihood, but for other Gibbs models, the pseudolikelihood is different from the likelihood (and the likelihood of a Gibbs model is hard to compute).

For model comparison and model selection, it is valid to compare the `logLik` values, or to compare the `AIC` values, but only when all the models are of class `"ppm"`.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### References

Varin, C. and Vidoni, P. (2005) A note on composite likelihood inference and model selection. Biometrika 92, 519–528.

### See Also

`ppm`, `as.owin`, `anova.ppm`, `coef.ppm`, `fitted.ppm`, `formula.ppm`, `model.frame.ppm`, `model.matrix.ppm`, `plot.ppm`, `predict.ppm`, `residuals.ppm`, `simulate.ppm`, `summary.ppm`, `terms.ppm`, `update.ppm`, `vcov.ppm`.

### Examples

      data(cells)
      fit <- ppm(cells, ~x)
      nobs(fit)
      logLik(fit)
      deviance(fit)
      extractAIC(fit)
      AIC(fit)
      step(fit)

[Package spatstat.core version 2.0-0 Index]