Stochastic Model Updating with Uncertainty Quantification: An Overview and Tutorial

This paper presents an overview of the theoretic framework of stochastic model updating, including critical aspects of model parameterisation, sensitivity analysis, surrogate modelling, test-analysis correlation, parameter calibration, etc. Special attention is paid to uncertainty analysis, which extends model updating from the deterministic domain to the stochastic domain. This extension is significantly promoted by uncertainty quantification metrics, no longer describing the model parameters as unknown-but-fixed constants but random variables with uncertain distributions, i.e. imprecise probabilities. As a result, the stochastic model updating no longer aims at a single model prediction with maximum fidelity to a single experiment, but rather a reduced uncertainty space of the simulation enveloping the complete scatter of multiple experiment data. Quantification of such an imprecise probability requires a dedicated uncertainty propagation process to investigate how the uncertainty space of the input is propagated via the model to the uncertainty space of the output. The two key aspects, forward uncertainty propagation and inverse parameter calibration, along with key techniques such as P-box propagation, statistical distance-based metrics, Markov chain Monte Carlo sampling, and Bayesian updating, are elaborated in this tutorial. The overall technical framework is demonstrated by solving the NASA Multidisciplinary UQ Challenge 2014, with the purpose of encouraging the readers to reproduce the result following this tutorial. The second practical demonstration is performed on a newly designed benchmark testbed, where a series of lab-scale aeroplane models are manufactured with varying geometry sizes, following pre-defined probabilistic distributions, and tested in terms of their natural frequencies and model shapes. Such a measurement database contains naturally not only measurement errors but also, more importantly, controllable uncertainties from the pre-defined distributions of the structure geometry. Finally, open questions are discussed to fulfil the motivation of this tutorial in providing researchers, especially beginners, with further directions on stochastic model updating with uncertainty treatment perspectives

Improving the convergence rate of seismic history matching with a proxy derived method to aid stochastic sampling

History matching is a very important activity during the continued development and management of petroleum reservoirs. Time-lapse (4D) seismic data provide information on the dynamics of fluids in reservoirs, relating variations of seismic signal to saturation and pressure changes. This information can be integrated with history matching to improve convergence towards a simulation model that predicts available data. The main aim of this thesis is to develop a method to speed up the convergence rate of assisted seismic history matching using proxy derived gradient method. Stochastic inversion algorithms often rely on simple assumptions for selecting new models by random processes. In this work, we improve the way that such approaches learn about the system they are searching and thus operate more efficiently. To this end, a new method has been developed called NA with Proxy derived Gradients (NAPG). To improve convergence, we use a proxy model to understand how parameters control the misfit and then use a global stochastic method with these sensitivities to optimise the search of the parameter space. This leads to an improved set of final reservoir models. These in turn can be used more effectively in reservoir management decisions. To validate the proposed approach, we applied the new approach on a number of analytical functions and synthetic cases. In addition, we demonstrate the proposed method by applying it to the UKCS Schiehallion field. The results show that the new method speeds up the rate of convergence by a factor of two to three generally. The performance of NAPG is much improved by updating the regression equation coefficients instead of keeping it fixed. In addition, we found that the initial number of models to start NAPG or NA could be reduced by using Experimental Design instead of using random initialization. Ultimately, with all of these approaches combined, the number of models required to find a good match reduced by an order of magnitude. We have investigated the criteria for stopping the SHM loop, particularly the use of a proxy model to help. More research is needed to complete this work but the approach is promising. Quantifying parameter uncertainty using NA and NAPG was studied using the NA-Bayes approach (NAB). We found that NAB is very sensitive to misfit magnitude but otherwise NA and NAPG produce similar uncertainty measures

A Geometric Variational Approach to Bayesian Inference

We propose a novel Riemannian geometric framework for variational inference in Bayesian models based on the nonparametric Fisher-Rao metric on the manifold of probability density functions. Under the square-root density representation, the manifold can be identified with the positive orthant of the unit hypersphere in L2, and the Fisher-Rao metric reduces to the standard L2 metric. Exploiting such a Riemannian structure, we formulate the task of approximating the posterior distribution as a variational problem on the hypersphere based on the alpha-divergence. This provides a tighter lower bound on the marginal distribution when compared to, and a corresponding upper bound unavailable with, approaches based on the Kullback-Leibler divergence. We propose a novel gradient-based algorithm for the variational problem based on Frechet derivative operators motivated by the geometry of the Hilbert sphere, and examine its properties. Through simulations and real-data applications, we demonstrate the utility of the proposed geometric framework and algorithm on several Bayesian models

