arXiv:2503.06381stat.MLcs.LG2025-03被引 1

用贝叶斯优化提升随机系统辨识精度,突破经典下界。

Adaptive Bayesian Optimization for Robust Identification of Stochastic Dynamical Systems

  • 用带加权核的集成高斯过程替代单核,增强建模能力。
  • 在多种参数与采样间隔下,均低于最大似然法的均方误差。
  • 无需人工先验,通过数据驱动隐式先验实现超精度估计。

本文研究线性随机动力系统的辨识问题,未知参数包括系统系数和噪声方差。传统基于最大似然估计(MLE)的方法需复杂梯度计算且易陷入局部最优。为此,提出一种基于贝叶斯优化(BO)的样本高效全局优化方法,采用由预定义字典中核函数加权构成的集成高斯过程(EGP)作为代理模型,丰富函数空间并提高鲁棒性。每次目标评估通过卡尔曼滤波递推高效完成。大量实验表明,在不同参数设置与采样间隔下,基于EGP的BO在均方误差(RMSE)和统计一致性方面持续优于基于稳态滤波与期望最大化(其推导为附带贡献)的MLE方法。相比之下,单核贝叶斯优化并不总具优势,凸显模型平均的价值。尤为显著的是,该方法在逆时间常数估计上实现了低于经典Cramer-Rao下界的均方误差,这一反直觉结果归因于高斯过程代理模型在贝叶斯优化中隐含引入的数据驱动先验。

原文摘要 · Abstract (English)

This paper deals with the identification of linear stochastic dynamical systems, where the unknowns include system coefficients and noise variances. Conventional approaches that rely on the maximum likelihood estimation (MLE) require nontrivial gradient computations and are prone to local optima. To overcome these limitations, a sample-efficient global optimization method based on Bayesian optimization (BO) is proposed, using an ensemble Gaussian process (EGP) surrogate with weighted kernels from a predefined dictionary. This ensemble enables a richer function space and improves robustness over single-kernel BO. Each objective evaluation is efficiently performed via Kalman filter recursion. Extensive experiments across parameter settings and sampling intervals show that the EGP-based BO consistently outperforms MLE via steady-state filtering and expectation-maximization (whose derivation is a side contribution) in terms of RMSE and statistical consistency. Unlike the ensemble variant, single-kernel BO does not always yield such gains, underscoring the benefits of model averaging. Notably, the BO-based estimator achieves RMSE below the classical Cramer-Rao bound, particularly for the inverse time constant, long considered difficult to estimate. This counterintuitive outcome is attributed to a data-driven prior implicitly induced by the GP surrogate in BO.

系统辨识贝叶斯优化高斯过程

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