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