Uncertainty in Engineering

This open access book provides an introduction to uncertainty quantification in engineering. Starting with preliminaries on Bayesian statistics and Monte Carlo methods, followed by material on imprecise probabilities, it then focuses on reliability theory and simulation methods for complex systems. The final two chapters discuss various aspects of aerospace engineering, considering stochastic model updating from an imprecise Bayesian perspective, and uncertainty quantification for aerospace flight modelling. Written by experts in the subject, and based on lectures given at the Second Training School of the European Research and Training Network UTOPIAE (Uncertainty Treatment and Optimization in Aerospace Engineering), which took place at Durham University (United Kingdom) from 2 to 6 July 2018, the book offers an essential resource for students as well as scientists and practitioners

Hydrological post-processing based on approximate Bayesian computation (ABC)

[EN] This study introduces a method to quantify the conditional predictive uncertainty in hydrological post-processing contexts when it is cumbersome to calculate the likelihood (intractable likelihood). Sometimes, it can be difficult to calculate the likelihood itself in hydrological modelling, specially working with complex models or with ungauged catchments. Therefore, we propose the ABC post-processor that exchanges the requirement of calculating the likelihood function by the use of some sufficient summary statistics and synthetic datasets. The aim is to show that the conditional predictive distribution is qualitatively similar produced by the exact predictive (MCMC post-processor) or the approximate predictive (ABC post-processor). We also use MCMC post-processor as a benchmark to make results more comparable with the proposed method. We test the ABC post-processor in two scenarios: (1) the Aipe catchment with tropical climate and a spatially-lumped hydrological model (Colombia) and (2) the Oria catchment with oceanic climate and a spatially-distributed hydrological model (Spain). The main finding of the study is that the approximate (ABC post-processor) conditional predictive uncertainty is almost equivalent to the exact predictive (MCMC post-processor) in both scenarios.This study was partially supported by the Departamento del Huila Scholarship Program No. 677 (Colombia) and Colciencias, by the Spanish Research Project TETIS-MED (ref. CGL2014-58127-C3-3-R) and TETIS-CHANGE (ref.RTI2018-093717-B-I00). Also, G. Adelfio's research has been supported by the national grant of the Italian Ministry of Education University and Research (MIUR) for the PRIN-2015 program, "Complex space-time modelling and functional analysis for probabilistic forecast of seismic events'. The authors also wish to thank the editor and the two anonymous reviewers for their thoughtful comments for the revision of the manuscript.Romero-Cuellar, J.; Abbruzzo, A.; Adelfio, G.; Francés, F. (2019). Hydrological post-processing based on approximate Bayesian computation (ABC). Stochastic Environmental Research and Risk Assessment. 33(7):1361-1373. https://doi.org/10.1007/s00477-019-01694-yS13611373337Beaumont MA, Zhang W, Balding DJ (2002) Approximate Bayesian computation in population genetics. Genetics 162(4):2025–2035Blackwell D, Dubins L (1962) Merging of opinions with increasing information. Ann Math Stat 33(3):882–886Bogner K, Liechti K, Zappa M (2016) Post-processing of stream flows in Switzerland with an emphasis on low flows and floods. Water 8(4):115Brown JD, Seo D-J (2010) A nonparametric postprocessor for bias correction of hydrometeorological and hydrologic ensemble forecasts. J Hydrometeorol 11(3):642–665Butts MB, Payne JT, Kristensen M, Madsen H (2004) An evaluation of the impact of model structure on hydrological modelling uncertainty for streamflow simulation. J Hydrol 298(1):242–266Coccia G, Todini E (2011) Recent developments in predictive uncertainty assessment based on the model conditional processor approach. Hydrol Earth Syst Sci 15:3253–3274Csillery K, Francois O, Blum MGB (2012) abc: an R package for approximate Bayesian computation (abc). Methods Ecol Evol 3:475–479Diaconis P, Freedman D (1986) On the consistency of bayes estimates. Ann Stat 14(1):1–26Diks CGH, Vrugt JA (2010) Comparison of point forecast accuracy of model averaging methods in hydrologic applications. Stoch Environ Res Risk Assess 24(6):809–820Drovandi CC, Pettitt AN (2011) Likelihood-free Bayesian estimation of multivariate quantile distributions. Comput Stat Data Anal 55(9):2541–2556Evin G, Thyer M, Kavetski D, McInerney D, Kuczera G (2014) Comparison of joint versus postprocessor approaches for hydrological uncertainty estimation accounting for error autocorrelation and heteroscedasticity. Water Resour Res 50(3):2350–2375Fearnhead P, Prangle D (2012) Constructing summary statistics for approximate bayesian