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Multivariate time series data appear often as realizations of non-stationaryprocesses where the covariance matrix or spectral matrix smoothly evolve overtime. Most of the current approaches estimate the time-varying spectralproperties only retrospectively - that is, after the entire data has beenobserved. Retrospective estimation is a major limitation in many adaptivecontrol applications where it is important to estimate these properties anddetect changes in the system as they happen in real-time. One major obstacle inonline estimation is the computational cost due to the high-dimensionality ofthe parameters. Existing methods such as the Kalman filter or local leastsquares are feasible. However, they are not always suitable because theyprovide noisy estimates and can become prohibitively costly as the dimension ofthe time series increases. In our brain signal application, it is critical todevelop a robust method that can estimate, in real-time, the properties of theunderlying stochastic process, in particular, the spectral brain connectivitymeasures. For these reasons we propose a new smooth online parameter estimationapproach (SOPE) that has the ability to control for the smoothness of theestimates with a reasonable computational complexity. Consequently, the modelsare fit in real-time even for high dimensional time series. We demonstrate thatour proposed SOPE approach is as good as the Kalman filter in terms ofmean-squared error for small dimensions. However, unlike the Kalman filter, theSOPE has lower computational cost and hence scalable for higher dimensions.Finally, we apply the SOPE method to a rat's local field potential data duringa hippocampus-dependent sequence-memory task. As demonstrated in the video, theproposed SOPE method is able to capture the dynamics of the connectivity as therat performs the sequence of non-spatial working memory tasks.

Smooth Online Parameter Estimation for time varying VAR models with application to rat's LFP data