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Title: Counter-checking uncertainty calculations in Bayesian operational modal analysis with EM techniques
Authors: Ma, Xinda
Au, Siu-Kui
Keywords: Engineering
Issue Date: 2024
Source: Ma, X. & Au, S. (2024). Counter-checking uncertainty calculations in Bayesian operational modal analysis with EM techniques. Probabilistic Engineering Mechanics, 75, 103542-.
Project: RG68/22 
Journal: Probabilistic Engineering Mechanics 
Abstract: 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.
ISSN: 0266-8920
DOI: 10.1016/j.probengmech.2023.103542
Schools: School of Civil and Environmental Engineering 
Rights: © 2024 Elsevier Ltd. All rights reserved. This article may be downloaded for personal use only. Any other use requires prior permission of the copyright holder. The Version of Record is available online at
Fulltext Permission: embargo_20260201
Fulltext Availability: With Fulltext
Appears in Collections:CEE Journal Articles

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