Learning a quantitative criterion for distinguishing chaos from noise

Abstract

Distinguishing chaos from noise using time-series data is fundamentally challenging because both exhibit irregular fluctuations and share many statistical and dynamical characteristics. Existing methods face two key limitations: temporally correlated noise can yield spurious signatures of chaos, and analyses of scalar time series often require explicit choices of embedding parameters. Here, we propose a purely data-driven method for distinguishing chaos and noise based on a reservoir-computing framework with a cross-prediction scheme. In the proposed approach, the model is trained to predict the future change of a variable from its current value, thereby combining a short-term predictability test with a test of the smoothness of deterministic flows. The recurrent structure of reservoir computing enables effective prediction of high-dimensional chaotic dynamics even from scalar time series without explicit delay-coordinate reconstruction, while the cross-prediction framework strongly suppresses spurious predictive correlations arising from noise. We apply the proposed method to diverse synthetic and empirical time series. Chaotic systems consistently yield strong correlations between the true and predicted future changes, whereas noise processes remain clearly separated in a low-correlation regime. The method also exhibits substantial robustness to practical limitations in empirical data, including measurement noise, limited data length, and increasing prediction lag. These results demonstrate that the squared Pearson correlation coefficient provides a simple quantitative criterion for distinguishing chaos from noise directly from observed time-series data.