Detecting and forecasting tipping points from sample variance alone

Anticipating tipping points in complex systems is a fundamental challenge across domains. Traditional early warning signals (EWSs) based on critical slowing down, such as increasing sample variance, are widely used, but their ability to reliably indicate imminent bifurcations and forecast their timing remains limited. Here, we introduce TIPMOC (TIpping via Power-law fits and MOdel Comparison), a parametric framework designed to statistically detect the approach of a bifurcation and estimate its future location using only the sample variance. TIPMOC exploits the mathematical property that variance diverges with a characteristic power-law form near codimension-one bifurcations. By sequentially monitoring system variance as a control parameter changes, TIPMOC statistically adjudicates between linear and power-law divergence at each step. When evidence favors power-law divergence, TIPMOC forecasts the impending tipping point and estimates its position; otherwise, it avoids false positives. Through numerical simulations, we demonstrate TIPMOC’s robustness and accuracy in both detection and timing prediction across different types of dynamics and bifurcation. TIPMOC shows low false positive rates and performs well even with uneven sampling and colored noise. This method thus enhances the interpretability and practical utility of classical EWSs, serving as both a transparent add-on and a stand-alone statistical tool for forecasting regime shifts in diverse complex systems.

💡 Research Summary

The paper introduces TIPMOC (Tipping via Power‑law fits and Model Comparison), a statistical framework that detects the approach of a bifurcation and forecasts its future location using only the sample variance, a classic early‑warning signal (EWS) based on critical slowing down. The authors start from the theoretical observation that, for codimension‑one bifurcations (saddle‑node, transcritical, pitchfork, Hopf), the true variance V(u) of a system observable diverges as an inverse power of the distance to the critical control parameter u_c: V(u) ∝ |u_c – u|^{‑γ}. Consequently, the observed sample variance \hat V(u) can be modeled by a four‑parameter power‑law function \hat V(u)=a (u_c – u)^{‑γ}+b, where a>0, γ>0, u_c exceeds all observed u values, and b is a lower bound.

TIPMOC proceeds sequentially as the control parameter u is varied. After an initial window of ℓ₀=8 observations, both the power‑law model and a simple linear model (\hat V = αu+β) are fitted to the accumulated data using least‑squares. Model quality is assessed with the corrected Akaike Information Criterion (AICc), which penalizes model complexity and corrects for small sample sizes. If the AICc of the power‑law model is at least 10 points lower than that of the linear model for three consecutive u values, TIPMOC declares that a bifurcation is imminent. At that detection point u_det, the fitted power‑law provides an estimate \hat u_c of the future bifurcation location, and the algorithm stops; otherwise it continues until the data are exhausted, concluding that no bifurcation is forthcoming.

The method is demonstrated on a stochastic double‑well system d x =