动力系统特征提取与矩重构的信息几何方法
An information-geometric approach to feature extraction and moment reconstruction in dynamical systems
摘要：
我们提出了一个降维框架的特征提取和时刻重建动力系统的操作空间上的概率测度引起的系统的观测值，而不是直接在原始数据空间的观测值本身在更传统的方法。我们的方法是基于这样一个事实，即轨道的动力系统诱导概率测度的可测空间定义的（部分）观察系统。我们为这些概率测度的空间配备一个发散，即，概率分布之间的距离，并使用该发散来定义核积分算子。该算子的本征函数创建了捕获动力系统的不同时间尺度的函数的正交基。我们的一个主要结果表明，动力学相关的概率测度的矩的演化可以与原动力系统上的时间平均算子有关。利用这一结果，我们表明，矩可以扩大的特征函数的基础上，从而开辟了途径的时刻的非参数预测。如果概率测度的集合本身是一个流形，我们可以另外用黎曼度量装备统计流形，并使用信息几何的技术。我们提出了应用于2-torus和Lorenz 63系统上的遍历动力系统，并在一个真实世界的例子中表明，少量的本征向量就足以重建大气时间序列的矩（这里是前四个矩），即，实时多变量马登-朱利安振荡指数。
英文摘要：
We propose a dimension reduction framework for feature extraction and moment reconstruction in dynamical systems that operates on spaces of probability measures induced by observables of the system rather than directly in the original data space of the observables themselves as in more conventional methods. Our approach is based on the fact that orbits of a dynamical system induce probability measures over the measurable space defined by (partial) observations of the system. We equip the space of these probability measures with a divergence, i.e., a distance between probability distributions, and use this divergence to define a kernel integral operator. The eigenfunctions of this operator create an orthonormal basis of functions that capture different timescales of the dynamical system. One of our main results shows that the evolution of the moments of the dynamics-dependent probability measures can be related to a time-averaging operator on the original dynamical system. Using this result, we show that the moments can be expanded in the eigenfunction basis, thus opening up the avenue for nonparametric forecasting of the moments. If the collection of probability measures is itself a manifold, we can in addition equip the statistical manifold with the Riemannian metric and use techniques from information geometry. We present applications to ergodic dynamical systems on the 2-torus and the Lorenz 63 system, and show on a real-world example that a small number of eigenvectors is sufficient to reconstruct the moments (here the first four moments) of an atmospheric time series, i.e., the realtime multivariate Madden-Julian oscillation index.
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