Counter-checking uncertainty calculations in Bayesian operational modal analysis with EM techniquesBayes 工作模态分析中基于期望最大化技术的不确定性计算检验
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Ma XD, Au SK, 2024. Counter-checking uncertainty calculations in Bayesian operational modal analysis with EM techniques. Probabilistic Engineering Mechanics, 75: 103542.DOI: 10.1016/j.probengmech.2023.103542
摘要 | Abstract
Bayes 工作模态分析利用 “仅输出” 的环境振动数据推断结构的模态特性 (例如固有频率、阻尼比)。在实际应用中，拥有足够的数据时，模态特性的后验概率密度函数 (probability density function, PDF) 可以通过 Gauss 概率密度函数来近似，其协方差矩阵由负对数似然函数 (negative log-likelihood function, NLLF) 在最可能值处的 Hesse 逆矩阵给出。现有的计算 Hesse 矩阵的方法基于半解析公式，为实际应用提供了一种高效且可靠的手段。然而，其计算机程序不可避免地会变得复杂，例如，变量的不同灵敏度混合、因约束导致的 Hesse 矩阵奇异性。由于缺乏基准测试的解析或数值 “精确” 结果，开发阶段的计算机程序验证也很困难。目前，通常采用有限差分法作为唯一且最后的验证手段，但其在步长选择和比较/收敛标准等方面也存在困难。基于此，本研究探索了期望最大化 (expectation-maximisation, EM) 算法理论中的一个恒等式，提供了一种评估负对数似然函数 Hesse 矩阵的替代方法。该恒等式可通过蒙特卡罗模拟，平均隐藏变量的随机样本来评估 Hesse 矩阵。虽然现有的半解析方法由于其高精度和效率仍是应用于 Hesse 计算的首选，但所提的蒙特卡罗方法在代码开发过程中提供了一种便捷的验证手段。本文将讨论该恒等式的理论意义，并通过数值算例展示其实现细节。关键词: 工作模态分析, Hesse 矩阵, 期望最大化算法, 隐变量, 蒙特卡罗模拟Bayesian operational modal analysis makes inference about the modal properties (e.g., natural frequency, damping ratio) of a structure using ‘output-only’ ambient vibration data. With sufficient data in applications, the posterior probability density function (PDF) of modal properties can be approximated by a Gaussian PDF, whose covariance matrix is given by the inverse of the Hessian of negative log-likelihood function (NLLF) at the most probable value. Existing methodologies for computing the Hessian are based on semi-analytical formulae that offer an efficient and reliable means for applications. Inevitably, their computer coding can be involved, e.g., a mix of variables with different sensitivities, singularity of Hessian due to constraints. In the absence of analytical or numerically ‘exact’ result for benchmarking, computer code verification during development stage is also non-trivial. Currently, finite difference method is often used as the only and last resort for verification, although there are also difficulties in, e.g., the choice of step size, and criterion for comparison/convergence. Motivated by these, this work explores an identity in the theory of Expectation-Maximisation (EM) algorithm to provide an alternative means for evaluating the Hessian of NLLF. Such identity allows one to evaluate the Hessian by means of Monte Carlo simulation, averaging over random samples of hidden variables. While the existing semi-analytical approach is still preferred for Hessian calculations in applications for its high definitive accuracy and speed, the proposed Monte Carlo solution offers a convenient means for counter-checking during code development. Theoretical implications of the identity will be discussed and numerical examples will be given to illustrate implementation aspects.Keywords: Operational modal analysis; Hessian; EM algorithm; Hidden variable; Monte Carlo simulation
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图 1: 功率谱密度与奇异值谱 (1 个模态的合成数据)Fig. 1. PSD and SV spectrum (synthetic data, m = 1)
