Non-stationary, online variational Bayesian learning, with circular variables

Abstract We introduce an online variational Bayesian model for tracking changes in a non-stationary, multivariate, temporal signal, using as an example the changing frequency and amplitude of a noisy sinusoidal signal over time. The model incorporates each observation as it arrives and then discards it, and places priors over precision hyperparameters to ensure that (i) the posterior probability distributions do not become overly tight, which would impede its ability to recognise and track changes, and (ii) no values in the system are able to continuously increase and hence exceed the numerical representation of the programming language. It is thus able to perform truly online processing for an infinitely long set of observations. Only a single round of updates in the variational Bayesian scheme per observation is used, and the complexity of the algorithm is constant in time. The proposed method is demonstrated on a large number of synthetic datasets, comparing the results from the full model (with precision hyperparameters as variables with priors) with those from the base model where the precision hyperparameters are fixed values. The full model is also demonstrated on a set of real climate data.