关于临界跃迁多尺度系统核方法的注记
A Note on Kernel Methods for Multiscale Systems with Critical Transitions
摘要：
我们研究的最大平均差异（MMD）的上下文中的临界过渡建模的快-慢随机动力系统。我们建立了一个新的临界跃迁的动力学理论与统计方面的MMD之间的联系。特别是，我们表明，一个正式的近似MMD附近的快速子系统的分歧点，可以计算到领先的顺序。特别是，这种领先的顺序近似表明，MMD复杂地依赖于快-慢系统的参数，人们只能期望在相当严格的条件下提取警告信号。然而，MMD原来是一个很好的二进制分类器，以检测由临界转变引起的变化点。我们交叉验证我们的结果通过数值模拟的范德波尔型模型。
英文摘要：
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.
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