Counter-checking uncertainty calculations in Bayesian operational modal analysis with EM techniques

The design and assessment of structural dynamics behaviour often rely on the ‘modal properties’ (primarily modal frequencies, damping ratios and mode shapes) of a structure. Modal properties are frequently identified in vibration related applications as their information can benefit design theory validation and in-situ property updating and monitoring. Given the prospect of flexibility and high implementation economy, operational modal analysis (OMA) aims to identify the modal properties using only ambient vibration response whose amplitude can be small. When instrument noise becomes a non-negligible part of measured data, the identified result can have significant uncertainty, calling for proper uncertainty quantification. Amidst the complex composition of ambient vibration data of civil structures, conventional noise assumption may not always apply. Motivated by issues about uncertainty modelling and computation, this thesis contributes to the theory and computational algorithm of OMA from a Bayesian perspective in the frequency domain. Focusing on the resonance band of vibration modes, a Bayesian probabilistic approach provides a fundamental mathematical framework for OMA. The ‘posterior’ (i.e., given data) most probable value (MPV) of modal properties maximising the likelihood of data informs about the best estimate. The uncertainty associated with the estimate is quantified by the posterior covariance matrix, which can be calculated as the inverse of the Hessian of negative log-likelihood function (NLLF) at the MPV. Existing formulations assume independent and identically distributed (i.i.d.) prediction errors among different channels, which account for both instrument noise and modelling error. While the assumption significantly simplifies theory and computation, and may not matter when the signal-to-noise (s/n) ratio is high, it is prone to be violated in OMA. For example, large disparity among noise intensities may exist, and significant correlation can occur because of the presence of unmodelled dynamics, which may affect identification results. To understand and manage the consequence of these potential violations, a comprehensive investigation on the effect of prediction error models with different statistical assumptions is presented in this work. Beyond the i.i.d. assumption, two generalised prediction error models have been considered to account for the disparity and correlation that may arise in applications. Such investigation was previously hindered by computational issues because the general prediction error models feature more parameters carrying somewhat different mathematical roles from the existing ones, making the computation of posterior MPV and covariance matrix more challenging in the first place. As a contribution to computation and enabling scientific investigation, new efficient algorithms applicable to general prediction error models are developed in this work for their computation. As a contribution to theory, a mathematical analogy for general noise is established with i.i.d. model for OMA problems. Not only does this provide a simple approximate way for handling general noise with existing tools based on i.i.d. assumption, it also allows the understanding and experience accumulated with i.i.d. model to be generalised. As a spin-off of this opportunity, asymptotic formulae for identification uncertainty are developed for general prediction error models in this work, which yield a generalised definition for s/n ratio that provides a fundamental quantification of data quality in OMA. The theory and algorithms are developed and tested against synthetic, laboratory and field data. As a further contribution to computation, motivated by existing overheads and hurdles in computer code development, this work proposes an alternative semi-analytical means for calculating the Hessian of NLLF in OMA. The proposed method explores a mathematical identity in the theory of expectation-maximisation (EM) algorithm to allow the Hessian matrix to be calculated via Monte Carlo simulation, which offers an alternative means for counter-checking/benchmarking during computer code development. Theoretical implications of the identity are also explored, and numerical examples are presented to illustrate implementation aspects.

A Bayesian model updating method using subset simulation optimization given ambient vibration data is proposed to improve the efficiency in sampling and avoid local optimums, which is applied to a full-scale high-rise structure. In the proposed method, the Bayesian fast Fourier transform method is used to extract the most probable values (MPVs) and coefficients of variation (COVs) of modal parameters of the structure, which are considered as the weighting factors in the likelihood function. The posterior probability density function (PDF) of the updating parameters is then derived considering the prior PDF as the regularization term. The subset simulation optimization algorithm is extended to find the global optimal solution of the posterior PDF. The MPVs and COVs of the updating parameters are finally presented. The proposed method is first verified by updating the 15 story stiffness scaling factors of a shear building model. Then, the proposed method is investigated through a numerical application of a four-story frame structure, in which accurate parameter estimations are observed independent of the prior PDF. Finally, the proposed method is applied to a real-world 13-story twin-tower masonry structure which is a modern heritage building. Ambient vibrations were measured through a series of accelerometers instrumented in the structure. The updating parameters are the moduli of elasticity of selected substructures with two model classes studied: three parameters and six parameters. The model-predicted modal parameters from the updated model are in good agreement with their identified counterparts. It is also shown that the case using three parameters provides larger evidence based on the Bayesian model class selection method. The updated refined finite element model provides a basis for the long-term structural health monitoring.

