On Data-Driven Unbiased Predictors using the Koopman Operator

The Koopman operator and its data-driven approximations, such as extended dynamic mode decomposition (EDMD), are widely used for analysing, modelling, and controlling nonlinear dynamical systems. However, when the true Koopman eigenfunctions cannot be identified from finite data, multi-step predictions may suffer from structural inaccuracies and systematic bias. To address this issue, we analyse the first and second moments of the multi-step prediction residual. By decomposing the residual into contributions from the one-step approximation error and the propagation of accumulated inaccuracies, we derive a closed-form expression characterising these effects. This analysis enables the development of a novel and computationally efficient algorithm that enforces unbiasedness and reduces variance in the resulting predictor. The proposed method is validated in numerical simulations, showing improved uncertainty properties compared to standard EDMD. These results lay the foundation for uncertainty-aware and unbiased Koopman-based prediction frameworks that can be extended to controlled and stochastic systems.

💡 Research Summary

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The paper investigates the statistical properties of data‑driven Koopman approximations, focusing on the bias and variance that arise when multi‑step predictions are performed with the Extended Dynamic Mode Decomposition (EDMD). Starting from a discrete‑time nonlinear system (x_{t+1}=f(x_t)), the authors lift the state into a higher‑dimensional observable space (\Psi(x)=