A Note on Kernel Methods for Multiscale Systems with Critical Transitions
This work addresses the detection of critical transitions in multiscale systems, which is incremental as it links dynamical theory with statistical methods.
The paper tackles the problem of detecting critical transitions in fast-slow stochastic systems using maximum mean discrepancy (MMD), showing that MMD serves as an excellent binary classifier for change points despite limited warning sign extraction under stringent conditions.
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.