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## 804Re: AI-GEOSTATS: Akaike's information criterion (AIC)

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• Dec 18, 2002
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>>The algebraic expression for the AIC results from the bias in the maximum
log-likelihood of a model as estimator of the mean expected log-likelihood,
this bias being a function of the number of free parameters in the model.
So it only covers those models fitted by maximum likelihood.
>
>Please, let me know. I'm interested in the AIC.

Check out 'Akaike Information Criterion Statistics', 1986, by Sakamoto,
Ishiguro, and Kitagawa (who are working associates to Akaike himself). KTK
Scientific Publishers, Tokyo. There is an English translation distributed
by Kluwer.

>If I have 3 models each one fitted with a least square method, are them
suitable for AIC application?

Yes if the models have different number of free parameters, they have an
additive stochastic component, and this component distributes normally.

>Are their SSRs the correct ones to use in the AIC?

Not quite. Compute the log likelihood under the normal assumption for each
model and use that in the AIC. If both the mean and variance of the normal
stochastic component are unknown, the log likelihood is

L(mu,sigma^2)=
-(n/2)ln(2*pi*sigma^2)-(1/2sigma^2)SUM_n(x_i-mu)^2

By taking the partial derivative of the log likelihood with respect to mu
and sigma^2, making it zero, solving for the MLE of mu and sigma^2, and
replacing these solutions into L, you get the maximum log likelihood of
each model,

L(mu_hat,sigma^2_hat)=-(n/2)ln(2*pi*sigma^2_hat)-n/2
=-(n/2)ln[(2*pi/n)SUM_n(x_i-mu_hat)^2]-n/2

Note that mu_hat would be each one of your models.
Cheers