Equation error versus output error methods

For the identification of linear processes on the basis of ARX-models, equation error least squares (EELS) (often indicated as theone step ahead prediction error method) is frequently used rather than output error least squares (OELS). This is mainly because the minimum of the convex EE-criterion can easily be found, in contrast to the DE-criterion, which often displays multiple local minima. Both methods lead to the correct parameter values when the system is in the model set chosen. But in many practical situations, such as human behaviour, the real process under study will be of infinite order causing essentially different models to be found from either EE or OE criteria. Various aspects of these differences are analysed in this study. Much attention has been paid to the performance of a simulation based on a model estimated with an EELS. This simulation performance can be predicted and bounds can be given without executing the simulation itself. Furthermore the simulation performance is very poor for systems where the energy in the initial impulse response samples is very small compared with the energy in the remainder of the response. For these systems an equation error estimate cannot even provide a proper initial guess for an OELS minimization algorithm. Examples are presented that illustrate this effect.