On Data-Driven Unbiased Predictors using the Koopman Operator

TL;DR

This paper develops a new data-driven Koopman operator-based prediction method that reduces bias and variance, improving multi-step predictions for nonlinear dynamical systems, especially when true eigenfunctions are unknown.

Contribution

It introduces a novel algorithm that enforces unbiasedness in Koopman predictions by analyzing residual moments and decomposing errors, validated through numerical simulations.

Findings

Improved prediction accuracy over standard EDMD.

Reduced bias and variance in multi-step forecasts.

Foundation for uncertainty-aware Koopman prediction frameworks.

Abstract

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.