An innovative Bayesian system identification method using autoregressive model

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.

Modal parameter identification via ambient vibration is popular but faces challenges from uncertainties due to unknown inputs and low signal‐to‐noise ratio. Bayesian methods are gaining increasing attention for operational modal identification due to their ability to quantify uncertainties. However, improvements in computational efficiency are needed, particularly when addressing numerous modes and degrees of freedom. To address this challenge, this study proposes an innovative approach, termed the “Bayesian spectral decomposition” method (BSD), employing the decompose‐and‐conquer strategy. This novel method, operating within the frequency domain, identifies each mode individually by exploiting their inherent separated modal characteristics. For each mode, the response spectrum matrix undergoes an eigenvalue decomposition, yielding crucial eigenvalues (incorporating frequency and damping information) and eigenvectors (containing mode shape information). Subsequently, statistical properties of the eigenvalues and eigenvectors are utilized to establish likelihood functions for Bayesian parameter identification. By combining prior information, the posterior probability distribution functions of modal parameters are derived. The optimal solution is then obtained by resolving the maximum posterior probability distribution function problem. To further quantify the uncertainty of modal parameters, Gaussian distributions are employed to approximate the posterior probability distribution functions. The adoption of the decomposition approach circumvents the joint identification of all modal parameters, substantially reducing the parameter dimensions for optimization. Consequently, this strategy leads to decreased computational complexity and significantly improved computational stability. The effectiveness of the BSD is confirmed through simulated data generated from an 8‐story shear building as well as measured data collected from both an experimental shear frame and the Canton Tower. The results demonstrate that the proposed method achieves high accuracy in identifying modal parameters, greatly improves computational efficiency, and effectively quantifies the uncertainties in modal parameters.

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.

Second order autoregressive [AR(2)] model has been adopted in the likelihood function to calibrate the soil and water assessment tool (SWAT) model for the Canard River watershed, southwestern Ontario, Canada. The Bayesian approach is used for uncertainty analysis of SWAT modeling. The performance of AR(2) model for uncertainty estimation is evaluated by the index called Percentage of observations bracketed by the unit confidence interval (PUCI) for 95% confidence limits. The results are compared with the simple least square (SLS) method of calibration. In the SLS method, the modeling errors are assumed to be uncorrelated. The study reveals that the model parameter uncertainty is high and there exists local optimum values in the parameter space. The reliability of streamflow simulation uncertainty due to parameter uncertainty is increased when AR(2) model is implemented in the calibration process. The comparison of PUCI values between AR(2) method and SLS method shows that the estimation of streamf...

This paper reports the structural health monitoring benchmark study results for the Canton Tower using Bayesian methods. In this study, output-only modal identification and finite element model updating are considered using a given set of structural acceleration measurements and the corresponding ambient conditions of 24 hours. In the first stage, the Bayesian spectral density approach is used for output-only modal identification with the acceleration time histories as the excitation to the tower is unknown. The modal parameters and the associated uncertainty can be estimated through Bayesian inference. Uncertainty quantification is important for determination of statistically significant change of the modal parameters and for weighting assignment in the subsequent stage of model updating. In the second stage, a Bayesian model updating approach is utilized to update the finite element model of the tower. The uncertain stiffness parameters can be obtained by minimizing an objective function that is a weighted sum of the square of the differences (residuals) between the identified modal parameters and the corresponding values of the model. The weightings distinguish the contribution of different residuals with different uncertain levels. They are obtained using the Bayesian spectral density approach in the first stage. Again, uncertainty of the stiffness parameters can be quantified with Bayesian inference. Finally, this Bayesian framework is applied to the 24-hour field measurements to investigate the variation of the modal and stiffness parameters under changing ambient conditions. Results show that the Bayesian framework successfully achieves the goal of the first task of this benchmark study.

The remaining useful life (RUL) estimation is an important metric that helps in condition-based maintenance. Damage data obtained from the diagnostics techniques are often noisy, and the RUL estimated from the data is less reliable. Estimating the probabilistic RUL by quantifying the uncertainty in the predictive model parameters using the noisy data increases confidence in the predicted values. In this work, Bayesian inference is used to inversely quantify the uncertainty in the model parameters of the predictive damage model in adhesively bonded joints, and quantified parameter uncertainty is represented by a probability distribution function (PDF). It is difficult to obtain a closed-form solution for the RUL from the PDF of model parameters due to the nonlinearity of the damage models. Instead, statistical simulations are used to generate the model parameter samples from the PDF, and the RUL is estimated from the samples. In the statistical simulations, the predictive damage model is evaluated for each parameter sample. Using a physics-based model as the predictive model increases computational time, which makes the statistical techniques to be computationally expensive. It is essential to reduce the computational time to estimate the RUL in a feasible time. In this work, real-time probabilistic RUL estimation is demonstrated in adhesively bonded joints using the Sequential Monte Carlo (SMC) sampling method and cloud-based computations. The SMC sampling method is an alternative to traditional Markov Chain Monte Carlo (MCMC) methods, which enables generating statistical parameter samples in parallel. The parallel computational capabilities of the SMC methods are exploited by running the SMC simulation on multiple cloud calls. This approach is demonstrated by estimating fatigue RUL in the adhesively bonded joint. The accuracy of probabilistic RUL estimated by SMC is validated by comparing it with RUL estimated by the MCMC and the experimental values. The SMC simulation is run on the cloud and the computational speedup of the SMC is demonstrated.

