Bayesian Variable Selection and Regularisation for Time- Frequency Surface Estimation

Here we describe novel Bayesian models for time-frequency analysis of nonstationary data. These models are based on the idea of a Gabor regression, in which a time series is represented as a superposition of time-frequency shifted versions of a simple window function whose essential support is well-localised in time and frequency. Specifically, we consider the case in which the set of potential predictors constitutes a frame rather than satisfying the more restrictive condition of a basis. Importantly, a key result in time-frequency theory (the Balian-Low Theorem) implies that in general, redundancy is a necessary consequence of good time-frequency localisation. Hence, in an inverse modelling context, the resultant models require careful regularisation through appropriate choices of variable selection schemes and prior distributions. Here, after constructing Bayesian models and prior distributions capable of taking into account the typical time-frequency characteristics of representative time series, we apply Markov chain Monte Carlo methods in order to sample from the resultant posterior distribution of interest. We provide examples using two of the most distinctive time-frequency surfaces—speech and music signals—as well as an EEG trace.