基于Koopman算子的数据驱动无偏预测器
On Data-Driven Unbiased Predictors using the Koopman Operator
AI总结：
本文提出一种基于Koopman算子的数据驱动无偏预测方法，通过分析残差矩来减少预测偏差和方差，提升预测的不确定性特性。
摘要：
Koopman算子及其数据驱动近似方法，如扩展动态模式分解（EDMD），被广泛用于分析、建模和控制非线性动力系统。然而，当真实的Koopman特征函数无法从有限数据中识别时，多步预测可能会受到结构性不准确和系统偏差的影响。为了解决这个问题，我们分析了多步预测残差的第一和第二矩。通过将残差分解为来自一步近似误差和累积不准确性的传播贡献，我们推导出一个闭式表达式，以表征这些影响。这种分析使我们能够开发出一种新颖且计算高效的算法，该算法强制无偏性并减少预测结果的方差。所提出的方法在数值模拟中得到验证，显示出比标准EDMD更好的不确定性特性。这些结果为具有不确定性意识和无偏Koopman预测框架奠定了基础，可以扩展到受控和随机系统。
英文摘要：
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.
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