# Economy Statistical Recurrent Units For Inferring Nonlinear Granger Causality

arXiv: https://arxiv.org/abs/1911.09879 · HTML: https://mlsanity.com/paper/1911.09879/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Nov 2019.

## Built on by (most cited first, 30 of 55)

- [Interpretable Models for Granger Causality Using Self-explaining Neural Networks](https://mlsanity.com/paper/2101.07600/) (2021)
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- [CUTS: Neural Causal Discovery from Irregular Time-Series Data](https://mlsanity.com/paper/2302.07458/) (2023)
- [Rhino: Deep Causal Temporal Relationship Learning With History-dependent Noise](https://mlsanity.com/paper/2210.14706/) (2022)
- [Neural Structure Learning with Stochastic Differential Equations](https://mlsanity.com/paper/2311.03309/) (2023)
- [GCAD: Anomaly Detection in Multivariate Time Series from the Perspective of Granger Causality](https://mlsanity.com/paper/2501.13493/) (2025)
- [Deep Causal Learning: Representation, Discovery and Inference](https://mlsanity.com/paper/2211.03374/) (2022)
- [Causality-Aware Spatiotemporal Graph Neural Networks for Spatiotemporal Time Series Imputation](https://mlsanity.com/paper/2403.11960/) (2024)
- [Discovering Mixtures of Structural Causal Models from Time Series Data](https://mlsanity.com/paper/2310.06312/) (2023)
- [Jacobian Regularizer-based Neural Granger Causality](https://mlsanity.com/paper/2405.08779/) (2024)
- [Deep Recurrent Modelling of Granger Causality with Latent Confounding](https://mlsanity.com/paper/2202.11286/) (2022)
- [Granger causal inference on DAGs identifies genomic loci regulating transcription](https://mlsanity.com/paper/2210.10168/) (2022)
- [CausalDynamics: A large-scale benchmark for structural discovery of dynamical causal models](https://mlsanity.com/paper/2505.16620/) (2025)
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- [Double Machine Learning Based Structure Identification from Temporal Data](https://mlsanity.com/paper/2311.06012/) (2023)
- [Learning Flexible Time-windowed Granger Causality Integrating Heterogeneous Interventional Time Series Data](https://mlsanity.com/paper/2406.10419/) (2024)
- [Granger Causality using Neural Networks](https://mlsanity.com/paper/2208.03703/) (2022)
- [Learning domain-specific causal discovery from time series](https://mlsanity.com/paper/2209.05598/) (2022)
- [Collective Relational Inference for learning heterogeneous interactions](https://mlsanity.com/paper/2305.00557/) (2023)
- [Discovering Causality for Efficient Cooperation in Multi-Agent Environments](https://mlsanity.com/paper/2306.11846/) (2023)
- [Causal Temporal Regime Structure Learning](https://mlsanity.com/paper/2311.01412/) (2023)
- [LOCAL: Learning with Orientation Matrix to Infer Causal Structure from Time Series Data](https://mlsanity.com/paper/2410.19464/) (2024)
- [UnCLe: Towards Scalable Dynamic Causal Discovery in Non-linear Temporal Systems](https://mlsanity.com/paper/2511.03168/) (2025)
- [Coarse-to-Fine Learning of Dynamic Causal Structures](https://mlsanity.com/paper/2602.22532/) (2026)

## Abstract (arXiv)

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.