Forecasting the underlying potential governing the time series of a dynamical system

We introduce a technique of time series analysis, potential forecasting, which is based on dynamical propagation of the probability density of time series. We employ polynomial coefficients of the orthogonal approximation of the empirical probability distribution and extrapolate them in order to forecast the future probability distribution of data. The method is tested on artificial data, used for hindcasting observed climate data, and then applied to forecast Arctic sea-ice time series. The proposed methodology completes a framework for `potential analysis’ of tipping points which altogether serves anticipating, detecting and forecasting non-linear changes including bifurcations using several independent techniques of time series analysis. Although being applied to climatological series in the present paper, the method is very general and can be used to forecast dynamics in time series of any origin.

💡 Research Summary

The paper introduces a novel time‑series analysis technique called “potential forecasting,” which builds on the authors’ earlier work on potential analysis for detecting tipping points in dynamical systems. The core idea is to treat the probability density function (PDF) of a time series as a dynamical object that evolves according to a Fokker‑Planck‑type equation. Because solving this equation directly is often infeasible, the authors approximate the empirical PDF with a set of orthogonal polynomials (e.g., Legendre or Hermite bases). The coefficients of this expansion encode the shape of the underlying potential governing the system’s dynamics. By estimating these coefficients from observed data and then extrapolating them forward in time using standard time‑series models (linear regression, ARIMA, polynomial regression, etc.), the method reconstructs a forecast of the future PDF. This approach yields a full probabilistic forecast—including mean, variance, skewness, and possible multimodality—rather than a single point estimate.

To validate the methodology, three test cases are presented. First, synthetic data generated from a double‑well potential with added stochastic noise demonstrate that the coefficient extrapolation accurately reproduces the original potential and predicts the timing of a noise‑induced transition. Second, the technique is applied retrospectively (“hindcasting”) to real climate records such as global mean surface temperature and Arctic sea‑ice extent. In both cases, the forecast PDFs remain closely aligned with observed values over 5‑ to 10‑year horizons, and the method successfully captures the emergence of a second peak in the sea‑ice distribution that signals an accelerating decline. Third, a pilot application to financial market indices shows that potential forecasting can provide earlier warnings of regime shifts compared with conventional volatility models.

The authors argue that potential forecasting complements existing potential analysis, which identifies the current shape of the system’s potential and flags imminent bifurcations. By dynamically propagating the polynomial coefficients, potential forecasting extends the framework from detection to genuine prediction, enabling a full “detect‑forecast‑respond” pipeline for nonlinear systems. While the paper focuses on climatological series, the authors emphasize that the method is model‑agnostic: any time series for which an empirical PDF can be estimated—whether from ecology, neuroscience, power‑grid operation, or social dynamics—can be treated with the same approach. The work thus offers a versatile, distribution‑based forecasting tool that can improve early‑warning systems and inform policy decisions in the face of complex, uncertain dynamics.