Résumé

State-space models (SSMs) with regime switching can efficiently identify recurring patterns of variation and recurring dynamics in nonstationary multivariate time series. These models have been successfully applied in various fields such as econometrics, signal processing, control engineering, and object tracking. In this work we focus on the implementation of switching SSMs in high dimension via the Expectation-Maximization (EM) algorithm. The EM algorithm provides a relatively simple way to compute the maximum likelihood estimator (MLE) of the model parameters. However, in switching SSMs, exact calculations are intractable as they grow exponentially with the time series length. Even approximate calculations are burdensome with high dimensional data. In addition, the EM algorithm has a tendency to get stuck in non-optimal stationary points of the likelihood function, a tendency further compounded in high-dimension. Considering two common switching SSMs, one with switching dynamics and the other with switching observation process, we make several practical contributions: 1) we propose novel robust initialization methods for the EM algorithm, 2) we develop a parametric bootstrap procedure for statistical inference, 3) we provide an efficient implementation of the EM algorithm for all discussed models in a comprehensive MATLAB package publicly available at https://github.com/ddegras/switch-ssm. These contributions make it possible to reliably calculate the MLE in a reasonable time, even with very long and/or high-dimensional time series. We evaluate the statistical performance of the MLE in a simulation study and compare it to a popular alternative approach (sliding windows correlation followed by k-means clustering). We also present applications to the study of dynamic functional connectivity in large electroencephalography (EEG) datasets.

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