Modelling Under Risk and Uncertainty

Etienne De Rocquigny is Senior Research fellow in Statistics in Risk and Environment at Electricite’ de France R&D He has 12 years of R&D and consulting experience in risk and environmental management. He has had consulting appointments and R&D contracts worldwide with the World Bank, the IMF, and UN as part of Sogreah consulting engineers. He is Chairman of the European Safety and Reliability & Data Society and Chairman of a consortium of French Industries on Uncertainty and Industry.

“In my opinion, reviewed book is well organized textbook for risk management.”  (Zentralblatt MATH, 1 December 2012)

• Preface xv Acknowledgements xviiIntroduction and reading guide xixNotation xxxiiiAcronyms and abbreviations xxxvii1 Applications and practices of modelling, risk and uncertainty 11.1 Protection against natural risk 11.1.1 The popular ‘initiator/frequency approach’ 31.1.2 Recent developments towards an ‘extended frequency approach’ 51.2 Engineering design, safety and structural reliability analysis (SRA) 71.2.1 The domain of structural reliability 81.2.2 Deterministic safety margins and partial safety factors 91.2.3 Probabilistic structural reliability analysis 101.2.4 Links and differences with natural risk studies 111.3 Industrial safety, system reliability and probabilistic risk assessment (PRA) 121.3.1 The context of systems analysis 121.3.2 Links and differences with structural reliability analysis 141.3.3 The case of elaborate PRA (multi-state, dynamic) 161.3.4 Integrated probabilistic risk assessment (IPRA) 171.4 Modelling under uncertainty in metrology, environmental/sanitary assessment and numerical analysis 201.4.1 Uncertainty and sensitivity analysis (UASA) 211.4.2 Specificities in metrology/industrial quality control 231.4.3 Specificities in environmental/health impact assessment 241.4.4 Numerical code qualification (NCQ), calibration and data assimilation 251.5 Forecast and time-based modelling in weather, operations research, economics or finance 271.6 Conclusion: The scope for generic modelling under risk and uncertainty 281.6.1 Similar and dissimilar features in modelling, risk and uncertainty studies 281.6.2 Limitations and challenges motivating a unified framework 30References 312 A generic modelling framework 342.1 The system under uncertainty 342.2 Decisional quantities and goals of modelling under risk and uncertainty 372.2.1 The key concept of risk measure or quantity of interest 372.2.2 Salient goals of risk/uncertainty studies and decision-making 382.3 Modelling under uncertainty: Building separate system and uncertainty models 412.3.1 The need to go beyond direct statistics 412.3.2 Basic system models 422.3.3 Building a direct uncertainty model on variable inputs 452.3.4 Developing the underlying epistemic/aleatory structure 462.3.5 Summary 492.4 Modelling under uncertainty – the general case 502.4.1 Phenomenological models under uncertainty and residual model error 502.4.2 The model building process 512.4.3 Combining system and uncertainty models into an integrated statistical estimation problem 552.4.4 The combination of system and uncertainty models: A key information choice 572.4.5 The predictive model combining system and uncertainty components 592.5 Combining probabilistic and deterministic settings 602.5.1 Preliminary comments about the interpretations of probabilistic uncertainty models 602.5.2 Mixed deterministic-probabilistic contexts 612.6 Computing an appropriate risk measure or quantity of interest and associated sensitivity indices 642.6.1 Standard risk measures or q.i. (single-probabilistic) 652.6.2 A fundamental case: The conditional expected utility 672.6.3 Relationship between risk measures, uncertainty model and actions 682.6.4 Double probabilistic risk measures 692.6.5 The delicate issue of propagation/numerical uncertainty 712.6.6 Importance ranking and sensitivity analysis 712.7 Summary: Main steps of the studies and later issues 73Exercises 74References 753 A generic tutorial example: Natural risk in an industrial installation 773.1 Phenomenology and motivation of the example 773.1.1 The hydro component 783.1.2 The system’s reliability component 803.1.3 The economic component 833.1.4 Uncertain inputs, data and expertise available 843.2 A short introduction to gradual illustrative modelling steps 863.2.1 Step one: Natural risk standard statistics 873.2.2 Step two: Mixing statistics and a QRA model 893.2.3 Step three: Uncertainty treatment of a physical/engineering model (SRA) 913.2.4 Step four: Mixing SRA and QRA 913.2.5 Step five: Level-2 uncertainty study on mixed SRA-QRA model 943.2.6 Step six: Calibration of the hydro component and updating of risk measure 963.2.7 Step seven: Economic assessment and optimisation under risk and/or uncertainty 973.3 Summary of the example 99Exercises 101References 1014 Understanding natures of uncertainty, risk margins and time bases for probabilistic decision-making 1024.1 Natures of uncertainty: Theoretical debates and practical implementation 1034.1.1 Defining uncertainty – ambiguity about the reference 1034.1.2 Risk vs. uncertainty – an impractical distinction 1044.1.3 The aleatory/epistemic distinction and the issue of reducibility 1054.1.4 Variability or uncertainty – the need for careful system specification 1074.1.5 Other distinctions 1094.2 Understanding the impact on margins of deterministic vs. probabilistic formulations 1104.2.1 Understanding probabilistic averaging, dependence issues and deterministic maximisation and in the linear case 1104.2.2 Understanding safety factors and quantiles in the monotonous case 1144.2.3 Probability limitations, paradoxes of the maximal entropy principle 1174.2.4 Deterministic settings and interval computation – uses and limitations 1194.2.5 Conclusive comments on the use of probabilistic and deterministic risk measures 1204.3 Handling time-cumulated risk measures through frequencies and probabilities 1214.3.1 The underlying time basis of the state of the system 1214.3.2 Understanding frequency vs. probability 1244.3.3 Fundamental risk measures defined over a period of interest 1264.3.4 Handling a time process and associated simplifications 1284.3.5 Modelling rare events through extreme value theory 1304.4 Choosing an adequate risk measure – decision-theory aspects 1354.4.1 The salient goal involved 1354.4.2 Theoretical debate and interpretations about the risk measure when selecting between risky alternatives (or controlling compliance with a risk target) 1364.4.3 The choice of financial risk measures 1374.4.4 The challenges associated with using double-probabilistic or conditional probabilistic risk measures 1384.4.5 Summary recommendations 140Exercises 140References 1415 Direct statistical estimation techniques 1435.1 The general issue 1435.2 Introducing estimation techniques on independent samples 1475.2.1 Estimation basics 1475.2.2 Goodness-of-fit and model selection techniques 1505.2.3 A non-parametric method: Kernel modelling 1545.2.4 Estimating physical variables in the flood example 1575.2.5 Discrete events and time-based statistical models (frequencies, reliability models, time series) 1595.2.6 Encoding phenomenological knowledge and physical constraints inside the choice of input distributions 1635.3 Modelling dependence 1655.3.1 Linear correlations 1655.3.2 Rank correlations 1685.3.3 Copula model 1725.3.4 Multi-dimensional non-parametric modelling 1735.3.5 Physical dependence modelling and concluding comments 1745.4 Controlling epistemic uncertainty through classical or Bayesian estimators 1755.4.1 Epistemic uncertainty in the classical approach 1755.4.2 Classical approach for Gaussian uncertainty models (small samples) 1775.4.3 Asymptotic covariance for large samples 1795.4.4 Bootstrap and resampling techniques 1855.4.5 Bayesian-physical settings (small samples with expert judgement) 1865.5 Understanding rare probabilities and extreme value statistical modelling 1945.5.1 The issue of extrapolating beyond data – advantages and limitations of the extreme value theory 1945.5.2 The significance of extremely low probabilities 201Exercises 203References 2046 Combined model estimation through inverse techniques 2066.1 Introducing inverse techniques 2066.1.1 Handling calibration data 2066.1.2 Motivations for inverse modelling and associated literature 2086.1.3 Key distinctions between the algorithms: The representation of time and uncertainty 2106.2 One-dimensional introduction of the gradual inverse algorithms 2166.2.1 Direct least square calibration with two alternative interpretations 2166.2.2 Bayesian updating, identification and calibration 2236.2.3 An alternative identification model with intrinsic uncertainty 2256.2.4 Comparison of the algorithms 2276.2.5 Illustrations in the flood example 2296.3 The general structure of inverse algorithms: Residuals, identifiability, estimators, sensitivity and epistemic uncertainty 2336.3.1 The general estimation problem 2336.3.2 Relationship between observational data and predictive outputs for decision-making 2336.3.3 Common features to the distributions and estimation problems associated to the general structure 2366.3.4 Handling residuals and the issue of model uncertainty 2386.3.5 Additional comments on the model-building process 2426.3.6 Identifiability 2436.3.7 Importance factors and estimation accuracy 2496.4 Specificities for parameter identification, calibration or data assimilation algorithms 2516.4.1 The BLUE algorithm for linear Gaussian parameter identification 2516.4.2 An extension with unknown variance: Multidimensional model calibration 2546.4.3 Generalisations to non-linear calibration 2556.4.4 Bayesian multidimensional model updating 2566.4.5 Dynamic data assimilation 2576.5 Intrinsic variability identification 2606.5.1 A general formulation 2606.5.2 Linearised Gaussian case 2616.5.3 Non-linear Gaussian extensions 2636.5.4 Moment methods 2646.5.5 Recent algorithms and research fields 2646.6 Conclusion: The modelling process and open statistical and computing challenges 267Exercises 267References 2687 Computational methods for risk and uncertainty propagation 2717.1 Classifying the risk measure computational issues 2727.1.1 Risk measures in relation to conditional