A Note on Kernel Methods for Multiscale Systems with Critical Transitions

Abstract

We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast-slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near fast subsystem bifurcation points can be computed to leading-order. In particular, this leading order approximation shows that the MMD depends intricately on the fast-slow systems parameters and one can only expect to extract warning signs under rather stringent conditions. However, the MMD turns out to be an excellent binary classifier to detect the change point induced by the critical transition. We cross-validate our results by numerical simulations for a van der Pol-type model.

15 pages, 4 figures; preprint, comments and suggestions very welcome!

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• Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system
• Wild binary segmentation for multiple change-point detection
• Change-Point Detection in Time-Series Data by Relative Density-Ratio Estimation
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Cited by in corpus (3)

• One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels
• Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series
• Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator