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

TL;DR

This paper extends Takens' theorem to linear least squares prediction of nonlinear time series using Koopman operators, showing that under certain conditions, exact linear prediction is achievable in the infinite-delay limit.

Contribution

It establishes a Koopman-Takens theorem demonstrating that linear least squares filters can exactly predict nonlinear dynamics in the infinite-delay limit for measure-preserving systems.

Findings

Linear least squares prediction can be exact for nonlinear systems in the infinite-delay limit.

The theorem applies to invertible measure-preserving maps and Koopman operators.

Predictability depends on the choice of observation functions and system properties.

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.