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Modelling Under Risk & Uncertainty: An Introduction to Statistical, Phenomenological & Computational Methods Book

Modelling Under Risk & Uncertainty: An Introduction to Statistical, Phenomenological & Computational Methods
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Modelling Under Risk & Uncertainty: An Introduction to Statistical, Phenomenological & Computational Methods, Modelling Under Risk and Uncertainty goes beyond the 'black-box' view that some risk analysts or statisticians develop the underlying phenomenology of the environmental or industrial processes, without valuing enough their physical properties and i, Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods
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  • Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods
  • Written by author Etienne de Rocquigny
  • Published by Wiley, John & Sons, Incorporated, 5/8/2012
  • Modelling Under Risk and Uncertainty goes beyond the 'black-box' view that some risk analysts or statisticians develop the underlying phenomenology of the environmental or industrial processes, without valuing enough their physical properties and i
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Authors

Preface 3

Acknowledgments 4

Table of Contents 5

Introduction and reading guide 8

Notation 22

Acronyms and abbreviations 25

1 Applications and practices of modelling, risk and uncertainty 26

1.1 Protection against natural risk 26

1.1.1 The popular “initiator / frequency approach” 27

1.1.2 Recent developments towards an “extended frequency approach” 30

1.2 Engineering design, safety and structural reliability analysis (SRA) 32

1.2.1 The domain of structural reliability 32

1.2.2 Deterministic safety margins and partial safety factors 33

1.2.3 Probabilistic structural reliability analysis 35

1.2.4 Links and differences with natural risk studies 36

1.3 Industrial safety, system reliability & probabilistic risk assessment (PRA) 36

1.3.1 The context of systems analysis 37

1.3.2 Links and differences with Structural Reliability Analysis 39

1.3.3 The case of elaborate PRA (multi-state, dynamic) 40

1.3.4 Integrated Probabilistic Risk Assessment (IPRA) 41

1.4 Modelling under uncertainty in metrology, environmental/sanitary assessment and numerical analysis 45

1.4.1 Uncertainty and Sensitivity Analysis (UASA) 45

1.4.2 Specificities in metrology / industrial quality control 47

1.4.3 Specificities in environmental / health impact assessment 49

1.4.4 Numerical code qualification (NCQ), calibration and data assimilation 49

1.5 Forecast and time-based modelling in weather, operations research, economics or finance 51

1.6 Conclusion: the scope for generic modelling under risk and uncertainty 52

1.6.1 Similar and dissimilar features in modelling, risk and uncertainty studies 52

1.6.2 Limitations and challenges motivating a unified framework 54

1.7 References 55

2 A generic modelling framework 58

2.1 The system under uncertainty 58

2.2 Decisional quantities and goals of modelling under risk and uncertainty 61

2.2.1 The key concept of risk measure or quantity of interest 61

2.2.2 Salient goals of risk/uncertainty studies and decision-making 62

2.3 Modelling under uncertainty: building separate system and uncertainty models 65

2.3.1 The need to go beyond direct statistics 65

2.3.2 Basic system models 66

2.3.3 Building a direct uncertainty model on variable inputs 68

2.3.4 Developing the underlying epistemic/aleatory structure 70

2.3.5 Summary 73

2.4 Modelling under uncertainty – the general case 73

2.4.1 Phenomenological models under uncertainty and residual model error 73

2.4.2 The model building process 75

2.4.3 Combining system and uncertainty models into an integrated statistical estimation problem 78

2.4.4 The combination of system and uncertainty models: a key information choice 80

2.4.5 The predictive model combining system and uncertainty components 81

2.5 Combining probabilistic and deterministic settings 82

2.5.1 Preliminary comments about the interpretations of probabilistic uncertainty models 82

2.5.2 Mixed deterministic-probabilistic contexts 84

2.6 Computing an appropriate risk measure or quantity of interest and associated sensitivity indices 86

2.6.1 Standard risk measures or q.i. (single-probabilistic) 87

2.6.2 A fundamental case: the conditional expected utility 88

2.6.3 Relationship between risk measures, uncertainty model and actions 89

2.6.4 Double probabilistic risk measures 90

2.6.5 The delicate issue of propagation/numerical uncertainty 91

2.6.6 Importance ranking and sensitivity analysis 92

2.7 Summary: main steps of the studies and later issues 93

2.8 Exercises 95

2.9 References 95

3 A generic tutorial example: natural risk on an industrial installation 98

3.1 Phenomenology and motivation of the example 98

