Takens meets Koopman: Linear least squares prediction of nonlinear time seriesMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

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 in an invertible ergodic-theoretic setting and the Koopman operator on square-integrable functions.