Variational Inference for Graphical Models of Multivariate Piecewise-Stationary Time Series

Graphical models provide a powerful formalism for statistical modeling of complex systems. Especially sparse graphical models have seen wide applications recently, as they allow us to infer network structure from multiple time series (e.g., functional brain networks from multichannel electroencephalograms). So far, most of the literature deals with stationary time series, whereas real-life time series often exhibit non-stationarity. In this paper, we focus on multivariate piecewise-stationary time series, and propose novel Bayesian techniques to infer the change points and the graphical models of stationary time segments. Concretely, we model the time series as a hidden Markov model whose hidden states correspond to different Gaussian graphical models. As such, the transition between different states represents a change point. We further impose a stick-breaking process prior on the hidden states and shrinkage priors on the inverse covariance matrices of different states. We then derive an efficient stochastic variational inference algorithm to learn the model with sublinear time complexity. As an important advantage of the proposed approach, the number and position of the change points as well as the graphical model structures are inferred in an automatic manner without tuning any parameters. The proposed method is validated through numerical experiments.