This study aims at identifying the modal characteristics and their uncertainties for a smart composite beam. The problem is addressed via Vector AutoRegressive with eXogenous excitation (VARX) models. The advantages of VARX modeling include simplicity of implementation, high accuracy, parsimony of representation, and capability of handling modal uncertainties. Two different approaches to assess the modal parameter uncertainties are investigated. The first is based upon linearizing the function that relates the VARX model parameters with the modal parameters, whereas the second is based upon computer simulations using the Monte Carlo and the bootstrap schemes. The results indicate that VARX modeling captures the system dynamics and provides accurate modal parameters with tight confidence intervals.

The hazard and increased risk associated with slope failures in the natural habitat are attributed to uncertainty in soil parameters. These soil parameters, that is, cohesion, internal friction angle, and shear modulus, are estimated using the back-analysis method. The method incorporates machine learning algorithms such as particle swarm optimization and support vector machine. However, to capture the uncertainties in the soil parameters, probabilistic analysis is required. This study implements support vector machine for estimating soil parameters as the function of displacement in the soil. The Bayesian method is used to account for uncertainty in soil parameters. The Noto Hanto earthquake in 2007 and the consequent slope failure at Noto Yuryo Road, Japan, is considered as the case study for the proposed model. The finite-element method is used to simulate this slope failure. The soil parameters are treated as random variables to account for uncertainties in these parameters. The predicted displacement is comparable with the measured displacement in the field, which validates the application of the proposed model. The proposed method can efficiently predict experiment soil parameters by capturing uncertainty in soil parameters through a probabilistic approach.

Satellite‐based quantitative precipitation estimates (QPE), such as NASA's Integrated Multi‐satellitE Retrievals for GPM (IMERG), provide easily accessible continental‐to‐global precipitation forcings for flood prediction and other hydrologic applications. Nevertheless, when used in hydrologic prediction, uncertainty in satellite‐based QPE often leads to significant bias. This forcing uncertainty is further blended with other error sources, including process representation, parameter values, and their interactions. The identification and decoupling of these uncertainties can enhance our understanding of their respective impacts, thereby improving hydrologic prediction. Addressing this issue worldwide is challenging, however, largely due to the scarcity of precipitation ground truth and complex uncertainty interactions. Therefore, we propose an efficient uncertainty quantification framework for ensemble streamflow prediction, which keeps different uncertainty sources separable through hierarchical Bayesian inference. Satellite‐based QPE uncertainty is characterized by a novel near‐realtime quasi‐global satellite‐only ensemble precipitation data set (STREAM‐Sat), which is completely independent of ground‐based precipitation measurements. Model parameter uncertainty in a distributed physics‐based hydrologic model is inferred by an Iterative Ensemble Smoother (IES). To illustrate the impact and limitations of precipitation uncertainty, we compared ensemble streamflow predictions driven by both model parameter and satellite precipitation uncertainties and ensemble streamflow predictions driven by model parameter uncertainty and deterministic QPE. We demonstrate that the quantification of satellite‐based QPE uncertainty notably improves the accuracy and reliability of near‐realtime streamflow predictions in data scarce regions. This study also lays a foundation for satellite‐based streamflow prediction in ungauged regions.

