Causality beyond the Reichenbach principle: the degrees-of-freedom method
Causal relationships underpin scientific reasoning across disciplines, yet reliable inference from observational data remains challenging. In many fields, from climate science to neuroscience, direct interventions are infeasible, and distinguishing genuine causality from correlations induced by hidden drivers is notoriously difficult. Classical approaches, from Granger causality to topological embedding methods, can be theoretically well founded for either stochastic or deterministic systems, but they usually do not cover both settings in the same framework, and only a few include an explicit hidden-driver class. In an earlier paper, we introduced a general framework for causal inference in dynamical systems based on the effective degrees of freedom of a system (df). The framework applies uniformly to deterministic and stochastic systems, in both discrete and continuous time. In the present paper, we develop an operational implementation of this framework based on conditional-entropy features and supervised classification, evaluate it on synthetic systems, and apply it to selected real time series as proof-of-applicability examples. Causal model classes are constructed using simulated coupled 0–1 Markov chains, and observed trajectories are mapped to binary processes and classified according to their df relationships. This procedure is designed to distinguish canonical causal classes, including directed driving, common cause, bidirectional coupling, and independence. The approach achieves high accuracy on controlled synthetic data, while performance under distribution shift depends on sample length and noise. In a small Hénon benchmark against Sauer–Sugihara, Granger causality, and transfer entropy, its advantage is not uniform, but is most visible in the full five-class task at longer sample length, where the common-driver class is retained. Applications to climate, ecological, and neural datasets demonstrate the breadth of the approach. In the climate example, different preprocessing and lag choices produced different class-score patterns, including a contemporaneous configuration and lagged configurations. We interpret these empirical examples as demonstrations of applicability rather than as definitive domain-specific causal conclusions. These findings suggest that df analysis can provide a broadly applicable and conceptually transparent route to disentangling causality from correlation in complex systems, provided that training design, preprocessing, and robustness checks are handled carefully.