In large collections of multivariate time series it is of interest to determine interactions between each pair of time series. We study methods for inferring time series interactions in three domains: 1) conditional independencies between time series, 2) Granger and instantaneous causality estimation in subsampled and mixed frequency time series, and 3) Granger causality estimation in multivariate categorical data.
First, we explore a Bayesian framework for inferring graphical models of time series. The goal is to determine conditional independence relations between entire time series, which for stationary series, are encoded by zeros in the inverse spectral density matrix. We place priors on (i) the graph structure and (ii) spectral matrices given the graph. We leverage a Whittle likelihood approximation and define a conjugate prior---the hyper complex inverse Wishart---on the complex-valued and graph-constrained spectral matrices. Due to conjugacy, we analytically marginalize the spectral matrices and obtain a closed-form marginal likelihood of the time series given a graph. We develop an inference procedure based on a stochastic search and apply our method to analyzing stock data and neuroimaging data of brain activity during various auditory tasks.
Second, we take a regularized likelihood approach and formulate a convex estimation procedure for the multiple transition distribution (MTD) model of multivariate categorical time series. Traditionally, the MTD model is plagued by a nonconvex objective, non-identifiability, and presence of many local optima. To circumvent these problems, we recast inference for MTD as a convex problem. The new formulation facilitates the application of MTD to high-dimensional multivariate time series using convex penalties. Our formulation also allows identifiability conditions to be stated and imposed. We perform inference by deriving a novel projected gradient algorithm for optimization. Developing high dimensional estimation theory for the convex MTD model is ongoing.
Third, we study identifiability and estimation of the structural vector autoregressive model under both subsampled and mixed frequency scenarios. In particular, we find that when the errors are non-Gaussian and independent, both the lagged linear effects and instantaneous causal effects are identifiable. This implies that the full DAG structure of the dynamic causal model is identifiable under arbitrary subsampling and mixed frequencies. Both EM and Gibbs sampling algorithms are developed for parameter estimation. Performing and evaluating simulations for inference in this framework is ongoing.
Taken together, these projects provide new methodology for inferring interactions in multivariate time series across data types, sampling regimes, and model classes.