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

摘要

Bayesian parameter estimation and prediction of a linear-in-parameters model with coloured noise is addressed, 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 coloured noise. The 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 the stability boundary. For on-line use, the ARMMAX model offers flexibility with respect to varying characteristics of the model noise. The flexibility gained is paid by a slight increase of the computational burden when compared with the single ARMAX with known MA part, which is, in this respect, close to recursive least-squares estimation. For off-line use, the ARMMAX model offers the possibility to estimate the unknown MA part in a novel way. Exploiting the natural parallelism of the ARMMAX model, a 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 modelling approach, describes algorithmic ingredients and illustrates the resulting algorithm using a simple example.