Bayesian Identification of Distributed Vector AutoRegressive Processes

The identification of vector autoregressive (VAR) processes from partial samples is a relevant problem motivated by several applications in finance, econometrics, and networked systems (including social networks). The literature proposes several estimation algorithms, leveraging on the fact that these models can be interpreted as random Markov processes with covariance matrices satisfying Yule-Walker equations. In this paper, we address the problem of identification of distributed vector autoregressive (DVAR) processes from partial samples. The DVAR theory builds on the assumption that several processes are evolving in time, and the transition matrices of each process share some common characteristics. First, we discuss different models for describing the coupling among single processes. Subsequently, we propose an estimator for the transition matrices of the DVAR processes adopting an Empirical Bayes approach. More precisely, the local parameters are treated as random variables with a partially-unknown a priori density function, chosen as the conjugate family of distributions defined over symmetric, nonnegative-definite matrix-valued random variables and parameterized by suitable unknown hyperparameters. We develop an optimization algorithm to obtain the maximum likelihood estimates of the hyperparameters. The main feature of the proposed approach is that it does not require exact knowledge of the model describing the coupling between the different VAR processes, and it proves particularly well suited in scenarios in which the number of samples are allowed to be highly inhomogeneous or incomplete. The proposed techniques are validated on a numerical problem arising in social networks estimation.