Generalized Stochastic Resilience for Early Warning Signals Based on Koopman Operator

TL;DR

This work addresses early warning of tipping points in nonlinear stochastic systems by tying tipping behavior to spectral properties of the stochastic Koopman operator. It introduces ResKMD, a residual signal that separates the projection error (dominant point-spectrum) from the noise-driven continuous-spectrum contribution, and proves that ResKMD diverges as a bifurcation is approached, thereby generalizing stochastic resilience beyond local linearization. The authors develop a practical, online framework using EDMD, kernel EDMD, and Residual DMD (ResDMD) with windowed DMD and time-delay coordinates to compute ResKMD from streaming data, demonstrating robust ROC performance across five synthetic/real datasets without pre-training. The results offer interpretable diagnostics via variance-spectrum separation and highlight the method’s robustness to noise and data scarcity, while outlining extensions to neural DMD and global tipping types for future work. Overall, ResKMD provides a theoretically grounded, data-driven, pre-training-free early warning signal that generalizes stochastic resilience and performs well across diverse domains.

Abstract

Developing methods for detecting tipping phenomena at an early stage is an important problem in various fields such as ecology, medicine, and economics. A tipping phenomenon is characterized by a rapid transition resulting from the accumulation of small parameter changes and is known to be related to bifurcations of dynamical systems. However, few studies have examined how nonlinear properties near bifurcation points affect early warning signal (EWS) performance. In this study, we apply the Koopman operator, which describes the time evolution of dynamical systems in an infinite-dimensional function space, to generalize stochastic resilience the theoretical basis of EWSs such as variance-based ones. As a result, we develop a novel signal capable of more accurately predicting tipping events by separately isolating stochastic fluctuations induced by noise and contributions from a continuous spectrum emerging immediately above tipping points. Our experimental results demonstrate that the proposed approach detects early signs of tipping phenomena more robustly than conventional methods.

Figures (5)

• Figure 1: Conceptual diagram of ResKMD
• Figure 2: Example of Subcritical Hopf Bifurcation: EWSs are computed by sliding a time window along the data. Moreover, suitable EWSs are expected to increase as the system approaches the bifurcation point. Namely, this implies that Kendall’s $\tau$ between the EWSs and time should approach $1$. (Residual(vanilla) means ResKMD calculated by Exact DMD. Also, residual(rbf, laplacian, polynomial) are ResKMD calculated by EDMD with RBF, Laplacian, and polynomial kernel.)
• Figure 3: ROC curve (The Results for Saddle-node, Hopf, Thermoacoustic, Paleoclimate and Anoxic datasets are shown from top left to right.)
• Figure 4: Behavior of EWSs against $1500$, $5000$, $10000$ Window Size This figure represents experimental results based on simulation data exhibiting saddle–node bifurcation.
• Figure 5: Behavior of against Observation Noise $\sigma_{\mathrm{obs}}^2 = 0.01, 0.1, 0.2$: This figure also represents experimental results based on simulation data exhibiting saddle–node bifurcation.

Theorems & Definitions (4)

• theorem 1
• corollary thmcountercorollary
• proposition thmcounterproposition
• theorem 2