# 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 class 'ppm':
logLik(object, ..., new.coef=NULL, warn=TRUE)
## S3 method for class 'ppm':
AIC(object, \dots, k=2, takeuchi=TRUE)
## S3 method for class 'ppm':
extractAIC(fit, scale=0, k=2, \dots, 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. - 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`

,
`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 $\mbox{edf}$ is 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 Rfunction `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.

##### 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`

,
`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)
extractAIC(fit)
AIC(fit)
step(fit)
```

*Documentation reproduced from package spatstat, version 1.42-2, License: GPL (>= 2)*