{"product_id":"modelling-under-risk-and-uncertainty-9780470695142","title":"Modelling Under Risk and Uncertainty","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eModelling has permeated virtually all areas of industrial, environmental, economic, bio-medical or civil engineering: yet the use of models for decision-making raises a number of issues to which this book is dedicated:\u003c\/p\u003e \u003cp\u003eHow uncertain is my model ? Is it truly valuable to support decision-making ? What kind of decision can be truly supported and how can I handle residual uncertainty ? How much refined should the mathematical description be, given the true data limitations ? Could the uncertainty be reduced through more data, increased modeling investment or computational budget ? Should it be reduced now or later ? How robust is the analysis or the computational methods involved ? Should \/ could those methods be more robust ? Does it make sense to handle uncertainty, risk, lack of knowledge, variability or errors altogether ? How reasonable is the choice of probabilistic modeling for rare events ? How rare are the events to be considered? How far does it make sense to handle ex\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003c\/p\u003e\u003cp\u003e“In my opinion, reviewed book is well organized textbook for risk management.”  (\u003ci\u003eZentralblatt MATH\u003c\/i\u003e, 1 December 2012)\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface xv  \u003cp\u003eAcknowledgements xvii\u003c\/p\u003e \u003cp\u003eIntroduction and reading guide xix\u003c\/p\u003e \u003cp\u003eNotation xxxiii\u003c\/p\u003e \u003cp\u003eAcronyms and abbreviations xxxvii\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Applications and practices of modelling, risk and uncertainty 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Protection against natural risk 1\u003c\/p\u003e \u003cp\u003e1.1.1 The popular ‘initiator\/frequency approach’ 3\u003c\/p\u003e \u003cp\u003e1.1.2 Recent developments towards an ‘extended frequency approach’ 5\u003c\/p\u003e \u003cp\u003e1.2 Engineering design, safety and structural reliability analysis (SRA) 7\u003c\/p\u003e \u003cp\u003e1.2.1 The domain of structural reliability 8\u003c\/p\u003e \u003cp\u003e1.2.2 Deterministic safety margins and partial safety factors 9\u003c\/p\u003e \u003cp\u003e1.2.3 Probabilistic structural reliability analysis 10\u003c\/p\u003e \u003cp\u003e1.2.4 Links and differences with natural risk studies 11\u003c\/p\u003e \u003cp\u003e1.3 Industrial safety, system reliability and probabilistic risk assessment (PRA) 12\u003c\/p\u003e \u003cp\u003e1.3.1 The context of systems analysis 12\u003c\/p\u003e \u003cp\u003e1.3.2 Links and differences with structural reliability analysis 14\u003c\/p\u003e \u003cp\u003e1.3.3 The case of elaborate PRA (multi-state, dynamic) 16\u003c\/p\u003e \u003cp\u003e1.3.4 Integrated probabilistic risk assessment (IPRA) 17\u003c\/p\u003e \u003cp\u003e1.4 Modelling under uncertainty in metrology, environmental\/sanitary assessment and numerical analysis 20\u003c\/p\u003e \u003cp\u003e1.4.1 Uncertainty and sensitivity analysis (UASA) 21\u003c\/p\u003e \u003cp\u003e1.4.2 Specificities in metrology\/industrial quality control 23\u003c\/p\u003e \u003cp\u003e1.4.3 Specificities in environmental\/health impact assessment 24\u003c\/p\u003e \u003cp\u003e1.4.4 Numerical code qualification (NCQ), calibration and data assimilation 25\u003c\/p\u003e \u003cp\u003e1.5 Forecast and time-based modelling in weather, operations research, economics or finance 27\u003c\/p\u003e \u003cp\u003e1.6 Conclusion: The scope for generic modelling under risk and uncertainty 28\u003c\/p\u003e \u003cp\u003e1.6.1 Similar and dissimilar features in modelling, risk and uncertainty studies 28\u003c\/p\u003e \u003cp\u003e1.6.2 Limitations and challenges motivating a unified framework 30\u003c\/p\u003e \u003cp\u003eReferences 31\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 A generic modelling framework 34\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 The system under uncertainty 34\u003c\/p\u003e \u003cp\u003e2.2 Decisional quantities and goals of modelling under risk and uncertainty 37\u003c\/p\u003e \u003cp\u003e2.2.1 The key concept of risk measure or quantity of interest 37\u003c\/p\u003e \u003cp\u003e2.2.2 Salient goals of risk\/uncertainty studies and decision-making 38\u003c\/p\u003e \u003cp\u003e2.3 Modelling under uncertainty: Building separate system and uncertainty models 41\u003c\/p\u003e \u003cp\u003e2.3.1 The need to go beyond direct statistics 41\u003c\/p\u003e \u003cp\u003e2.3.2 Basic system models 42\u003c\/p\u003e \u003cp\u003e2.3.3 Building a direct uncertainty model on variable inputs 45\u003c\/p\u003e \u003cp\u003e2.3.4 Developing the underlying epistemic\/aleatory structure 46\u003c\/p\u003e \u003cp\u003e2.3.5 Summary 49\u003c\/p\u003e \u003cp\u003e2.4 Modelling under uncertainty – the general case 50\u003c\/p\u003e \u003cp\u003e2.4.1 Phenomenological models under uncertainty and residual model error 50\u003c\/p\u003e \u003cp\u003e2.4.2 The model building process 51\u003c\/p\u003e \u003cp\u003e2.4.3 Combining system and uncertainty models into an integrated statistical estimation problem 55\u003c\/p\u003e \u003cp\u003e2.4.4 The combination of system and uncertainty models: A key information choice 57\u003c\/p\u003e \u003cp\u003e2.4.5 The predictive model combining system and uncertainty components 59\u003c\/p\u003e \u003cp\u003e2.5 Combining probabilistic and deterministic settings 60\u003c\/p\u003e \u003cp\u003e2.5.1 Preliminary comments about the interpretations of probabilistic uncertainty models 60\u003c\/p\u003e \u003cp\u003e2.5.2 Mixed deterministic-probabilistic contexts 61\u003c\/p\u003e \u003cp\u003e2.6 Computing an appropriate risk measure or quantity of interest and associated sensitivity indices 64\u003c\/p\u003e \u003cp\u003e2.6.1 Standard risk measures or q.i. (single-probabilistic) 65\u003c\/p\u003e \u003cp\u003e2.6.2 A fundamental case: The conditional expected utility 67\u003c\/p\u003e \u003cp\u003e2.6.3 Relationship between risk measures, uncertainty model and actions 68\u003c\/p\u003e \u003cp\u003e2.6.4 Double probabilistic risk measures 69\u003c\/p\u003e \u003cp\u003e2.6.5 The delicate issue of propagation\/numerical uncertainty 71\u003c\/p\u003e \u003cp\u003e2.6.6 Importance ranking and sensitivity analysis 71\u003c\/p\u003e \u003cp\u003e2.7 Summary: Main steps of the studies and later issues 73\u003c\/p\u003e \u003cp\u003eExercises 74\u003c\/p\u003e \u003cp\u003eReferences 75\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 A generic tutorial example: Natural risk in an industrial installation 77\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Phenomenology and motivation of the example 77\u003c\/p\u003e \u003cp\u003e3.1.1 The hydro component 78\u003c\/p\u003e \u003cp\u003e3.1.2 The system’s reliability component 80\u003c\/p\u003e \u003cp\u003e3.1.3 The economic component 83\u003c\/p\u003e \u003cp\u003e3.1.4 Uncertain inputs, data and expertise available 84\u003c\/p\u003e \u003cp\u003e3.2 A short introduction to gradual illustrative modelling steps 86\u003c\/p\u003e \u003cp\u003e3.2.1 Step one: Natural risk standard statistics 87\u003c\/p\u003e \u003cp\u003e3.2.2 Step two: Mixing statistics and a QRA model 89\u003c\/p\u003e \u003cp\u003e3.2.3 Step three: Uncertainty treatment of a physical\/engineering model (SRA) 91\u003c\/p\u003e \u003cp\u003e3.2.4 Step four: Mixing SRA and QRA 91\u003c\/p\u003e \u003cp\u003e3.2.5 Step five: Level-2 uncertainty study on mixed SRA-QRA model 94\u003c\/p\u003e \u003cp\u003e3.2.6 Step six: Calibration of the hydro component and updating of risk measure 96\u003c\/p\u003e \u003cp\u003e3.2.7 Step seven: Economic assessment and optimisation under risk and\/or uncertainty 97\u003c\/p\u003e \u003cp\u003e3.3 Summary of the example 99\u003c\/p\u003e \u003cp\u003eExercises 101\u003c\/p\u003e \u003cp\u003eReferences 101\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Understanding natures of uncertainty, risk margins and time bases for probabilistic decision-making 102\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Natures of uncertainty: Theoretical debates and practical implementation 103\u003c\/p\u003e \u003cp\u003e4.1.1 Defining uncertainty – ambiguity about the reference 103\u003c\/p\u003e \u003cp\u003e4.1.2 Risk vs. uncertainty – an impractical distinction 104\u003c\/p\u003e \u003cp\u003e4.1.3 The aleatory\/epistemic distinction and the issue of reducibility 105\u003c\/p\u003e \u003cp\u003e4.1.4 Variability or uncertainty – the need for careful system specification 107\u003c\/p\u003e \u003cp\u003e4.1.5 Other distinctions 109\u003c\/p\u003e \u003cp\u003e4.2 Understanding the impact on margins of deterministic vs. probabilistic formulations 110\u003c\/p\u003e \u003cp\u003e4.2.1 Understanding probabilistic averaging, dependence issues and deterministic maximisation and in the linear case 110\u003c\/p\u003e \u003cp\u003e4.2.2 Understanding safety factors and quantiles in the monotonous case 114\u003c\/p\u003e \u003cp\u003e4.2.3 Probability limitations, paradoxes of the maximal entropy principle 117\u003c\/p\u003e \u003cp\u003e4.2.4 Deterministic settings and interval computation – uses and limitations 119\u003c\/p\u003e \u003cp\u003e4.2.5 Conclusive comments on the use of probabilistic and deterministic risk measures 120\u003c\/p\u003e \u003cp\u003e4.3 Handling time-cumulated risk measures through frequencies and probabilities 121\u003c\/p\u003e \u003cp\u003e4.3.1 The underlying time basis of the state of the system 121\u003c\/p\u003e \u003cp\u003e4.3.2 Understanding frequency vs. probability 124\u003c\/p\u003e \u003cp\u003e4.3.3 Fundamental risk measures defined over a period of interest 126\u003c\/p\u003e \u003cp\u003e4.3.4 Handling a time process and associated simplifications 128\u003c\/p\u003e \u003cp\u003e4.3.5 Modelling rare events through extreme value theory 130\u003c\/p\u003e \u003cp\u003e4.4 Choosing an adequate risk measure – decision-theory aspects 135\u003c\/p\u003e \u003cp\u003e4.4.1 The salient goal involved 135\u003c\/p\u003e \u003cp\u003e4.4.2 Theoretical debate and interpretations about the risk measure when selecting between risky alternatives (or controlling compliance with a risk target) 136\u003c\/p\u003e \u003cp\u003e4.4.3 The choice of financial risk measures 137\u003c\/p\u003e \u003cp\u003e4.4.4 The challenges associated with using double-probabilistic or conditional probabilistic risk measures 138\u003c\/p\u003e \u003cp\u003e4.4.5 Summary recommendations 140\u003c\/p\u003e \u003cp\u003eExercises 140\u003c\/p\u003e \u003cp\u003eReferences 141\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Direct