System Identification Toolbox™

aic - Akaike Information Criterion for estimated model

Syntax

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

Arguments

model: Name of an idarx, idgrey, idpoly, idproc, or idss model object. These model objects belong to the idmodel abstract class.

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

Use Akaike Information Criterion (AIC) to perform a relative comparison of models with different structures. Smaller value of AIC indicates a better model.

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.

For

The loss function V is

where represents the estimated parameters.

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.

To obtain the AIC expression from the last result, remove the constants and normalize.

References