Granger causality is a widely-used criterion for analyzing interactions in large-scale networks. As most physical interactions are inherently nonlinear, we consider the problem of inferring the existence of pairwise Granger causality between nonlinearly interacting stochastic processes from their time series measurements. Our proposed approach relies on modeling the embedded nonlinearities in the measurements using a component-wise time series prediction model based on Statistical Recurrent Units (SRUs). We make a case that the network topology of Granger causal relations is directly inferrable from a structured sparse estimate of the internal parameters of the SRU networks trained to predict the processes’ time series measurements. We propose a variant of SRU, called economy-SRU, which, by design has considerably fewer trainable parameters, and therefore less prone to overfitting. The economy-SRU computes a low-dimensional sketch of its high-dimensional hidden state in the form of random projections to generate the feedback for its recurrent processing. Additionally, the internal weight parameters of the economy-SRU are strategically regularized in a group-wise manner to facilitate the proposed network in extracting meaningful predictive features that are highly time-localized to mimic real-world causal events. Extensive experiments are carried out to demonstrate that the proposed economy-SRU based time series prediction model outperforms the MLP, LSTM and attention-gated CNN-based time series models considered previously for inferring Granger causality. The physical mechanisms behind the functioning of any large-scale system can be understood in terms of the networked interactions between the underlying system processes. Granger causality is one widely-accepted criterion used in building network models of interactions between large ensembles of stochastic processes. While Granger causality may not necessarily imply true causality, it has proven effective in qualifying pairwise interactions between stochastic processes in a variety of system identification problems, e.g., gene regulatory network mapping (Fujita et al. (2007) ), and the mapping of human brain connectome (Seth et al. (2015) ). This perspective has given rise to the canonical problem of inferring pairwise Granger causal relationships between a set of stochastic processes from their time series measurements. At present, the vast majority of Granger causal inference methods adopt a model-based inference approach whereby the measured time series data is modeled using with a suitable parameterized data generative model whose inferred parameters ultimately reveal the true topology of pairwise Granger causal relationships. Such methods typically rely on using linear regression models for inference. However, as illustrated in the classical bivariate example by Baek & Brock (1992) , linear model-based Granger causality tests can fail catastrophically in the presence of even mild nonlinearities in the measurements, thus making a strong case for our work which tackles the nonlinearities in the measurements by exploring new generative models of the time series measurements based on recurrent neural networks. In this work, we addressed the problem of inferring pairwise Granger causal relationships between stochastic processes that interact nonlinearly. We showed that the such causality between the processes can be robustly inferred from the regularized internal parameters of the proposed eSRU-based recurrent models trained to predict the time series measurements of the individal processes. Future work includes: i Investigating the use of other loss functions besides the mean-square error loss which can capture the exogenous and instantaneous causal effects in a more realistic way. ii Incorporating unobserved confounding variables/processes in recurrent models. iii Inferring Granger causality from multi-rate time series measurements.
A new recurrent neural network architecture for detecting pairwise Granger causality between nonlinearly interacting time series. 
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