A Note on Kernel Methods for Multiscale Systems with Critical Transitions

TL;DR

This paper explores the use of maximum mean discrepancy (MMD) to detect critical transitions in fast-slow stochastic dynamical systems, establishing theoretical links and validating with numerical simulations.

Contribution

It introduces a formal approximation of MMD near bifurcation points and demonstrates its effectiveness as a binary classifier for critical transitions.

Findings

MMD depends on system parameters near bifurcations

Approximation of MMD can predict critical transitions

MMD effectively detects change points in simulations

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.