computation: semi-automatic approximate Bayesian computation. J R Stat Soc Ser B Stat Methodol 74(3):419–474Fenicia F, Kavetski D, Reichert P, Albert C (2018) Signature-domain calibration of hydrological models using approximate Bayesian computation: empirical analysis of fundamental properties. Water Resour Res 54:3958–3987Francés F, Vélez JI, Vélez JJ (2007) Split-parameter structure for the automatic calibration of distributed hydrological models. J Hydrol 332(1):226–240Frazier DT, Maneesoonthorn W, Martin GM, McCabe BP (2019) Approximate Bayesian forecasting. Int J Forecast 35(2):521–539Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7(4):457–472Gelman A, Stern HS, Carlin JB, Dunson DB, Vehtari A, Rubin DB (2013) Bayesian data analysis. Chapman and Hall/CRC, Boca RatonGlahn HR, Lowry DA (1972) The use of model output statistics (mos) in objective weather forecasting. J Appl Meteorol 11(8):1203–1211Gupta HV, Kling H, Yilmaz KK, Martinez GF (2009) Decomposition of the mean squared error and nse performance criteria: implications for improving hydrological modelling. J Hydrol 377(1):80–91Haario H, Saksman E, Tamminen J (2001) An adaptive metropolis algorithm. Bernoulli 7(2):223–242Kavetski D, Fenicia F, Reichert P, Albert C (2018) Signature-domain calibration of hydrological models using approximate Bayesian computation: theory and comparison to existing applications. Water Resour Res 54:4059–4083Khajehei S, Moradkhani H (2017) Towards an improved ensemble precipitation forecast: a probabilistic post-processing approach. J Hydrol 546:476–489Klein B, Meissner D, Kobialka H-U, Reggiani P (2016) Predictive uncertainty estimation of hydrological multi-model ensembles using pair-copula construction. Water 8(4):125Krzysztofowicz R, Kelly KS (2000) Hydrologic uncertainty processor for probabilistic river stage forecasting. Water Resour Res 36(11):3265–3277Laio F, Tamea S (2007) Verification tools for probabilistic forecasts of continuous hydrological variables. Hydrol Earth Syst Sci 11(4):1267–1277Li B, Liang Z, He Y, Hu L, Zhao W, Acharya K (2017) Comparison of parameter uncertainty analysis techniques for a topmodel application. Stoch Environ Res Risk Assess 31(5):1045–1059Liang Z, Chang W, Li B (2012) Bayesian flood frequency analysis in the light of model and parameter uncertainties. Stoch Environ Res Risk Assess 26(5):721–730Lindley DV, Smith AFM (1972) Bayes estimates for the linear model. J R Stat Soc Ser B Methodol 34(1):1–41Liu Y, Gupta HV (2007) Uncertainty in hydrologic modeling: toward an integrated data assimilation framework. Water Resour Res 43(7):W07401Madadgar S, Moradkhani H (2014) Improved Bayesian multimodeling: integration of copulas and Bayesian model averaging. Water Resour Res 50(12):9586–9603Marin J-M, Pudlo P, Robert CP, Ryder RJ (2012) Approximate Bayesian computational methods. Stat Comput 22(6):1167–1180Marjoram P, Molitor J, Plagnol V, Tavaré S (2003) Markov chain monte carlo without likelihoods. Proc Natl Acad Sci 100(26):15324–15328Marshall L, Nott D, Sharma A (2004) A comparative study of Markov chain Monte Carlo methods for conceptual rainfall-runoff modeling. Water Resour Res 40(2):W02501Mengersen KL, Pudlo P, Robert CP (2013) Bayesian computation via empirical likelihood. Proc Natl Acad Sci 110(4):1321–1326Montanari A, Brath A (2004) A stochastic approach for assessing the uncertainty of rainfall-runoff simulations. Water Resour Res 40:W01106. https://doi.org/10.1029/2003WR002540Montanari A, Grossi G (2008) Estimating the uncertainty of hydrological forecasts: a statistical approach. Water Resour Res 44:W00B08. https://doi.org/10.1029/2008WR006897Montanari A, Koutsoyiannis D (2012) A blueprint for process-based modeling of uncertain hydrological systems. Water Resour Res 48(9):W09555Moriasi DN, Arnold JG, Van Liew MW, Bingner RL, Harmel RD, Veith TL (2007) Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans ASABE 50(3):885–900Nott DJ, Marshall L, Brown J (2011) Generalized likelihood uncertainty estimation (glue) and approximate Bayesian computation: what’s the connection? Water Resour Res 48(12):W12602Price LF, Drovandi CC, Lee A, Nott DJ (2018) Bayesian synthetic likelihood. J Comput Graph Stat 27(1):1–11Pritchard JK, Seielstad MT, Perez-Lezaun A, Feldman MW (1999) Population growth of human y chromosomes: a study of y chromosome microsatellites. Mol Biol Evol 16(12):1791–1798R Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, AustriaRaftery AE, Gneiting T, Balabdaoui F, Polakowski M (2005) Using Bayesian model averaging to calibrate forecast ensembles. Mon Weather Rev 133(5):1155–1174Reichert P, Langhans SD, Lienert J, Schuwirth N (2015) The conceptual foundation of environmental decision support. J Environ Manag 154:316–332Robert CP (2016) Approximate bayesian computation: A survey on recent results. In: Cools R, Nuyens D (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer, Cham, pp 185–205Romero-Cuéllar J, Buitrago-Vargas