# logLik.ppm

##### Log Likelihood and AIC for Point Process Model

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 ppm
logLik(object, …, new.coef=NULL, warn=TRUE, absolute=FALSE)
```# S3 method for ppm
deviance(object, …)

# S3 method for ppm
AIC(object, …, k=2, takeuchi=TRUE)

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

# S3 method for 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"`

.

##### 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

```
# NOT RUN {
data(cells)
fit <- ppm(cells, ~x)
nobs(fit)
logLik(fit)
deviance(fit)
extractAIC(fit)
AIC(fit)
step(fit)
# }
```

*Documentation reproduced from package spatstat, version 1.62-2, License: GPL (>= 2)*