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The Akaike Information Criterion

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The Akaike Information Criterion is one of a range of ways of choosing between different types of models that seek an appropriate trade-off between goodness of fit and model complexity. The more complicated a model is the better generally will be its apparent goodness of fit, if the parameters are selected to optimise goodness of fit, but this does not necessarily make it a ‘better’ model overall for identifying how new data might behave.

 

A simple example of this is that if we have  datapoints, i.e.  for , relating to some unknown function then we can exactly fit all of these points with a polynomial of order , i.e.   where the  are fixed, but a smoother polynomial with lower order or some other function with few parameters may actually be a better guide as to what the value of  might be for the ’th datapoint even though it is unlikely to fit the first  points as well as the exact fit polynomial of order .

 

As explained in e.g. Billah, Hyndman and Koehler (2003), a common way of handling this trade-off in the context of statistics is to choose the model (out of say  model types, each of which is characterised by a vector of the  unknown free parameters where  varies between the different model types) that provides the highest ‘information criterion’ of the form:

 

 

where  is the maximised log-likelihood function,  is the vector of the  unknown free parameters within the relevant model,  is a penalty function that penalises more complex models and we are using a data series of length  for fitting purposes.

 

A range of information criteria have been proposed for this purpose including:

 

Criterion

Penalty function

AIC (Akaike’s Information Criterion)

BIC (Bayesian Information Criterion)

HQ (Hannan & Quinn’s Criterion)

MCp (Mallow’s Criterion)

GCV (Generalized Cross Validation Criterion)

FPE (Finite Prediction Error Criterion)

 

where (for MCp)  and  is the number of free parameters in the smallest model that nests all models under consideration. Billah, Hyndman and Koehler’s innovation is seek to estimate an ‘ideal’  for the purpose in hand, thus deriving an ‘empirical’ information criterion rather than necessarily adopting a fixed penalty functional form.


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