Fast Bayesian ambient modal identification in the frequency domain, Part II: Posterior uncertainty

Operational modal analysis of a high-rise multi-function building with dampers by a Bayesian approach

Asymptotic identification uncertainty of well-separated modes in operational modal analysis with multiple setups

Purpose:The purpose of this project is to develop a robust method of parameter estimation for a Poisson‐based TCP model using Bayesian inference.Methods:Bayesian inference was performed using the PyMC3 probabilistic programming framework written in Python. A Poisson‐based TCP regression model that accounts for clonogen proliferation was fit to observed rates of local relapse as a function of equivalent dose in 2 Gy fractions for a population of 623 stage‐I non‐small‐cell lung cancer patients. The Slice Markov Chain Monte Carlo sampling algorithm was used to sample the posterior distributions, and was initiated using the maximum of the posterior distributions found by optimization. The calculation of TCP with each sample step required integration over the free parameter α, which was performed using an adaptive 24‐point Gauss‐Legendre quadrature. Convergence was verified via inspection of the trace plot and posterior distribution for each of the fit parameters, as well as with comparisons of the most probable parameter values with their respective maximum likelihood estimates.Results:Posterior distributions for α, the standard deviation of α (σ), the average tumour cell‐doubling time (Td), and the repopulation delay time (Tk), were generated assuming α/β = 10 Gy, and a fixed clonogen density of 10⁷ cm−3. Posterior predictive plots generated from samples from these posterior distributions are in excellent agreement with the observed rates of local relapse used in the Bayesian inference. The most probable values of the model parameters also agree well with maximum likelihood estimates.Conclusion:A robust method of performing Bayesian inference of TCP data using a complex TCP model has been established.

Bayesian structural parameter identification from ambient vibration in cultural heritage buildings: The case of the San Jerónimo monastery in Granada, Spain

Bayesian operational modal analysis with buried modes

Identification of the modal properties of the UCLA Factor Building, a 15‐story steel moment‐resisting frame, is performed using low‐amplitude earthquake and ambient vibration data. The numerical algorithm for subspace state‐space system identification is employed to identify the structural frequencies, damping ratios, and mode shapes corresponding to the first nine modes. The frequencies and mode shapes identified based on the data recorded during the 2004 Parkfield earthquake ( Mw =6.0) are used to update a three‐dimensional finite element model of the building to improve correlation between analytical and identified modal properties and responses. A linear dynamic analysis of the updated model excited by the 1994 Northridge earthquake is performed to assess the likelihood of structural damage.

This chapter discusses the computational issues in Bayesian operational modal analysis and presents their solutions in a general context. The main theoretical issue lies in the computation of the posterior covariance matrix of modal parameters under constraints, which can be addressed systematically using Lagrange multiplier concepts. The uncertainty quantification of mode shape, which is of vectorial nature under scaling constraint, is also discussed. The computational framework in this chapter will be followed in Chaps. 12– 14 where specific algorithms are developed.

The Alfred Zampa Memorial Bridge (AZMB), a newly built long-span suspension bridge, is located 32km northeast of San Francisco on interstate Highway I-80. A set of dynamic field tests were conducted on the AZMB in November 2003, just before the bridge opening to traffic. These tests provided a unique opportunity to identify the modal properties of the bridge in its as-built condition with no previous traffic loads or seismic excitation. A benchmark study on modal identification of the AZMB is performed using three different state-of-the-art system identification algorithms based on ambient, as well as forced vibration measurements. These system identification methods consist of: (1) the multiple-reference natural excitation technique combined with the eigensystem realization algorithm; (2) the data-driven stochastic subspace identification method; and (3) the enhanced frequency domain decomposition method. Overall, the modal parameters identified using these system identification methods are found to be in very good agreement for each type of tests (ambient and forced vibration tests). For most vibration modes, the natural frequencies and mode shapes identified using the two different types of test data also match very well. However, the modal damping ratios identified from forced vibration test data are, in general, higher than those estimated from ambient vibration data. The identified natural frequencies and mode shapes are finally compared with their analytical counterparts from a three-dimensional finite-element model of the AZMB. The modal properties of the AZMB presented in this paper can be used as baseline in future health monitoring studies of this bridge.

