Optimal filtering of multivariate noisy AR processes

Autoregressive (AR) models play a role of paramount importance in the description of scalar and multivariate time series and find many applications in prediction and filtering. Themain limit of AR models is associated with their elementary description of the misfit between observations and model (equation error considered as a white noise). A more realistic family of autoregressive models is given by “AR+noise” ones where besides a white equation error also additive white noise on the observations is considered. Noisy AR models have given very good results in practical applications and lead to more realistic descriptions of the underlying processes; for these reasons, they are intrinsically more suitable than AR models for filtering applications. The use of AR+noise models in filtering requires the construction of a state-space realization and Kalman filtering. This article proposes an efficient innovation-based filtering approach whose computational burden is lower than that of Kalman filtering. The proposed algorithm relies directly on polynomial input–output models and on Cholesky factorization.