Model structure selection for multivariable systems by cross-validation methods

Using cross-validation ideas, two procedures are proposed for making a choice between different model structures used for (approximate) modelling of multivariable systems. The procedures are derived under fairly general conditions: the 'true' system does not need to be contained in the model set; model structures do not need to be nested and different criteria may be used for model estimation and validation. The proposed structure selection rules are shown to be invariant to parameter scaling. Under certain conditions (essentially requiring that the system belongs to the model set and that the maximum likelihood method is used for parameter estimation) they are shown to be asymptotically equivalent to the (generalized) Akaike structure selection criteria.