3.1.1 The hydro component 99

3.1.2 The system’s reliability component 100

3.1.3 The economic component 102

3.1.4 Uncertain inputs, data and expertise available 104

3.2 A short introduction to gradual illustrative modelling steps 105

3.2.1 Step one: natural risk standard statistics 106

3.2.2 Step two: mixing statistics and a QRA model 108

3.2.3 Step three: uncertainty treatment of a physical/engineering model (SRA) 110

3.2.4 Step four: mixing SRA and QRA 112

3.2.5 Step five: level-2 uncertainty study on mixed SRA-QRA model 113

3.2.6 Step six: calibration of the hydro component and updating of risk measure 115

3.2.7 Step seven: economic assessment and optimization under risk and/or uncertainty 116

3.3 Summary of the example 118

3.4 Exercises 118

3.5 References 119

4 Understanding natures of uncertainty, risk margins and time bases for probabilistic decision-making

4.1 Natures of uncertainty: theoretical debates and practical implementation 120

4.1.1 Defining uncertainty – ambiguity about the reference 120

4.1.2 Risk vs. uncertainty – an impractical distinction 121

4.1.3 The aleatory/epistemic distinction and the issue of reducibility 122

4.1.4 Variability or uncertainty – the need for careful system specification 124

4.1.5 Other distinctions 126

4.1.6 Conclusions – handling various natures of uncertainty 127

4.2 Understanding the impact on margins of deterministic vs. probabilistic formulations 127

4.2.1 Understanding probabilistic averaging, dependence issues and deterministic maximisation and in the linear case 127

4.2.2 Understanding safety factors and quantiles in the monotonous case 132

4.2.3 Probability limitations, paradoxes of the maximal entropy principle 134

4.2.4 Deterministic settings and interval computation – uses and limitations 137

4.2.5 Conclusive comments on the use of probabilistic and deterministic risk measures 138

4.3 Handling time-cumulated risk measures through frequencies and probabilities 138

4.3.1 The underlying time basis of the state of the system 139

4.3.2 Understanding frequency vs. probability 142

4.3.3 Fundamental risk measures defined over a period of interest 144

4.3.4 Handling a time process and associated simplifications 146

4.3.5 Modeling rare events through extreme value theory 147

4.4 Choosing an adequate risk measure – decision-theory aspects 153

4.4.1 The salient goal involved 153

4.4.2 Theoretical debate and interpretations about the risk measure when selecting between risky alternatives (or controlling compliance with a risk target) 153

4.4.3 The choice of financial risk measures 155

4.4.4 The challenges associated with using double-probabilistic or conditional probabilistic risk measures 156

4.4.5 Summary recommendations 157

4.5 Exercises 157

4.6 References 158

5 Direct statistical estimation techniques 5

5.1 The general issue 5

5.2 Introducing estimation techniques on independent samples 8

5.2.1 Estimation basics 8

5.2.2 Estimating physical variables in the flood example 17

5.2.3 Discrete events and time-based statistical models (frequencies, reliability models, time series) 20

5.2.4 Encoding phenomenological knowledge and physical constraints inside the choice of input distributions 23

5.3 Modelling dependence 25

5.3.1 Linear correlations 25

5.3.2 Rank correlations 28

5.3.3 Copula model 32

5.3.4 Physical dependence modeling & concluding comments 33

5.4 Controlling epistemic uncertainty through classical or Bayesian estimators 35

5.4.1 Epistemic uncertainty in the classical approach 35

5.4.2 Classical approach for Gaussian uncertainty models (small samples) 37

5.4.3 Asymptotic covariance for large samples 38

5.4.4 Bootstrap and resampling techniques 44

5.4.5 Bayesian-physical settings (small samples with expert judgment) 45

5.5 Understanding rare probabilities and extreme value statistical modelling 52

5.5.1 The issue of extrapolating beyond data – advantages and limitations of the extreme value theory 53

5.5.2 The significance of extremely low probabilities 58

5.6 Exercises 61

5.7 References 62

6 Combined model estimation through inverse techniques 64

6.1 Introducing inverse techniques 64

6.1.1 Handling calibration data 64

6.1.2 Motivations for inverse modelling and associated litterature 66

6.1.3 Key distinctions between the algorithms: the representation of time and uncertainty 67

6.2 One-dimensional introduction of the gradual inverse algorithms 74

6.2.1 Direct least square calibration with two alternative interpretations 74

6.2.2 Bayesian updating, identification and calibration 81

6.2.3 An alternative identification model with intrinsic uncertainty 83

6.2.4 Comparison of the algorithms 86

6.2.5 Illustrations in the flood example 87

6.3 The general structure of inverse algorithms: residuals, identifiability, estimators, sensitivity and epistemic uncertainty 90

