Open Access Publications from the University of California

Estimation of Sparse State Transition Parameters in State Space Models

• Author(s): Khayambashi, Misagh
• et al.
Abstract

Motivated by the many applications associated with sparse multivariate models, the estimation of the directional interactions between imperfectly measured nodes of a network is studied.

First, the node dynamics and interactions are assumed to follow a linear multivariate autoregressive (MVAR) model. The observations consist of noisy linear combinations of the underlying node activities. Maximum a posteriori (MAP) criterion is adopted for parameter estimation. Due to the intractability of the MAP problem, the Expectation Maximization (EM) algorithm is used to iteratively solve the MAP problem. To impose sparsity on state transition parameters, the EM algorithm is augmented with an $\ell_1$ regularization of the connectivity matrix. Multiple techniques have been used to lower the computational complexity. Importantly, an efficient coordinate descent algorithm utilizing a closed-form solution is designed to solve the $\ell_1$-regularized EM. For noise covariance estimation, the Cholesky factors of the unknown covariance matrices are used directly in the optimization process in order to impose positive definiteness and guarantee the functionality of the $\ell_1$ optimization.

The algorithm is first applied to synthetic data to evaluate the estimation accuracy. Comparison with previous work over an extensive set of configurations shows that our method is superior under moderate to high sparsity.

The algorithm is then evaluated on real data for two different applications: temperature prediction and estimation of effective brain connectivity. Applied to real temperature data obtained from 98 stations across the U.S. mainland, the algorithm is able to identify the predictive interactions between the time series that not only are consistent with previous work, but also reveal predictive power for coastal stations.

The algorithm, however, does not perform well in the estimation of effective brain connectivity from real electroencephalography (EEG) data. We show that this shortcoming is due to the inflexibility of the linear model to capture EEG dynamics. The Neural Mass Model (NMM) is then adopted to realistically model the underlying mechanisms of EEG signals. The estimation algorithm is tailored to the nonlinearity of the NMM model. The modified algorithm is then applied to a simple synthetic system, and it is observed that the results are insensitive to the source of the connection. The root cause of the problem is then analyzed and the challenges facing the future work are discussed.