A, Quintero-Ruiz T, Francés F (2018) Modelling the potential impacts of climate change on the hydrology of the Aipe river basin in Huila, Colombia. Ribagua 5(1):63–78Schefzik R, Thorarinsdottir TL, Gneiting T (2013) Uncertainty quantification in complex simulation models using ensemble copula coupling. Stat Sci 28(4):616–640Schoups G, Vrugt JA (2010) A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and non-Gaussian errors. Water Resour Res 46(10):W10531Schoups G, van de Giesen NC, Savenije HHG (2008) Model complexity control for hydrologic prediction. Water Resour Res 44(12):W00B03Shafii M, Tolson B, Matott LS (2014) Uncertainty-based multi-criteria calibration of rainfall-runoff models: a comparative study. Stoch Environ Res Risk Assess 28(6):1493–1510Sikorska AE, Montanari A, Koutsoyiannis D (2015) Estimating the uncertainty of hydrological predictions through data-driven resampling techniques. J Hydrol Eng 20(1):A4014009Sisson SA, Fan Y, Tanaka MM (2007) Sequential Monte Carlo without likelihoods. Proc Natl Acad Sci 104(6):1760–1765Solomatine DP, Shrestha DL (2009) A novel method to estimate model uncertainty using machine learning techniques. Water Resour Res 45:W00B11. https://doi.org/10.1029/2008WR006839Tavaré S, Balding DJ, Griffiths RC, Donnelly P (1997) Inferring coalescence times from DNA sequence data. Genetics 145(2):505–518Thomas H (1981) Improved methods for national water assessment, water resources contract: WR15249270. Technical report, Harvard University, CambridgeThyer M, Renard B, Kavetski D, Kuczera G, Franks SW, Srikanthan S (2009) Critical evaluation of parameter consistency and predictive uncertainty in hydrological modeling: a case study using Bayesian total error analysis. Water Resour Res 45:W00B14. https://doi.org/10.1029/2008WR006825Tian Y, Nearing GS, Peters-Lidard CD, Harrison KW, Tang L (2016) Performance metrics, error modeling, and uncertainty quantification. Mon Weather Rev 144(2):607–613Todini E (2008) A model conditional processor to assess predictive uncertainty in flood forecasting. Int J River Basin Manag 6(2):123–137Tran M-N, Nott DJ, Kohn R (2017) Variational bayes with intractable likelihood. J Comput Graph Stat 26(4):873–882Turner BM, Van Zandt T (2012) A tutorial on approximate Bayesian computation. J Math Psychol 56(2):69–85van Oijen M (2017) Bayesian methods for quantifying and reducing uncertainty and error in forest models. Curr For Rep 3(4):269–280Vélez JJ, Puricelli M, López Unzu F, Francés F (2009) Parameter extrapolation to ungauged basins with a hydrological distributed model in a regional framework. Hydrol Earth Syst Sci 13(2):229–246Vrugt JA, Robinson BA (2007) Treatment of uncertainty using ensemble methods: comparison of sequential data assimilation and Bayesian model averaging. Water Resour Res 43(1):W01411Vrugt JA, Sadegh M (2013) Toward diagnostic model calibration and evaluation: approximate Bayesian computation. Water Resour Res 49:4335–4345Waerden BVD (1953) Order tests for the two-sample problem and their power. Indag Math Proc 56:80Wagener T, Gupta HV (2005) Model identification for hydrological forecasting under uncertainty. Stoch Environ Res Risk Assess 19(6):378–387Wang Q, Robertson D, Chiew FS (2009) A bayesian joint probability modeling approach for seasonal forecasting of streamflows at multiple sites. Water Resour Res 45(5):W05407Weerts AH, Winsemius HC, Verkade JS (2011) Estimation of predictive hydrological uncertainty using quantile regression: examples from the national flood forecasting system (england and wales). Hydrol Earth Syst Sci 15(1):255–265Wentao L, Qingyun D, Chiyuan M, Aizhong Y, Wei G, Zhenhua D (2017) A review on statistical postprocessing methods for hydrometeorological ensemble forecasting. Wiley Interdiscip Rev Water 4(6):e1246Wilby RL, Harris I (2006) A framework for assessing uncertainties in climate change impacts: low-flow scenarios for the river thames, UK. Water Resour Res 42(2):W02419Woldemeskel F, McInerney D, Lerat J, Thyer M, Kavetski D, Shin D, Tuteja N, Kuczera G (2018) Evaluating post-processing approaches for monthly and seasonal streamflow forecasts. Hydrol Earth Syst Sci 22:6257–6278. https://doi.org/10.5194/hess-22-6257-2018Ye A, Duan Q, Yuan X, Wood EF, Schaake J (2014) Hydrologic post-processing of MOPEX streamflow simulations. J Hydrol 508:147–156Yoon S, Cho W, Heo J-H, Kim CE (2010) A full bayesian approach to generalized maximum likelihood estimation of generalized extreme value distribution. Stoch Environ Res Risk Assess 24(5):761–770Zhang X, Zhao K (2012) Bayesian neural networks for uncertainty analysis of hydrologic modeling: a comparison of two schemes. Water Resour Manag 26(8):2365–2382Zhao L, Duan Q, Schaake J, Ye A, Xia J (2011) A hydrologic post-processor for ensemble streamflow predictions. Adv Geosci 29:51–59Zhu W, Marin JM, Leisen F (2016) A bootstrap likelihood approach to Bayesian computation. Aust N Z J Stat 58(2):227–24