BackgroundBayesian phylogenetic inference holds promise as an alternative to maximum likelihood, particularly for large molecular-sequence data sets. We have investigated the performance of Bayesian inference with empirical and simulated protein-sequence data under conditions of relative branch-length differences and model violation.ResultsWith empirical protein-sequence data, Bayesian posterior probabilities provide more-generous estimates of subtree reliability than does the nonparametric bootstrap combined with maximum likelihood inference, reaching 100% posterior probability at bootstrap proportions around 80%. With simulated 7-taxon protein-sequence datasets, Bayesian posterior probabilities are somewhat more generous than bootstrap proportions, but do not saturate. Compared with likelihood, Bayesian phylogenetic inference can be as or more robust to relative branch-length differences for datasets of this size, particularly when among-sites rate variation is modeled using a gamma distribution. When the (known) correct model was used to infer trees, Bayesian inference recovered the (known) correct tree in 100% of instances in which one or two branches were up to 20-fold longer than the others. At ratios more extreme than 20-fold, topological accuracy of reconstruction degraded only slowly when only one branch was of relatively greater length, but more rapidly when there were two such branches. Under an incorrect model of sequence change, inaccurate trees were sometimes observed at less extreme branch-length ratios, and (particularly for trees with single long branches) such trees tended to be more inaccurate. The effect of model violation on accuracy of reconstruction for trees with two long branches was more variable, but gamma-corrected Bayesian inference nonetheless yielded more-accurate trees than did either maximum likelihood or uncorrected Bayesian inference across the range of conditions we examined. Assuming an exponential Bayesian prior on branch lengths did not improve, and under certain extreme conditions significantly diminished, performance. The two topology-comparison metrics we employed, edit distance and Robinson-Foulds symmetric distance, yielded different but highly complementary measures of performance.ConclusionsOur results demonstrate that Bayesian inference can be relatively robust against biologically reasonable levels of relative branch-length differences and model violation, and thus may provide a promising alternative to maximum likelihood for inference of phylogenetic trees from protein-sequence data.

An overview of the recent accomplishments of the Computational Benchmark Problems Committee of the American Nuclear Society Mathematics and Computation Division is presented. Solutions of computational benchmark problems in the following eight areas are presented and discussed: (a) high-temperature gas-cooled reactor neutronics, (b) pressurized water reactor (PWR) thermal hydraulics, (c) PWR neutronics, (d) neutron transport in a cylindrical “black” rod, (e) neutron transport in a boiling water reactor (BWR) rod bundle, (f) BWR transient neutronics with thermal feedback, (g) neutron depletion in a heavy water reactor, and (h) heavy water reactor transient neutronics.It is concluded that these problems and solutions are of considerable value to the nuclear industry because they have been and will continue to be useful in the development, evaluation, and verification of computer codes and numerical-solution methods.

Bayesian statistical methods have recently been combined with conventional methods for fisheries stock assessment (e.g. catch-age analysis) to provide a conceptually elegant approach for providing fishery management advice under uncertainty. Uncertainties in the advice provided can be conveyed using posterior probability distributions (or “posteriors”) for the potential outcomes of each policy option. Posteriors can be estimated using data (e.g. catch-age data and relative abundance indices) for the fish population of interest and prior probability distributions for population model parameters (e.g. stock-recruit function parameters) based on data from similar fish populations. Despite growing interest, Bayesian methods remain accessible to relatively few. To increase the accessibility of these methods, the conceptual basis for Bayesian statistical estimation is reviewed and set in the context of fisheries stock assessment. The use of Bayesian methods is illustrated by fitting a logistic model to relative abundance indices for Namibian hake (Merlucius capensis and M. paradoxus) and presenting a decision analysis of alternative harvest policy options. Some alternative approaches are outlined for constructing prior and posterior probability distributions and some recent applications in fisheries. Some of the problems that can be encountered while implementing Bayesian methods are also discussed.

To evaluate the usefulness of the Bayesian method for hepatic fibrosis staging with magnetic resonance elastography (MRE). The sample of this retrospective study comprised patients with chronic liver disease (n = 309), in whom histopathological fibrosis staging and MRE using either a 1.5T (n = 214) or a 3T magnetic resonance imaging (MRI) system (n = 95) had been performed. The optimal cutoff stiffness value was determined and used to calculate the discrimination ability of fibrosis staging by the cutoff method. The Bayesian method calculated post-MRE probability of each fibrosis stage, yielding MRE-based fibrosis staging without a cutoff value as well as the confidence of staging. We compared the discrimination ability in all patients and in a subgroup of patients with high (≥90%) posterior probability. The discrimination ability for hepatic fibrosis staging was comparable between the Bayesian method and the cutoff method in all patients because the accuracy of staging with the Bayesian method and the cutoff method in all patients was not different (P = 1.0000). However, in patients with high posterior probability by the Bayesian method, the accuracy of staging with the Bayesian method was significantly improved compared with that of the cutoff method in all patients; for discriminating stage ≥F2 from F0-F1 (98.9% vs. 94.8%, P = 0.0069); for ≥F3 (99.6% vs. 92.6%, P < 0.0001); and for F4 (100% vs. 94.2%, P = 0.0002). The Bayesian method has a highly accurate discrimination ability for noninvasive hepatic fibrosis staging using MRE, if the posterior probability is high. 3 Technical Efficacy: Stage 2 J. MAGN. RESON. IMAGING 2017;46:375-382.