SURD-based causal decomposition for nonlinear modeling from multivariate time series

Natural phenomena, such as air–sea interactions, result from complex interactions among many components. Faithfully capturing such interactions is essential for improving predictive and inferential skills. However, identifying variable relationships directly in multivariate time series remains highly challenging. To address this, we specifically prioritize causality, defined as temporal precedence and directed influence, not mere correlation. This is key to identifying mechanisms governing time evolution.

In our previous work, we used transfer entropy (TE) to infer causal structure and reconstruct nonlinear dynamical models exemplified by the Lorenz system from multivariate time series [1]. Through this approach, linear terms were accurately recovered, but multiplicative nonlinear terms proved difficult to reconstruct. We attribute this limitation to the fact that TE primarily quantifies directed information flow between two variables [2], but does not explicitly decompose multi-variable interaction effects (e.g., multiplicative couplings) within the information transfer. This shortcoming motivates the use of a causality framework that can separate redundant, unique, and synergistic contributions, thereby isolating nonlinear interaction effects. Synergistic-Unique-Redundant Decomposition (SURD) decomposes causality into redundant, unique, and synergistic information components in multivariate time series [3]. We use SURD as a causal analysis tool to validate its applicability to data-driven modeling and event-focused observational analysis relevant to tipping or critical transition dynamics.

In this study, we apply SURD-based causal decomposition to time series data generated from low-dimensional nonlinear differential equations. Synergy-dominant driver pairs are used to screen candidate multiplicative terms, which then constrain a sparse model-identification step (e.g., SINDy), improving recovery of nonlinear terms compared with pairwise TE-guided screening alone. While Martínez-Sánchez et al. (2024) primarily established SURD as a causality decomposition framework [3], the present study examines how SURD outputs can be leveraged to support reconstruction of nonlinear model equations, following our previous work, in which we implemented a data-driven approach to identify basis functions to reproduce multivariate time series [1].

We demonstrate this approach on the Lorenz63 and Rössler systems and satellite observation datasets. Concretely, we define the effect variables in the causal analysis as instantaneous tendencies (e.g., dx/dt  = (x(t+∆t) − x(t))/∆t) rather than one-step-ahead states. We then quantify how candidate drivers contribute to each tendency using an appropriate lag time ∆t corresponding to causal delay. In this setting, synergistic components highlight interaction effects as nonlinear terms that require joint knowledge of multiple drivers. Unique components support single-source linear term contributions, whereas redundant components capture shared explanatory information among drivers. Moreover, we find that applying SURD to time windows extracted immediately before and after tipping yields more discriminative synergy signatures and further improves the reconstruction accuracy of multiplicative nonlinear terms.

Acknowledgments
We thank ALD Lab for the SURD framework (https://github.com/ALD-Lab/SURD) used in this study.

References
[1] K. Kohyama, R. Irie, and M. Hisada, Causal analysis of time series data for modeling nonlinear phenomena, EGU General Assembly 2025, EGU25-3480 (2025).
[2] T. Schreiber, Measuring information transfer, Physical Review Letters, 85(2), 461 (2000).
[3] C. Martínez-Sánchez, G. Arranz, and A. Lozano-Durán, Decomposing causality into its synergistic, unique, and redundant components, Nature Communications, 15(1), 9296 (2024).