statistical estimation techniques 143\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 The general issue 143\u003c\/p\u003e \u003cp\u003e5.2 Introducing estimation techniques on independent samples 147\u003c\/p\u003e \u003cp\u003e5.2.1 Estimation basics 147\u003c\/p\u003e \u003cp\u003e5.2.2 Goodness-of-fit and model selection techniques 150\u003c\/p\u003e \u003cp\u003e5.2.3 A non-parametric method: Kernel modelling 154\u003c\/p\u003e \u003cp\u003e5.2.4 Estimating physical variables in the flood example 157\u003c\/p\u003e \u003cp\u003e5.2.5 Discrete events and time-based statistical models (frequencies, reliability models, time series) 159\u003c\/p\u003e \u003cp\u003e5.2.6 Encoding phenomenological knowledge and physical constraints inside the choice of input distributions 163\u003c\/p\u003e \u003cp\u003e5.3 Modelling dependence 165\u003c\/p\u003e \u003cp\u003e5.3.1 Linear correlations 165\u003c\/p\u003e \u003cp\u003e5.3.2 Rank correlations 168\u003c\/p\u003e \u003cp\u003e5.3.3 Copula model 172\u003c\/p\u003e \u003cp\u003e5.3.4 Multi-dimensional non-parametric modelling 173\u003c\/p\u003e \u003cp\u003e5.3.5 Physical dependence modelling and concluding comments 174\u003c\/p\u003e \u003cp\u003e5.4 Controlling epistemic uncertainty through classical or Bayesian estimators 175\u003c\/p\u003e \u003cp\u003e5.4.1 Epistemic uncertainty in the classical approach 175\u003c\/p\u003e \u003cp\u003e5.4.2 Classical approach for Gaussian uncertainty models (small samples) 177\u003c\/p\u003e \u003cp\u003e5.4.3 Asymptotic covariance for large samples 179\u003c\/p\u003e \u003cp\u003e5.4.4 Bootstrap and resampling techniques 185\u003c\/p\u003e \u003cp\u003e5.4.5 Bayesian-physical settings (small samples with expert judgement) 186\u003c\/p\u003e \u003cp\u003e5.5 Understanding rare probabilities and extreme value statistical modelling 194\u003c\/p\u003e \u003cp\u003e5.5.1 The issue of extrapolating beyond data – advantages and limitations of the extreme value theory 194\u003c\/p\u003e \u003cp\u003e5.5.2 The significance of extremely low probabilities 201\u003c\/p\u003e \u003cp\u003eExercises 203\u003c\/p\u003e \u003cp\u003eReferences 204\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Combined model estimation through inverse techniques 206\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Introducing inverse techniques 206\u003c\/p\u003e \u003cp\u003e6.1.1 Handling calibration data 206\u003c\/p\u003e \u003cp\u003e6.1.2 Motivations for inverse modelling and associated literature 208\u003c\/p\u003e \u003cp\u003e6.1.3 Key distinctions between the algorithms: The representation of time and uncertainty 210\u003c\/p\u003e \u003cp\u003e6.2 One-dimensional introduction of the gradual inverse algorithms 216\u003c\/p\u003e \u003cp\u003e6.2.1 Direct least square calibration with two alternative interpretations 216\u003c\/p\u003e \u003cp\u003e6.2.2 Bayesian updating, identification and calibration 223\u003c\/p\u003e \u003cp\u003e6.2.3 An alternative identification model with intrinsic uncertainty 225\u003c\/p\u003e \u003cp\u003e6.2.4 Comparison of the algorithms 227\u003c\/p\u003e \u003cp\u003e6.2.5 Illustrations in the flood example 229\u003c\/p\u003e \u003cp\u003e6.3 The general structure of inverse algorithms: Residuals, identifiability, estimators, sensitivity and epistemic uncertainty 233\u003c\/p\u003e \u003cp\u003e6.3.1 The general estimation problem 233\u003c\/p\u003e \u003cp\u003e6.3.2 Relationship between observational data and predictive outputs for decision-making 233\u003c\/p\u003e \u003cp\u003e6.3.3 Common features