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
tradeoff 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 tradeoff 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
loglikelihood 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.