Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques
Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques
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DOI：
10.1016/j.physd.2020.132520
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发表时间：
2020-08-01
影响因子：
4
通讯作者：
Giannakis, Dimitrios
中科院分区：
文献类型：
--
作者：
Alexander, Romeo;Giannakis, Dimitrios
核模拟预测（KAF），也称为核主成分回归，是一种用于动态生成的时间序列数据的非参数统计预测的核方法。本文综合了核方法和Koopman算子理论的描述，以提供一个单一的一致的帐户KAF。这里提出的框架阐明了KAF方法的性质，即在保持测量和遍历动力学的情况下，它始终近似于由动力系统的Koopman算子作用的观测量的条件期望，并以预测初始化时的观测数据为条件。更确切地说，KAF产生最佳的预测，在这个意义上的最小均方根误差相对于不变的措施，在大数据的渐近极限。此外，所提出的框架有利于分析泛化误差和量化的不确定性。KAF的条件方差和条件概率函数的建设，以及非对称内核的扩展，也示出。KAF的各个方面的说明提供了应用程序的简单的例子，即周期性的循环和混沌洛伦兹63系统的流量。(C)2020爱思唯尔B.V.保留所有权利。
Kernel analog forecasting (KAF), alternatively known as kernel principal component regression, is a kernel method used for nonparametric statistical forecasting of dynamically generated time series data. This paper synthesizes descriptions of kernel methods and Koopman operator theory in order to provide a single consistent account of KAF. The framework presented here illuminates the property of the KAF method that, under measure-preserving and ergodic dynamics, it consistently approximates the conditional expectation of observables that are acted upon by the Koopman operator of the dynamical system and are conditioned on the observed data at forecast initialization. More precisely, KAF yields optimal predictions, in the sense of minimal root mean square error with respect to the invariant measure, in the asymptotic limit of large data. The presented framework facilitates, moreover, the analysis of generalization error and quantification of uncertainty. Extensions of KAF to the construction of conditional variance and conditional probability functions, as well as to non-symmetric kernels, are also shown. Illustrations of various aspects of KAF are provided with applications to simple examples, namely a periodic flow on the circle and the chaotic Lorenz 63 system. (C) 2020 Elsevier B.V. All rights reserved.