图 2: 精确值 (x 轴) 与 100 万个样本蒙特卡罗估计值 (y 轴): 对角元值 (左) 与交叉项灵敏度系数 (右) Fig. 2. Exact value (x-axis) v.s. estimate by MCS of 1,000,000 samples (y-axis): diagonal entry values (left) and sensitivity coefficients of cross entries (right) in the Hessian
图 3: Hesse 矩阵对角元估计值与精确值之比的模拟时程Fig. 3. Simulation histories of ratios between estimates and exact values for diagonal entries in the Hessian
图 4: Hesse 矩阵交叉项灵敏度系数的模拟时程: (a) 仅关于自振频率、阻尼比与模态功率谱密度的交叉项; (b) 关于噪声功率谱密度与振型向量参数的交叉项Fig. 4. Simulation histories of sensitivity coefficients for cross entries in the Hessian: (a) Cross term w.r.t. only f, ζ and S; (b) Cross terms involve S_e and parameters in φ
图 5: 精确协方差 (x 轴) 与下界 (y 轴): (a) 自振频率与阻尼比; (b) 模态功率谱密度与噪声功率谱密度; (c) 振型向量参数Fig. 5. Exact c.o.v. (x-axis) v.s. lower bound (y-axis): (a) For f and ζ; (b) For S and S_e; (c) For parameters in φ
图 6: 功率谱密度与奇异值谱 (2 个模态的合成数据)Fig. 6. PSD and SV spectrum (synthetic data, m = 2)
图 7: 精确值 (x 轴) 与 100 万个样本蒙特卡罗估计值 (y 轴): 对角元值 (左) 与交叉项灵敏度系数 (右) Fig. 7. Exact value (x-axis) v.s. estimate by MCS of 1,000,000 samples (y-axis): diagonal entry values (left) and sensitivity coefficients of cross entries (right) in the Hessian
图 8: Hesse 矩阵对角元估计值与精确值之比的模拟时程Fig. 8. Simulation histories of ratios between estimates and exact values for diagonal entries in the Hessian
图 9: Hesse 矩阵交叉项灵敏度系数的模拟时程: (a) 仅关于自振频率、阻尼比与模态功率谱密度的交叉项; (b) 关于噪声功率谱密度与振型矩阵参数的交叉项Fig. 9. Simulation histories of sensitivity coefficients for cross entries in the Hessian: (a) Cross term w.r.t. only f, ζ and S; (b) Cross terms involve S_e and parameters in Φ
图 10: 精确协方差 (x 轴) 与下界 (y 轴): (a) 自振频率与阻尼比参数; (b) 噪声功率谱密度与模态功率谱密度参数; (c) 振型矩阵参数Fig. 10. Exact c.o.v. (x-axis) v.s. lower bound (y-axis): (a) For parameters in f and ζ; (b) For S_e and parameters in S; (c) For parameters in Φ
图 11: 不同样本大小下估计灵敏度系数与精确灵敏度系数之间的最大偏差Fig. 11. Largest discrepancy between estimated and exact sensitivity coefficients for each sample size
Fig. 12. PSD and SV spectrum (field data)
图 13: 1 Hz 左右两个模态的振型现场数据 (江阴大桥) 实例Fig. 13. Mode shapes of two modes around 1 Hz, field data (Jiangyin Bridge) example
图 14: 精确值 (x 轴) 与 100 万个样本蒙特卡罗估计值 (y 轴): 对角元值 (左) 与交叉项灵敏度系数 (右) Fig. 14. Exact value (x-axis) v.s. estimate by MCS of 1,000,000 samples (y-axis): diagonal entry values (left) and sensitivity coefficients of cross entries (right) in the Hessian
图 15: Hesse 矩阵对角元估计值与精确值之比的模拟时程Fig. 15. Simulation histories of ratios between estimates and exact values for diagonal entries in the Hessian
图 16: Hesse 矩阵交叉项灵敏度系数的模拟时程: (a) 仅关于自振频率、阻尼比与模态功率谱密度的交叉项; (b) 关于噪声功率谱密度与振型向量参数的交叉项Fig. 16. Simulation histories of sensitivity coefficients for cross entries in the Hessian: (a) Cross term w.r.t. only f, ζ and S; (b) Cross terms involve S_e and parameters in φ
图 17: 精确协方差 (x 轴) 与下界 (y 轴): (a) 自振频率与阻尼比参数; (b) 模态功率谱密度与噪声功率谱密度; (c) 振型值Fig. 17. Exact c.o.v. (x-axis) v.s. lower bound (y-axis): (a) For parameters in f and ζ; (b) For S and S_e; (c) For mode shape values
图 18: 现场数据不同样本大小下估计灵敏度系数与精确灵敏度系数之间的最大偏差Fig. 18. Largest discrepancy between estimated and exact sensitivity coefficients for each sample size, field data
作者信息 | Authors
马欣达 Xin-Da Ma, 通讯作者 (Corresp.) 新加坡南洋理工大学 (Nanyang Technological University) 土木与环境工程学院Email: xinda001@e.ntu.edu.sg
区兆驹 Siu-Kui Au
新加坡南洋理工大学 (Nanyang Technological University) 土木与环境工程学院
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
