Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques

Abstract

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.

arXiv comment34 pages, 8 figures

References in corpus (20)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Kernel methods in machine learning
• Applied Koopmanism
• Chaos as an Intermittently Forced Linear System
• Consistency of spectral clustering
• Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator
• Data-driven model reduction and transfer operator approximation
• On the numerical approximation of the Perron-Frobenius and Koopman operator
• Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
• Nonparametric forecasting of low-dimensional dynamical systems
• Kernel Flows: from learning kernels from data into the abyss
• The Lorenz attractor is mixing
• Universality, Characteristic Kernels and RKHS Embedding of Measures
• Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach
• Delay-coordinate maps and the spectra of Koopman operators
• Koopman spectra in reproducing kernel Hilbert spaces
• Data-driven prediction strategies for low-frequency patterns of North Pacific climate variability
• Predicting regional and pan-Arctic sea ice anomalies with kernel analog forecasting
• Bi-stochastic kernels via asymmetric affinity functions
• Diffusion forecasting model with basis functions from QR decomposition

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