Deciphering interventional dynamical causality from non-intervention complex systems.

Detecting and quantifying causal influence is a focal topic in the fields of science, engineering, and interdisciplinary studies. However, causal studies on non-intervention systems attract much attention but remain extremely challenging. The delay-embedding technique provides a promising approach. In this study, we propose a framework named interventional dynamical causality (IntDC) in contrast to the traditional constructive dynamical causality (ConDC). ConDC, including Granger causality, transfer entropy, and convergence of cross-mapping, measures the causality by constructing a dynamical model without considering interventions. A computational criterion, interventional embedding entropy (IEE), is proposed to measure causal strengths in an interventional manner. IEE is an intervened causal information flow but in the delay-embedding space. Further, the IEE theoretically and numerically enables the deciphering of IntDC solely from observational (non-interventional) time-series data, without requiring any knowledge of dynamical models or real interventions in the considered system. In particular, IEE can be applied to rank causal effects according to their importance and construct causal networks from data. We conducted numerical experiments on logistic dynamics, coupled-Hénon maps, and chaotic neural networks to demonstrate that IEE can find causal edges accurately, eliminate effects of confounding, and quantify causal strength robustly over traditional indices. We also applied IEE to real-world tasks, including estimating neural connectomes of