Real-time people tracking in a camera network

Visual tracking is a fundamental key to the recognition and analysis of human behaviour. In this thesis we present an approach to track several subjects using multiple cameras in real time. The tracking framework employs a numerical Bayesian estimator, also known as a particle lter, which has been developed for parallel implementation on a Graphics Processing Unit (GPU). In order to integrate multiple cameras into a single tracking unit we represent the human body by a parametric ellipsoid in a 3D world. The elliptical boundary can be projected rapidly, several hundred times per subject per frame, onto any image for comparison with the image data within a likelihood model. Adding variables to encode visibility and persistence into the state vector, we tackle the problems of distraction and short-period occlusion. However, subjects may also disappear for longer periods due to blind spots between cameras elds of view. To recognise a desired subject after such a long-period, we add coloured texture to the ellipsoid surface, which is learnt and retained during the tracking process. This texture signature improves the recall rate from 60% to 70-80% when compared to state only data association. Compared to a standard Central Processing Unit (CPU) implementation, there is a signi cant speed-up ratio

Detection of regulator genes and eQTLs in gene networks

Genetic differences between individuals associated to quantitative phenotypic traits, including disease states, are usually found in non-coding genomic regions. These genetic variants are often also associated to differences in expression levels of nearby genes (they are "expression quantitative trait loci" or eQTLs for short) and presumably play a gene regulatory role, affecting the status of molecular networks of interacting genes, proteins and metabolites. Computational systems biology approaches to reconstruct causal gene networks from large-scale omics data have therefore become essential to understand the structure of networks controlled by eQTLs together with other regulatory genes, and to generate detailed hypotheses about the molecular mechanisms that lead from genotype to phenotype. Here we review the main analytical methods and softwares to identify eQTLs and their associated genes, to reconstruct co-expression networks and modules, to reconstruct causal Bayesian gene and module networks, and to validate predicted networks in silico.Comment: minor revision with typos corrected; review article; 24 pages, 2 figure