6.3.1 The general estimation problem 91

6.3.2 Relationship between observational data and predictive outputs for decision-making 92

6.3.3 Common features to the distributions and estimation problems associated to the general structure 93

6.3.4 Handling residuals and the issue of model uncertainty 96

6.3.5 Additional comments on the model-building process 100

6.3.6 Identifiability 101

6.3.7 Importance factors and estimation accuracy 108

6.4 Specificities for parameter identification, calibration or data assimilation algorithms 110

6.4.1 The BLUE algorithm for linear Gaussian parameter identification 110

6.4.2 An extension with unknown variance: multidimensional model calibration 113

6.4.3 Generalisations to non-linear calibration 114

6.4.4 Bayesian multidimensional model updating 115

6.4.5 Dynamic data assimilation 116

6.5 Intrinsic variability identification 118

6.5.1 A general formulation 119

6.5.2 Linearised Gaussian case 120

6.5.3 Non-linear Gaussian extensions 121

6.5.4 Moment methods 122

6.5.5 Recent algorithms and research fields 123

6.6 Conclusion: the modelling process and open statistical & computing challenges 125

6.7 Exercises 126

6.8 References 126

7 Computational methods for risk and uncertainty propagation 129

7.1 Classifying the risk measure computational issues 130

7.1.1 Risk measures in relation with conditional and combined uncertainty distributions 130

7.1.2 Expectation-based single probabilistic risk measures 132

7.1.3 Simplified integration of sub-parts with discrete inputs 134

7.1.4 Non-expectation based single probabilistic risk measures 137

7.1.5 Other risk measures (double probabilistic, mixed deterministic-probabilistic) 138

7.2 The generic Monte-Carlo simulation method and associated error control 139

7.2.1 Undertaking Monte-Carlo simulation on a computer 139

7.2.2 Dual interpretation and probabilistic properties of Monte-Carlo simulation 141

7.2.3 Control of propagation uncertainty: asymptotic results 146

7.2.4 Control of propagation uncertainty: robust results for quantiles (Wilks formula) 149

7.2.5 Sampling double-probabilistic risk measures 153

7.2.6 Sampling mixed deterministic-probabilistic measures 155

7.3 Classical alternatives to direct Monte-Carlo sampling 155

7.3.1 Overview of the computation alternatives to MCS 155

7.3.2 Taylor approximation (linear or polynomial system models) 157

7.3.3 Numerical integration 160

7.3.4 Accelerated sampling (or variance reduction) 160

7.3.5 Reliability methods (FORM-SORM and derived methods) 167

7.3.6 Polynomial chaos and stochastic developments 170

7.3.7 Response surface or meta-models 170

7.4 Monotony, regularity and robust risk measure computation 172

7.4.1 Simple examples of monotonous behaviours 172

7.4.2 Direct consequences of monotony onto computing the risk measure 173

7.4.3 Robust computation of exceedance probability in the monotonous case 175

7.4.4 Use of other forms of system model regularity 183

7.5 Sensitivity analysis and importance ranking 183

7.5.1 Elementary indices and importance measures and their equivalence in linear system models 184

7.5.2 Sobol sensitivity indices 190

7.5.3 Specificities of Boolean input or output events – importance measures in risk assessment 194

7.5.4 Concluding remarks and further research 195

7.6 Numerical challenges, distributed computing and use of direct or adjoint differentiation of codes 196

7.7 Exercises 197

7.8 References 198

8 Optimising under uncertainty: economics and computational challenges 201

8.1 Getting the costs inside risk modelling – from engineering economics to financial modelling 201

8.1.1 Moving to costs as output variables of interest – elementary engineering economics 201

8.1.2 Costs of uncertainty and the value of information 205

8.1.3 The expected utility approach for risk aversion 206

8.1.4 Non-linear transformations 209

8.1.5 Robust design and alternatives mixing cost expectation and variance inside the optimisation procedure 210

8.2 The role of time – cash flows and associated risk measures 212

8.2.1 Costs over a time period – the cash flow model 212

8.2.2 The issue of discounting 214

8.2.3 Valuing time flexibility of decision-making and stochastic optimisation 217

8.3 Computational challenges associated to optimisation 219

8.3.1 Static optimization (utility-based) 219

8.3.2 Stochastic dynamic programming 220

8.3.3 Computation and robustness challenges 220

8.4 The promise of high performance computing 221

8.4.1 The computational load of risk and uncertainty modeling 221

8.4.2 The potential of high-performance computing 223

8.5 Exercises 224

8.6 References 224

9 Conclusion: perspectives of modelling in the context of risk and uncertainty and further research 226

9.1 Open scientific challenges 226

9.2 Challenges involved by the dissemination of advanced modelling in the context of risk and uncertainty 228

9.3 References 229

Epilogue 230

Index 232


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Modelling Under Risk & Uncertainty: An Introduction to Statistical, Phenomenological & Computational Methods, <i>Modelling Under Risk and Uncertainty</i> goes beyond the 'black-box' view that some risk analysts or statisticians develop the underlying phenomenology of the environmental or industrial processes, without valuing enough their physical properties and i, Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods

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Modelling Under Risk & Uncertainty: An Introduction to Statistical, Phenomenological & Computational Methods, <i>Modelling Under Risk and Uncertainty</i> goes beyond the 'black-box' view that some risk analysts or statisticians develop the underlying phenomenology of the environmental or industrial processes, without valuing enough their physical properties and i, Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods

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