Monte Carlo Bayesian filtering and smoothing for TVAR signals in symmetric α-stable noise

In this paper we propose an on-line Bayesian filtering and smoothing method for time series models with heavy-tailed alpha-stable noise, with a particular focus on TVAR models. We first point out how a filter that fails to take into account the heavy-tailed character of the noise performs poorly and then examine how an α-stable based particle filter can be devised to overcome this problem. The filtering methodology is based on a scale mixtures of normals (SMiN) representation of the α-stable distribution, which allows efficient Rao-Blackwellised implementation within a conditionally Gaussian framework, and requires no direct evaluation of the α-stable density, which is in general unavailable in closed form. The methodology is shown to work well, outperforming the traditional Gaussian methods both on simulated and real audio data. The analysis of real degraded audio samples highlights the fact that α-stable distributions are particularly well suited for noise modelling in a realistic scenario.