Sparse Causal Discovery in Multivariate Time Series

Our goal is to estimate causal interactions in multivariate time series. Using vector autoregressive (VAR) models, these can be defined based on non-vanishing coefficients belonging to respective time-lagged instances. As in most cases a parsimonious causality structure is assumed, a promising approach to causal discovery consists in fitting VAR models with an additional sparsity-promoting regularization. Along this line we here propose that sparsity should be enforced for the subgroups of coefficients that belong to each pair of time series, as the absence of a causal relation requires the coefficients for all time-lags to become jointly zero. Such behavior can be achieved by means of l1-l2-norm regularized regression, for which an efficient active set solver has been proposed recently. Our method is shown to outperform standard methods in recovering simulated causality graphs. The results are on par with a second novel approach which uses multiple statistical testing.

💡 Research Summary

The paper addresses the problem of discovering causal relationships in multivariate time‑series by leveraging vector autoregressive (VAR) models with a sparsity‑inducing regularizer that respects the natural grouping of coefficients across time lags. In a standard VAR(p) representation, the influence of variable i on variable j is encoded by a set of p coefficients β_{ij}^{(1)},…,β_{ij}^{(p)}. Conventional sparse VAR approaches apply an ℓ1 penalty to each individual coefficient, which can lead to situations where some lags are set to zero while others remain non‑zero, thereby violating the intuitive notion that a causal link should be either present for all lags or absent altogether.

To resolve this, the authors propose a group‑sparsity formulation: each ordered pair (i, j) constitutes a group of coefficients β_{ij} =