Monitoring Structural Health Using a Probabilistic Measure

Identifying the burden of disease and its inequality between geographical regions is an important issue to study health priorities. Estimating burden of diseases using statistical models is inevitable especially in the context of rare data availability. To this purpose, the spatio-temporal model can provide a statistically sound approach for explaining the response variable observed over a region and various times. However, there are some methodological challenges in analysis of these complex data. Our primary objective is to provide some remedies to overcome these challenges. Data from nationally representative surveys and systematic reviews have been gathered across contiguous areal units over a period of more than 20 years (1990 - 2013). Generally, observations of areal units are spatially and temporally correlated in such a way that observations closer in space and time tend to be more correlated than observations farther away. It is critical to determine the correlation structure in space-time process which has been observed over a set of irregular regions. Moreover, these data sets are subject to high percentage of missing, including misaligned areal units, areas with small sample size, and may have nonlinear trends over space and time. Furthermore, the Gaussian assumption might be overly restrictive to represent the data. In this setting, the traditional statistical techniques are not appropriate and more flexible and comprehensive methodology is required. Particularly, we focus on approaches that allow extending spatio-temporal models proposed previously in the literature.Since statistical models include both continuous and categorical outcomes, we assume a latent variable framework for describing the underlying structure in mixed outcomes and use a conditionally autoregressive (CAR) prior for the random effects. In addition, we will employ misalignment modeling to combine incompatible areal units between data sources and/or over the years to obtain a unified clear picture of population health status over this period. In order to take parameter uncertainties into account, we pursue a Bayesian sampling-based inference. Hence, a hierarchical Bayes approach is constructed to model the data. The hierarchical structure enables us to "borrow information" from neighboring areal units to improve estimates for areas with missing values and small number of observations. For their general applicability and ease of implementation, the MCMC methods are the most adapted tool to perform Bayesian inference. This study aims to combine different available data sources and produce precise and reliable evidences for Iranian burden of diseases and risk factors and their disparities among geographical regions over time. Providing appropriate statistical methods and models for analyzing the data is undoubtedly crucial to circumvent the problems and obtain satisfactory estimates of model parameters and reach accurate assessment.

As all experimental procedures, Experimental Modal Analysis (EMA) is subject to a wide range of potential testing and processing errors. The modal identification methods are sensitive to these errors, yielding modal results which are uncertain up to certain error bounds. The question hence is what these error bounds on test data and modal parameters are. In this paper, the studied source of uncertainty is related to the variance (noise) on the Frequency Response Function (FRF) measurements. Under the H1 assumptions and in single-input cases, the FRF variances can be computed from the coherences and the FRFs. In multiple-input cases, some more measurement functions are required. Advanced system identification methods like the Maximum Likelihood Estimator (MLE) and PolyMAX Plus have the possibility to take the uncertainty on the measurement data into account and to propagate the data uncertainty to (modal) parameter uncertainty. This paper will review FRF variance estimation techniques, including some pragmatic approaches. The basic concepts of Maximum Likelihood Estimation and the calculation of confidence bounds will be discussed. Some typical structural testing and modal analysis cases will be used as illustration of the discussed concepts.

New techniques for the analysis of stochastic volatility models in which the logarithm of conditional variance follows an autoregressive model are developed. A cyclic Metropolis algorithm is used to construct a Markov-chain simulation tool. Simulations from this Markov chain converge in distribution to draws from the posterior distribution enabling exact finite-sample inference. The exact solution to the filtering/smoothing problem of inferring about the unobserved variance states is a by-product of our Markov-chain method. In addition, multistep-ahead predictive densities can be constructed that reflect both inherent model variability and parameter uncertainty. We illustrate our method by analyzing both daily and weekly data on stock returns and exchange rates. Sampling experiments are conducted to compare the performance of Bayes estimators to method of moments and quasi-maximum likelihood estimators proposed in the literature. In both parameter estimation and filtering, the Bayes estimators outperform these other approaches.

Development and comparison in uncertainty assessment based Bayesian modularization method in hydrological modeling

Abstract. The future of the Antarctic Ice Sheet in response to climate warming is one of the largest sources of uncertainty in estimates of future changes in global mean sea level (ΔGMSL). Mass loss is currently concentrated in regions of warm circumpolar deep water, but it is unclear how ice shelves currently surrounded by relatively cold ocean waters will respond to climatic changes in the future. Studies suggest that warm water could flush the Filchner–Ronne (FR) ice shelf cavity during the 21st century, but the inland ice sheet response to a drastic increase in ice shelf melt rates is poorly known. Here, we use an ice flow model and uncertainty quantification approach to project the GMSL contribution of the FR basin under RCP emissions scenarios, and we assess the forward propagation and proportional contribution of uncertainties in model parameters (related to ice dynamics and atmospheric/oceanic forcing) on these projections. Our probabilistic projections, derived from an extensive sample of the parameter space using a surrogate model, reveal that the FR basin is unlikely to contribute positively to sea level rise by the 23rd century. This is primarily due to the mitigating effect of increased accumulation with warming, which is capable of suppressing ice loss associated with ocean-driven increases in sub-shelf melt. Mass gain (negative ΔGMSL) from the FR basin increases with warming, but uncertainties in these projections also become larger. In the highest emission scenario RCP8.5, ΔGMSL is likely to range from −103 to 26 mm, and this large spread can be apportioned predominantly to uncertainties in parameters driving increases in precipitation (30 %) and sub-shelf melting (44 %). There is potential, within the bounds of our input parameter space, for major collapse and retreat of ice streams feeding the FR ice shelf, and a substantial positive contribution to GMSL (up to approx. 300 mm), but we consider such a scenario to be very unlikely. Adopting uncertainty quantification techniques in future studies will help to provide robust estimates of potential sea level rise and further identify target areas for constraining projections.