A Koopman-Takens theorem: Linear least squares prediction of nonlinear time series
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911819752275968 |
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| author | Koltai, Péter Kunde, Philipp |
| author_facet | Koltai, Péter Kunde, Philipp |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02175 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Koopman-Takens theorem: Linear least squares prediction of nonlinear time series Koltai, Péter Kunde, Philipp Dynamical Systems Functional Analysis 37A30, 37A46, 37C20, 37M10, 37M25, 47A16 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. |
| title | A Koopman-Takens theorem: Linear least squares prediction of nonlinear time series |
| topic | Dynamical Systems Functional Analysis 37A30, 37A46, 37C20, 37M10, 37M25, 47A16 |
| url | https://arxiv.org/abs/2308.02175 |