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Characteristic values of spatially varying material properties in existing structures
Published in Airong Chen, Xin Ruan, Dan M. Frangopol, Life-Cycle Civil Engineering: Innovation, Theory and Practice, 2021
S. Geyer, I. Papaioannou, D. Straub
This contribution presents a study on a novel approach for calculating characteristic values of spatially variable parameters. A hierarchical Bayesian model is used to derive the point-wise marginal distributions accounting for measurements and their location. By appropriate choice of the prior distribution and likelihood function, the posterior predictive distribution can be calculated analytically at any location of the domain of interest. Characteristic values to be used for structural verification are then directly available through applying the inverse cumulative distribution function on the posterior predictive distribution. Comparison with the commonly used approach to calculate characteristic values from available samples as included in Annex D.7 of EN 1990 shows that the proposed approach can result in lower characteristic values at certain locations. However, it is more realistic as it accounts for the spatial correlation of the measurements and thus avoids the over-estimation of the available amount of information. Furthermore, the illustrative nature of spatial effects when using the proposed model can be used for efficient planning of inspections in the future to maximize the information gained from measurements. Additional studies are necessary for choosing an informative prior distribution and on the appropriateness of the chosen spatial correlation model. Especially the scale of fluctuation and the assumed isotropic behavior need to be investigated in more detail before using the method for practical applications, as these parameters control the spatial fluctuation of the random field and thus the correlation to and among the measurements.
Improvement of the risk-based approach for evaluation of permanently plugged and abandoned oil and gas wells
Published in Stein Haugen, Anne Barros, Coen van Gulijk, Trond Kongsvik, Jan Erik Vinnem, Safety and Reliability – Safe Societies in a Changing World, 2018
H. Langdalen, E.B. Abrahamsen, J.T. Selvik, H.P. Lohne
By virtue of Bayes’ theorem, the posterior predictive distribution is based on the posterior distribution of some parameters θ. The true values of such parameters are unknown. By using an alternative prior distribution, or probability distribution, a different posterior predictive distribution would be generated. Paramount for the assessment is to understand the basis for the resulting predictive distribution. Other technical difficulties with Bayesian analysis are summarized by e.g. Ferson (2005).
Response surfaces, blocking, and split plots: A predictive distribution case study
Published in Quality Engineering, 2023
Before proceeding with specific inferences, it is prudent to do a posterior predictive check to see if the posterior predictions are in reasonable alignment with the actual observations. Figure 3 shows a 3-by-3 lattice panel of overlay scatter plots, one for each of the nine experimental design points. The actual observation-pair points are in (solid) black, while the posterior prediction points are gray. Here, there are thirty posterior prediction points for each panel plot. The swarms of posterior predictive points seem to hover around the actual points reasonably well. However, the posterior predictive response points appear to have a somewhat wider spread. This is because each posterior predicted response represents a new future observation involving a new oven run and a new batch of raw materials. In addition, the posterior predictive distribution also encompasses the model parameter uncertainty.
A Bayesian hierarchical model for quantitative and qualitative responses
Published in Journal of Quality Technology, 2018
Lulu Kang, Xiaoning Kang, Xinwei Deng, Ran Jin
To measure the prediction accuracy we consider the root mean squared prediction error for the quantitative response Y, where represents the predicted value (the mean, median, or mode of the posterior predictive distribution) of yi in the testing data set. The prediction performance for the qualitative response Z is measured by the misclassification error , where is the predicted value of zi and I( · ) is an indicator function. The prediction value is set as one if the corresponding posterior predictive sample mean or median of Pr ( is the MCMC chain) is larger than 0.5.
A Bayesian approach to reliability of MSE walls
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2021
Nezam Bozorgzadeh, Richard J. Bathurst
Continuing with the discussion of Bayesian data analysis, the posterior predictive distribution:is the distribution of future observations of the data (y*) given the current state of knowledge, i.e. conditional on already observed data. This integrates over the estimated but uncertain parameters θ. Similar to this concept, Der Kiureghian (2008) defines the predictive failure probability as the expectation of the conditional failure probability p(Θ) over the outcome space of the uncertain parameters Θ: