Estimation and prediction with ARMMAX model: a mixture of ARMAX models with common ARX part

Bayesian parameter estimation and prediction of a linear-in-parameters model with colored noise is addressed. It is based on a novel mixture model called ARMMAX. ARMMAX is a finite mixture with its ARMAX components having a common ARX part. It assumes that the common ARX part describes a fixed deterministic input-output relationship and allows for varying characteristics of the driving colored noise. ARMMAX model with fixed MA parts is estimated by a specific version of recursive Quasi-Bayes (ARMMAX-QB) algorithm. It rests on a classical Bayesian solution that requires no restrictions on MA part allowing it to be even at stability boundary. For on line use, ARMMAX model offers flexibility with respect to varying characteristics of the model noise. The gained flexibility is paid by a slight increase of the computational burden comparing to single ARMAX with known MA part, which is, in this respect, close to recursive least squares. For off line use, ARMMAX model offers the possibility to estimate unknown MA parts in a novel way. Exploiting the natural parallelism of ARMMAX model, robust, derivative free multi-directional search (MDS) is selected to deal with extensive data sets for which universal optimization tools are too cumbersome. The paper motivates the model, describes algorithmic ingredients and illustrates the resulting algorithm on a simple example. Copyright c © 2003 John Wiley & Sons, Ltd.