A Koopman-Takens theorem: Linear least squares prediction of nonlinear time series

2023/08/04 by Koltai, Péter, Kunde, Philipp · 2 citations
#37A30 #37A46 #37C20 #37M10 #37M25 #47A16 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2308.02175

Abstract

The least squares linear filter, also called the Wiener filter, is a popular tool to predict the next element(s) of time series by linear combination of time-delayed observations. We consider observation sequences of deterministic dynamics, and ask: Which pairs of observation function and dynamics are predictable? If one allows for nonlinear mappings of time-delayed observations, then Takens' well-known theorem implies that a set of pairs, large in a specific topological sense, exists for which an exact prediction is possible. We show that a similar statement applies for the linear least squares filter in the infinite-delay limit, by considering the forecast problem for invertible measure-preserving maps and the Koopman operator on square-integrable functions.

Cited by

• An overview of Koopman-based control: From error bounds to closed-loop guarantees
• Learning Dynamical Systems with the Spectral Exterior Calculus

Related

• Probability of constructing prediction model for observable of a dynamical process via time series
• SLIP: Learning to Predict in Unknown Dynamical Systems with Long-Term Memory
• Time-Delay Observables for Koopman: Theory and Applications
• Linear Prediction of Long-Memory Processes: Asymptotic Results on Mean-squared Errors
• Deep Variational Koopman Models: Inferring Koopman Observations for Uncertainty-Aware Dynamics Modeling and Control