Operator theoretic measures of causality from linearised dynamics

TL;DR

This paper introduces LOCA, an operator-theoretic framework for analyzing causality in linearised dynamical systems like fluid flows, providing a physics-based alternative to statistical causality measures.

Contribution

The paper develops a novel operator-based causality measure from linearised equations and introduces a data-driven estimation method similar to DMD, improving robustness and interpretability.

Findings

LOCA encodes causality via matrix exponential of linearised dynamics.

LOCA provides a global causality measure across all time horizons.

Application to Couette flow demonstrates robustness and effectiveness.

Abstract

This paper presents an operator-theoretic framework Linear Operator Causality Analysis (LOCA), for analysing causality in linearised dynamical systems, focusing here on fluid flows. We demonstrate that the matrix exponential of the linearised differential equations fundamentally encodes the causal relationships between system modes at any future time. We further develop a global measure of causality that quantifies the presence and extent of global causality across all time horizons. This approach provides a physics-based alternative to statistical and information-theoretic causality measures such as Granger causality and transfer entropy. Unlike these data-driven techniques that infer causality from time-series data, LOCA leverages the linearised governing equations, yielding a more rigorous and interpretable measure of causal interactions. We show that LOCA gives equivalent results to data-driven methods under certain assumptions, and further discuss connections to key system properties such as controllability, observability, and graph-theoretic transitive closure. To complement this operator-based approach, we introduce a data-driven methodology akin to Dynamic Mode Decomposition (DMD) that estimates causal connections directly from time series data by approximating the matrix exponential. LOCA also mitigates common issues in data-driven causality analyses, such as misleading inferences due to correlated variables or state truncation. We demonstrate our method on linearised Couette flow, demonstrating how our framework captures both direct and indirect causal interactions among flow structures. Through this example, we highlight the advantages of our approach, including its robustness to correlation-induced biases and its ability to identify causally significant modes.