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

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Koltai, Peter ; Kunde, Philipp:
A Koopman-Takens theorem : Linear least squares prediction of nonlinear time series.
In: Communications in Mathematical Physics. Bd. 405 (2024) . - 120.
ISSN 1432-0916
DOI: https://doi.org/10.1007/s00220-024-05004-8

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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.