Bayesian approach to parameter estimation and interpolation of time-varying autoregressive processes using the Gibbs sampler

A nonstationary time series is one in which the statistics of the process are a function of time; this time dependency makes it impossible to utilise standard analytically defined statistical estimators to parameterise the process. To overcome this difficulty, the time series is considered within a finite time interval and is modelled as a time-varying autoregressive (AR) process. The AR coefficients that characterise this process are functions of time, represented by a family of basis vectors. The corresponding basis coefficients are invariant over the time window and have stationary statistical properties. A method is described for applying a Markov chain Monte Carlo method known as the Gibbs sampler to the problem of estimating the parameters of such a time-varying autoregressive (TVAR) model, whose time dependent coefficients are modelled by basis functions. The Gibbs sampling scheme is then extended to include a stage which may be used for interpolation. Results on synthetic and real audio signals show that the model is flexible, and that a Gibbs sampling framework is a reasonable scheme for estimating and characterising a time-varying AR process