and combined uncertainty distributions 2737.1.2 Expectation-based single probabilistic risk measures 2757.1.3 Simplified integration of sub-parts with discrete inputs 2777.1.4 Non-expectation based single probabilistic risk measures 2807.1.5 Other risk measures (double probabilistic, mixed deterministic-probabilistic) 2817.2 The generic Monte-Carlo simulation method and associated error control 2837.2.1 Undertaking Monte-Carlo simulation on a computer 2837.2.2 Dual interpretation and probabilistic properties of Monte-Carlo simulation 2857.2.3 Control of propagation uncertainty: Asymptotic results 2907.2.4 Control of propagation uncertainty: Robust results for quantiles (Wilks formula) 2927.2.5 Sampling double-probabilistic risk measures 2987.2.6 Sampling mixed deterministic-probabilistic measures 2997.3 Classical alternatives to direct Monte-Carlo sampling 2997.3.1 Overview of the computation alternatives to MCS 2997.3.2 Taylor approximation (linear or polynomial system models) 3007.3.3 Numerical integration 3057.3.4 Accelerated sampling (or variance reduction) 3067.3.5 Reliability methods (FORM-SORM and derived methods) 3127.3.6 Polynomial chaos and stochastic developments 3167.3.7 Response surface or meta-models 3167.4 Monotony, regularity and robust risk measure computation 3177.4.1 Simple examples of monotonous behaviours 3177.4.2 Direct consequences of monotony for computing the risk measure 3197.4.3 Robust computation of exceedance probability in the monotonous case 3227.4.4 Use of other forms of system model regularity 3297.5 Sensitivity analysis and importance ranking 3307.5.1 Elementary indices and importance measures and their equivalence in linear system models 3307.5.2 Sobol sensitivity indices 3367.5.3 Specificities of Boolean input/output events – importance measures in risk assessment 3397.5.4 Concluding remarks and further research 3417.6 Numerical challenges, distributed computing and use of direct or adjoint differentiation of codes 342Exercises 342References 3438 Optimising under uncertainty: Economics and computational challenges 3478.1 Getting the costs inside risk modelling – from engineering economics to financial modelling 3478.1.1 Moving to costs as output variables of interest – elementary engineering economics 3478.1.2 Costs of uncertainty and the value of information 3518.1.3 The expected utility approach for risk aversion 3538.1.4 Non-linear transformations 3558.1.5 Robust design and alternatives mixing cost expectation and variance inside the optimisation procedure 3568.2 The role of time – cash flows and associated risk measures 3588.2.1 Costs over a time period – the cash flow model 3588.2.2 The issue of discounting 3618.2.3 Valuing time flexibility of decision-making and stochastic optimisation 3628.3 Computational challenges associated to optimisation 3668.3.1 Static optimisation (utility-based) 3678.3.2 Stochastic dynamic programming 3688.3.3 Computation and robustness challenges 3688.4 The promise of high performance computing 3698.4.1 The computational load of risk and uncertainty modelling 3698.4.2 The potential of high-performance computing 371Exercises 372References 3729 Conclusion: Perspectives of modelling in the context of risk and uncertainty and further research 3749.1 Open scientific challenges 3749.2 Challenges involved by the dissemination of advanced modelling in the context of risk and uncertainty 377References 37710 Annexes 37810.1 Annex 1 – refresher on probabilities and statistical modelling of uncertainty 37810.1.1 Modelling through a random variable 37810.1.2 The impact of data and the estimation uncertainty 38010.1.3 Continuous probabilistic distributions 38210.1.4 Dependence and stationarity 38210.1.5 Non-statistical approach of probabilistic modelling 38410.2 Annex 2 – comments about the probabilistic foundations of the uncertainty models 38610.2.1 The overall space of system states and the output space 38610.2.2 Correspondence to the Kaplan/Garrick risk analysis triplets 38910.2.3 The model and model input space 38910.2.4 Estimating the uncertainty model through direct data 39110.2.5 Model calibration and estimation through indirect data and inversion techniques 39310.3 Annex 3 – introductory reflections on the sources of macroscopic uncertainty 39410.4 Annex 4 – details about the pedagogical example 39710.4.1 Data samples 39710.4.2 Reference probabilistic model for the hydro component 39910.4.3 Systems reliability component – expert information on elementary failure probabilities 39910.4.4 Economic component – cost functions and probabilistic model 40310.4.5 Detailed results on various steps 40410.5 Annex 5 – detailed mathematical demonstrations 41410.5.1 Basic results about vector random variables and matrices 41410.5.2 Differentiation results and solutions of quadratic likelihood maximisation 41510.5.3 Proof of the Wilks formula 41910.5.4 Complements on the definition and chaining of monotony 42010.5.5 Proofs on level-2 quantiles of monotonous system models 42210.5.6 Proofs on the estimator of adaptive Monte-Carlo under monotony (section 7.4.3) 423References 426Epilogue 427Index 429