Sparse autoregressive model estimation for learning granger causality in time series

This paper considers a problem of estimating multivariate autoregressive (AR) models with sparse coefficient matrices. A joint zero pattern of AR coefficients reveals a Granger causality structure of the variables, which is typically depicted as a graphical model. The problem of estimating the graph topology is then formulated as a least-squares problem with an ℓ₁-type regularization to promote a sparsity in the AR coefficients. We obtain a convex framework of the estimation problem which can become challenging to solve in a large scale setting due to the nondifferentiability of the cost function. We apply a recent powerful algorithm, namely, the alternating direction method of multipliers (ADMM) for solving topology selection problems in Granger graphical models of AR processes. We illustrate the idea and verify the performance of the ADMM algorithm on randomly generated data sets. This approach is finally applied on Google Flu Trends data learn a causal structure of flu activities in the 51 states of the USA.