Estimation of Sparse Directional Connectivity With Expectation Maximization

Motivated by the many applications associated with estimation of sparse multivariate models, the estimation of sparse directional connectivity between the imperfectly measured nodes of a network is studied. Node dynamics and interactions are assumed to follow a multivariate autoregressive model driven by noise, and the observations are a noisy linear combination of the underlying node activities. The corresponding maximum a posteriori (MAP) problem is derived to estimate system parameters. Due to the intractability of the MAP problem, the expectation maximization (EM) framework is used to iteratively implement the MAP estimation. To impose sparsity, 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 sub-problem of the EM. 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. A comparison with the previous work over an extensive set of configurations shows that our method is superior under moderate to high sparsity. Finally, the algorithm is applied to real temperature data obtained from 98 stations across the U.S. mainland to identify the predictive interactions between the time series.