JMASM 51: Bayesian Reliability Analysis of Binomial Model – Application to Success/Failure Data

Reliability data are generated in the form of success/failure. An attempt was made to model such type of data using binomial distribution in the Bayesian paradigm. For fitting the Bayesian model both analytic and simulation techniques are used. Laplace approximation was implemented for approximating posterior densities of the model parameters. Parallel simulation tools were implemented with an extensive use of R and JAGS. R and JAGS code are developed and provided. Real data sets are used for the purpose of illustration

Probabilistic Identification and Prognosis of Nonlinear Dynamic Systems with applications in Structural Control and Health Monitoring

A Bayesian approach to system identification for structural control and health monitoring contains three main levels of inference, namely model assessment, joint state/parameter estimation and noise estimation. All of them have individually, or as a whole, been studied extensively for offline applications. In an online setting, the middle level of inference (joint state/parameter estimation) is performed using various algorithms such as the Kalman filter (KF), the extended Kalman filter (EKF), the Unscented Kalman filter (UKF), or particle filter (PF) methods. This problem has been explored in depth for structural dynamics. This dissertation focuses on the other two levels of inference, in particular on developing methods to perform them online, simultaneously to the joint state/parameter estimation. The quality of structural parameter estimates depends heavily on the choice of noise characteristics involved in the aforementioned online inference algorithms, hence the need for simultaneous online noise estimation. Model assessment, on the other hand, is an integral part of many engineering applications, since any analytical or numerical mathematical model used for predictive purposes is only an approximation of the real system. An online implementation of model assessment is valuable, amongst others, for structural control applications, and for identifying several models in parallel, some of which might be of deteriorating nature, thus generating some sort of alert. The performance of the proposed online techniques is evaluated using simulated and experimental data sets generated by nonlinear hysteretic systems. Upon completion of the study of hierarchical online system identification (diagnostic phase/estimation), a system/damage prognostic analysis (prognostic phase/prediction) is attempted using a gamma deterioration process. Prognostic analysis is still at a relatively early stage of development in the field of structural dynamics, but it can potentially provide useful insights regarding the lifetime of a dynamically excited structural system. The technique is evaluated on a data set recorded during an experiment involving a full-scale bridge pier under base excitation, tested to impending collapse

Forward uncertainty quantification with special emphasis on a Bayesian active learning perspective

Uncertainty quantification (UQ) in its broadest sense aims at quantitatively studying all sources of uncertainty arising from both computational and real-world applications. Although many subtopics appear in the UQ field, there are typically two major types of UQ problems: forward and inverse uncertainty propagation. The present study focuses on the former, which involves assessing the effects of the input uncertainty in various forms on the output response of a computational model. In total, this thesis reports nine main developments in the context of forward uncertainty propagation, with special emphasis on a Bayesian active learning perspective. The first development is concerned with estimating the extreme value distribution and small first-passage probabilities of uncertain nonlinear structures under stochastic seismic excitations, where a moment-generating function-based mixture distribution approach (MGF-MD) is proposed. As the second development, a triple-engine parallel Bayesian global optimization (T-PBGO) method is presented for interval uncertainty propagation. The third contribution develops a parallel Bayesian quadrature optimization (PBQO) method for estimating the response expectation function, its variable importance and bounds when a computational model is subject to hybrid uncertainties in the form of random variables, parametric probability boxes (p-boxes) and interval models. In the fourth research, of interest is the failure probability function when the inputs of a performance function are characterized by parametric p-boxes. To do so, an active learning augmented probabilistic integration (ALAPI) method is proposed based on offering a partially Bayesian active learning perspective on failure probability estimation, as well as the use of high-dimensional model representation (HDMR) technique. Note that in this work we derive an upper-bound of the posterior variance of the failure probability, which bounds our epistemic uncertainty about the failure probability due to a kind of numerical uncertainty, i.e., discretization error. The fifth contribution further strengthens the previously developed active learning probabilistic integration (ALPI) method in two ways, i.e., enabling the use of parallel computing and enhancing the capability of assessing small failure probabilities. The resulting method is called parallel adaptive Bayesian quadrature (PABQ). The sixth research presents a principled Bayesian failure probability inference (BFPI) framework, where the posterior variance of the failure probability is derived (not in closed form). Besides, we also develop a parallel adaptive-Bayesian failure probability learning (PA-BFPI) method upon the BFPI framework. For the seventh development, we propose a partially Bayesian active learning line sampling (PBAL-LS) method for assessing extremely small failure probabilities, where a partially Bayesian active learning insight is offered for the classical LS method and an upper-bound for the posterior variance of the failure probability is deduced. Following the PBAL-LS method, the eighth contribution finally obtains the expression of the posterior variance of the failure probability in the LS framework, and a Bayesian active learning line sampling (BALLS) method is put forward. The ninth contribution provides another Bayesian active learning alternative, Bayesian active learning line sampling with log-normal process (BAL-LS-LP), to the traditional LS. In this method, the log-normal process prior, instead of a Gaussian process prior, is assumed for the beta function so as to account for the non-negativity constraint. Besides, the approximation error resulting from the root-finding procedure is also taken into consideration. In conclusion, this thesis presents a set of novel computational methods for forward UQ, especially from a Bayesian active learning perspective. The developed methods are expected to enrich our toolbox for forward UQ analysis, and the insights gained can stimulate further studies

