提出一种贝叶斯方法，高效估计威纳模型的非线性部分并主动设计输入信号。
Optimal Bayesian Affine Estimator and Active Learning for the Wiener Model
- 基于动态基统计量，推导出参数的闭式最优仿射估计器。
- 在单轨迹下傅里叶基存在固有不一致性，且估计误差随轨迹增长而单调递减。
- 设计主动学习算法，通过优化输入信号降低估计误差，优于传统正则化最小二乘法。
本文提出针对威纳模型的贝叶斯估计框架，聚焦于已知线性状态动力学下的非线性输出函数学习。推导出未知参数的闭式最优仿射估计器，其特性由所谓的“动态基统计量”（DBS）刻画。研究了该估计器的贝叶斯无偏性、闭式后验统计量、误差随轨迹长度单调递减，以及一致性条件（即持续激励）。在傅里叶基函数的特殊情形下，证明闭式表达可计算，因傅里叶型DBS具有显式形式。此外，发现无论输入激励如何，单轨迹测量下傅里叶基存在固有不一致性。基于闭式估计误差，开发了一种主动学习算法，用于合成输入信号以最小化估计误差。数值实验验证了该方法的有效性，显著优于传统正则化最小二乘法。
原文摘要 · Abstract (English)
This paper presents a Bayesian estimation framework for Wiener models, focusing on learning nonlinear output functions under known linear state dynamics. We derive a closed-form optimal affine estimator for the unknown parameters, characterized by the so-called "dynamic basis statistics" (DBS). Several features of the proposed estimator are studied, including Bayesian unbiasedness, closed-form posterior statistics, error monotonicity in trajectory length, and consistency condition (also known as persistent excitation). In the special case of Fourier basis functions, we demonstrate that the closed-form description is computationally available, as the Fourier DBS enjoys explicit expressions. Furthermore, we identify an inherent inconsistency in the Fourier bases for single-trajectory measurements, regardless of the input excitation. Leveraging the closed-form estimation error, we develop an active learning algorithm synthesizing input signals to minimize estimation error. Numerical experiments validate the efficacy of our approach, showing significant improvements over traditional regularized least-squares methods.
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