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Hierarchical Bayesian operational modal analysis: Theory and computations

机译：贝叶斯层次操作模态分析：理论与计算

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This paper presents a hierarchical Bayesian modeling framework for the uncertainty quantification in modal identification of linear dynamical systems using multiple vibration data sets. This novel framework integrates the state-of-the-art Bayesian formulations into a hierarchical setting aiming to capture both the identification precision and the variability prompted due to modeling errors. Such developments have been absent from the modal identification literature, sustained as a long-standing problem at the research spotlight. Central to this framework is a Gaussian hyper probability model, whose mean and covari-ance matrix are unknown, encapsulating the uncertainty of the modal parameters. Detailed computation of this hierarchical model is addressed under two major algorithms using Markov chain Monte Carlo (MCMC) sampling and Laplace asymptotic approximation methods. Since for a small number of data sets the hyper covariance matrix is often unidentifiable, a practical remedy is suggested through the eigenbasis transformation of the covariance matrix, which effectively reduces the number of unknown hyper-parameters. It is also proved that under some conditions the maximum a posteriori (MAP) estimation of the hyper mean and covariance coincide with the ensemble mean and covariance computed using the optimal estimations corresponding to multiple data sets. This interesting finding addresses relevant concerns related to the outcome of the mainstream Bayesian methods in capturing the stochastic variability from dissimilar data sets. Finally, the dynamical response of a prototype structure tested on a shaking table subjected to Gaussian white noise base excitation and the ambient vibration measurement of a cable footbridge are employed to demonstrate the proposed framework.
机译：本文提出了一个分层贝叶斯建模框架，用于使用多个振动数据集的线性动力学系统模态识别中的不确定性量化。这个新颖的框架将最新的贝叶斯公式集成到一个层次结构中，旨在捕获由于建模错误而引起的识别精度和可变性。模态识别文献缺乏这种发展，这一直是研究的一个长期问题。该框架的中心是高斯超概率模型，其均值和协方差矩阵未知，从而封装了模态参数的不确定性。使用Markov链蒙特卡洛（MCMC）采样和Laplace渐近逼近方法，在两种主要算法下解决了该层次模型的详细计算问题。由于对于少量数据集，超协方差矩阵通常是无法识别的，因此建议通过协方差矩阵的本征基变换来进行实际的补救，从而有效减少未知超参数的数量。还证明了在某些条件下，超均值和协方差的最大后验（MAP）估计与使用对应于多个数据集的最佳估计所计算的集合均值和协方差一致。这个有趣的发现解决了与主流贝叶斯方法的结果有关的相关问题，这些方法从不同的数据集中捕获了随机变异性。最后，采用振动台在高斯白噪声基准激励下测试的原型结构的动力响应以及电缆人行天桥的环境振动测量来论证所提出的框架。

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