In many real-world applications where the system dynamics has an underlying interdependency among its variables (such as power grid, economics, neuroscience, omics networks, environmental ecosystems, and others), one is often interested in knowing whether the past values of one time series influences the future of another, known as Granger causality, and the associated underlying dynamics. This paper introduces a Koopman-inspired framework that leverages neural networks for data-driven learning of the Koopman bases, termed NeuroKoopman Dynamic Causal Discovery (NKDCD), for reliably inferring the Granger causality along with the underlying nonlinear dynamics. NKDCD employs an autoencoder architecture that lifts the nonlinear dynamics to a higher dimension using data-learned bases, where the lifted time series can be reliably modeled linearly. The lifting function, the linear Granger causality lag matrices, and the projection function (from lifted space to base space) are all represented as multilayer perceptrons and are all learned simultaneously in one go. NKDCD also utilizes sparsity-inducing penalties on the weights of the lag matrices, encouraging the model to select only the needed causal dependencies within the data. Through extensive testing on practically applicable datasets, it is shown that the NKDCD outperforms the existing nonlinear Granger causality discovery approaches.

Department of Statistical Science Cornell University The Vector AutoRegressive (VAR) model is fundamental to the study of multivariate time series. Although VAR models are intensively investigated by many researchers, practitioners often show more interest in analyzing VARX models that incorporate the impact of unmodeled exogenous variables (X) into the VAR. However, since the parameter space grows quadratically with the number of time series, estimation quickly becomes challenging. While several proposals have been made to sparsely estimate large VAR models, the estimation of large VARX models is under-explored. Moreover, typically these sparse proposals involve a lasso-type penalty and do not incorporate lag selection into the estimation procedure. As a consequence, the resulting models may be difficult to interpret. In this paper, we propose a lag-based hierarchically sparse estimator, called "HVARX", for large VARX models. We illustrate the usefulness of HVARX on a cross-category management marketing application. Our results show how it provides a highly interpretable model, and improves out-of-sample forecast accuracy compared to a lasso-type approach.