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aic - Akaike Information Criterion for estimated model

Syntax

am = aic(model) am = aic(model1,model2,...)

Arguments

model: Name of an idarx, idgrey, idpoly, idproc, idss, idnlarx, idnlhw, or idnlgrey model object.

Description

am = aic(model) returns a scalar value of the Akaike's Information Criterion (AIC) for the estimated model.

am = aic(model1,model2,...) returns a row vector containing AIC values for the estimated models model1,model2,....

Remarks

Akaike's Information Criterion (AIC) provides a measure of model quality by simulating the situation where the model is tested on a different data set. After computing several different models, you can compare them using this criterion. According to Akaike's theory, the most accurate model has the smallest AIC.

Note If you use the same data set for both model estimation and validation, the fit always improves as you increase the model order and, therefore, the flexibility of the model structure.

Akaike's Information Criterion (AIC) is defined by the following equation:

where V is the loss function, d is the number of estimated parameters, and N is the number of values in the estimation data set.

The loss function V is defined by the following equation:

where represents the estimated parameters.

For d<<N:

Note AIC is approximately equal to log(FPE).

AIC is formally defined as the negative log-likelihood function , evaluated at the estimated parameters, plus the number of estimated parameters. You can derive AIC from this definition, as follows:

If the disturbance source is Gaussian with the covariance matrix , the logarithm of the likelihood function is

Maximizing this analytically with respect to , and then maximizing the result with respect to , gives

where p is the number of outputs.

References

Ljung, L. System Identification: Theory for the User, Upper Saddle River, NJ, Prentice-Hal PTR, 1999. See sections about the statistical framework for parameter estimation and maximum likelihood method and comparing model structures.