A multi-fidelity Bayesian framework for robust seismic fragility analysis

Fragility analysis of structures via numerical methods involves a complex trade-off between the desired accuracy, the explicit consideration of uncertainties (both epistemic and aleatory) related to the numerical structural model and the available computational performance. This paper introduces a framework for deriving numerical fragility relationships based on multi-fidelity non-linear models of the structure under investigation and response-analysis types. The proposed framework aims to reduce the computational burden while achieving a desired accuracy of the fragility estimates without neglecting aleatory and epistemic uncertainties. The proposed approach is an extension of the well-known robust fragility (RF) analysis framework. Different model classes, each characterised by increasing refinement, are used to define multi-fidelity polynomial expansions of the fragility model parameters. Each analysis result is then considered as a ‘new observation’ in a Bayesian framework and used to update the coefficients of the polynomial expansions. An adaptive sampling algorithm is also proposed to futher improve the performance of the multi-fidelity framework. Specifically, such an adaptive sampling algorithm relies on partitioning the sample space and the Kullback–Leibler divergence to find the optimal sampling path. The sample space partitioning allows an analyst to specify different criteria and parameters of the algorithm for different regions, thus further improving the performance of the procedure. The proposed approach is illustrated for an archetype reinforced concrete (RC) frame for which two model classes are developed/analysed: the simple lateral mechanism analysis (SLaMA), coupled with the capacity spectrum method, and non-linear dynamic analysis. Both model classes involve a cloud-based approach employing unscaled real (i.e. recorded) ground motions. The fragility relationships derived with the proposed procedure are finally compared to those calculated by using only the most advanced/high-fidelity (HF) model class, thus quantifying the performance of the proposed approach and highlighting further research needs

Observing the Observer (II): Deciding When to Decide

In a companion paper [1], we have presented a generic approach for inferring how subjects make optimal decisions under uncertainty. From a Bayesian decision theoretic perspective, uncertain representations correspond to “posterior” beliefs, which result from integrating (sensory) information with subjective “prior” beliefs. Preferences and goals are encoded through a “loss” (or “utility”) function, which measures the cost incurred by making any admissible decision for any given (hidden or unknown) state of the world. By assuming that subjects make optimal decisions on the basis of updated (posterior) beliefs and utility (loss) functions, one can evaluate the likelihood of observed behaviour. In this paper, we describe a concrete implementation of this meta-Bayesian approach (i.e. a Bayesian treatment of Bayesian decision theoretic predictions) and demonstrate its utility by applying it to both simulated and empirical reaction time data from an associative learning task. Here, inter-trial variability in reaction times is modelled as reflecting the dynamics of the subjects' internal recognition process, i.e. the updating of representations (posterior densities) of hidden states over trials while subjects learn probabilistic audio-visual associations. We use this paradigm to demonstrate that our meta-Bayesian framework allows for (i) probabilistic inference on the dynamics of the subject's representation of environmental states, and for (ii) model selection to disambiguate between alternative preferences (loss functions) human subjects could employ when dealing with trade-offs, such as between speed and accuracy. Finally, we illustrate how our approach can be used to quantify subjective beliefs and preferences that underlie inter-individual differences in behaviour