logLik.ppm: Log Likelihood and AIC for Point Process Model

In spatstat.model: Parametric Statistical Modelling and Inference for the 'spatstat' Family

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 \times \mbox{penalty}

where L is the maximised likelihood of the fitted model, and \mbox{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 \mbox{penalty} term is calculated as follows. If takeuchi=FALSE then \mbox{penalty} is the number of fitted parameters. If takeuchi=TRUE then \mbox{penalty} = \mbox{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)

\spatstatAuthors

.

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

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

spatstat.model documentation built on Sept. 30, 2024, 9:26 a.m.