Selecting optimal ARMA order by a minimum spectrum distance criterion

In general, an optimal model which has the best effect in a specific application among the different estimated models may not necessarily have the same order as the actual model. The objective here is to propose a criterion for selecting the optimal order model which gives the minimum ARMA spectral estimation error. The model spectral distance (MSD) is proposed to measure the difference between two models in terms of their spectral characteristics. In order to select the optimal order model, the modelling error is decomposed into two components using the measurements of MSD. One component is caused by the parameter estimation (PE ME), and the other is due to order insufficiency (OI ME). Based on the error decomposition, the model spectral distance criterion (MSDC) is proposed to select the order by comparing the changes in the PE ME and OI ME. Theoretically, orders selected by the MSDC are optimal in minimizing the MSD from the estimated model to the actual model. However, in implementation, approximations must be made. To minimize their influence on the order selection, the approximations have been designed so that the selected models are optimal or nearly optimal. Simulations have been conducted to demonstrate the new approach.