Operator theoretic measure of causality in linear dynamical systems

Abstract: This paper introduces Linear Operator Causality Analysis (LOCA), a physics-based framework for analyzing causality in linear dynamical systems, with a focus on fluid mechanics. Unlike data-driven statistical methods like Granger causality, LOCA uses the matrix exponential from the system's governing equations to provide a more rigorous and interpretable measure of causal interactions. Our framework reveals that, under certain conditions, common data-driven measures effectively approximate the squared magnitude of the matrix exponential. LOCA also mitigates common analysis issues, such as misleading inferences from correlated variables. We validate the methodology using a linearized Couette flow, demonstrating its ability to robustly identify both direct and indirect causal relationships between flow structures. This approach offers a powerful tool for understanding modal interactions and perturbation propagation in complex systems.