Smooth online parameter estimation for time varying VAR models with application to rat s LFP data

Multivariate time series data appear often as realizations of nonstationary 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 especially in situations where it is important to estimate these properties and detect changes in the system as they happen in real-time. Thus, the goal of this paper is to develop a method that estimates the statistical properties in real-time. Online estimation allows real-time update of the time-varying parameters as new observations arrive, it is a critical task in many adaptive control applications. One approach to modeling non-stationary time series is to fit time-varying VAR models. However, one major obstacle in online estimation of such models is the computational cost due to the high-dimensionality of the parameters. Existing methods such as the Kalman filter, parametric models or local least squares are feasible but, for some particular situations, they are not always suitable because they provide noisy estimates and can become prohibitively costly as the dimension of the time series increases. We propose a new smooth online parameter estimation approach that has the ability to control for the smoothness of the estimates with a reasonable computational complexity. Consequently, we are able to fit models, in real-time, for high dimensional time series. We demonstrate that our proposed SOPE approach can be as good as the Kalman filter in terms of mean-squared error for small dimensions. However, unlike the Kalman filter, the SOPE has much smaller computational cost and hence scalable for higher dimensions.