to the distributions and estimation problems associated to the general structure 236\u003c\/p\u003e \u003cp\u003e6.3.4 Handling residuals and the issue of model uncertainty 238\u003c\/p\u003e \u003cp\u003e6.3.5 Additional comments on the model-building process 242\u003c\/p\u003e \u003cp\u003e6.3.6 Identifiability 243\u003c\/p\u003e \u003cp\u003e6.3.7 Importance factors and estimation accuracy 249\u003c\/p\u003e \u003cp\u003e6.4 Specificities for parameter identification, calibration or data assimilation algorithms 251\u003c\/p\u003e \u003cp\u003e6.4.1 The BLUE algorithm for linear Gaussian parameter identification 251\u003c\/p\u003e \u003cp\u003e6.4.2 An extension with unknown variance: Multidimensional model calibration 254\u003c\/p\u003e \u003cp\u003e6.4.3 Generalisations to non-linear calibration 255\u003c\/p\u003e \u003cp\u003e6.4.4 Bayesian multidimensional model updating 256\u003c\/p\u003e \u003cp\u003e6.4.5 Dynamic data assimilation 257\u003c\/p\u003e \u003cp\u003e6.5 Intrinsic variability identification 260\u003c\/p\u003e \u003cp\u003e6.5.1 A general formulation 260\u003c\/p\u003e \u003cp\u003e6.5.2 Linearised Gaussian case 261\u003c\/p\u003e \u003cp\u003e6.5.3 Non-linear Gaussian extensions 263\u003c\/p\u003e \u003cp\u003e6.5.4 Moment methods 264\u003c\/p\u003e \u003cp\u003e6.5.5 Recent algorithms and research fields 264\u003c\/p\u003e \u003cp\u003e6.6 Conclusion: The modelling process and open statistical and computing challenges 267\u003c\/p\u003e \u003cp\u003eExercises 267\u003c\/p\u003e \u003cp\u003eReferences 268\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Computational methods for risk and uncertainty propagation 271\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Classifying the risk measure computational issues 272\u003c\/p\u003e \u003cp\u003e7.1.1 Risk measures in relation to conditional and combined uncertainty distributions 273\u003c\/p\u003e \u003cp\u003e7.1.2 Expectation-based single probabilistic risk measures 275\u003c\/p\u003e \u003cp\u003e7.1.3 Simplified integration of sub-parts with discrete inputs 277\u003c\/p\u003e \u003cp\u003e7.1.4 Non-expectation based single probabilistic risk measures 280\u003c\/p\u003e \u003cp\u003e7.1.5 Other risk measures (double probabilistic, mixed deterministic-probabilistic) 281\u003c\/p\u003e \u003cp\u003e7.2 The generic Monte-Carlo simulation method and associated error control 283\u003c\/p\u003e \u003cp\u003e7.2.1 Undertaking Monte-Carlo simulation on a computer 283\u003c\/p\u003e \u003cp\u003e7.2.2 Dual interpretation and probabilistic properties of Monte-Carlo simulation 285\u003c\/p\u003e \u003cp\u003e7.2.3 Control of propagation uncertainty: Asymptotic results 290\u003c\/p\u003e \u003cp\u003e7.2.4 Control of propagation uncertainty: Robust results for quantiles (Wilks formula) 292\u003c\/p\u003e \u003cp\u003e7.2.5 Sampling double-probabilistic risk measures 298\u003c\/p\u003e \u003cp\u003e7.2.6 Sampling mixed deterministic-probabilistic measures 299\u003c\/p\u003e \u003cp\u003e7.3 Classical alternatives to direct Monte-Carlo sampling 299\u003c\/p\u003e \u003cp\u003e7.3.1 Overview of the computation alternatives to MCS 299\u003c\/p\u003e \u003cp\u003e7.3.2 Taylor approximation (linear or polynomial system models) 300\u003c\/p\u003e \u003cp\u003e7.3.3 Numerical integration 305\u003c\/p\u003e \u003cp\u003e7.3.4 Accelerated sampling (or variance reduction) 306\u003c\/p\u003e \u003cp\u003e7.3.5 Reliability methods (FORM-SORM and derived methods) 312\u003c\/p\u003e \u003cp\u003e7.3.6 Polynomial chaos and stochastic developments 316\u003c\/p\u003e \u003cp\u003e7.3.7 Response surface or meta-models 316\u003c\/p\u003e \u003cp\u003e7.4 Monotony, regularity and robust risk measure computation 317\u003c\/p\u003e \u003cp\u003e7.4.1 Simple examples of monotonous behaviours 317\u003c\/p\u003e \u003cp\u003e7.4.2 Direct consequences of monotony for computing the risk measure 319\u003c\/p\u003e \u003cp\u003e7.4.3 Robust computation of exceedance probability in the monotonous case 322\u003c\/p\u003e \u003cp\u003e7.4.4 Use of other forms of system model regularity 329\u003c\/p\u003e \u003cp\u003e7.5 Sensitivity analysis and importance ranking 330\u003c\/p\u003e \u003cp\u003e7.5.1 Elementary indices and importance measures and their equivalence in linear system models 330\u003c\/p\u003e \u003cp\u003e7.5.2 Sobol sensitivity indices 336\u003c\/p\u003e \u003cp\u003e7.5.3 Specificities of Boolean input\/output events – importance measures in risk assessment 339\u003c\/p\u003e \u003cp\u003e7.5.4 Concluding remarks and further research 341\u003c\/p\u003e \u003cp\u003e7.6 Numerical challenges, distributed computing and use of direct or adjoint differentiation of codes 342\u003c\/p\u003e \u003cp\u003eExercises 342\u003c\/p\u003e \u003cp\u003eReferences 343\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Optimising under uncertainty: Economics and computational challenges 347\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Getting the costs inside risk modelling – from engineering economics to financial modelling 347\u003c\/p\u003e \u003cp\u003e8.1.1 Moving to costs as output variables of interest – elementary engineering economics 347\u003c\/p\u003e \u003cp\u003e8.1.2 Costs of uncertainty and the value of information 351\u003c\/p\u003e \u003cp\u003e8.1.3 The expected utility approach for risk aversion 353\u003c\/p\u003e \u003cp\u003e8.1.4 Non-linear transformations 355\u003c\/p\u003e \u003cp\u003e8.1.5 Robust design and alternatives mixing cost expectation and variance inside the optimisation procedure 356\u003c\/p\u003e \u003cp\u003e8.2 The role of time – cash flows and associated risk measures 358\u003c\/p\u003e \u003cp\u003e8.2.1 Costs over a time period – the cash flow model 358\u003c\/p\u003e \u003cp\u003e8.2.2 The issue of discounting 361\u003c\/p\u003e \u003cp\u003e8.2.3 Valuing time flexibility of decision-making and stochastic optimisation 362\u003c\/p\u003e \u003cp\u003e8.3 Computational challenges associated to optimisation 366\u003c\/p\u003e \u003cp\u003e8.3.1 Static optimisation (utility-based) 367\u003c\/p\u003e \u003cp\u003e8.3.2 Stochastic dynamic programming 368\u003c\/p\u003e \u003cp\u003e8.3.3 Computation and robustness challenges 368\u003c\/p\u003e \u003cp\u003e8.4 The promise of high performance computing 369\u003c\/p\u003e \u003cp\u003e8.4.1 The computational load of risk and uncertainty modelling 369\u003c\/p\u003e \u003cp\u003e8.4.2 The potential of high-performance computing 371\u003c\/p\u003e \u003cp\u003eExercises 372\u003c\/p\u003e \u003cp\u003eReferences 372\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Conclusion: Perspectives of modelling in the context of risk and uncertainty and further research 374\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Open scientific challenges 374\u003c\/p\u003e \u003cp\u003e9.2 Challenges involved by the dissemination of advanced modelling in the context of risk and uncertainty 377\u003c\/p\u003e \u003cp\u003eReferences 377\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Annexes 378\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Annex 1 – refresher on probabilities and statistical modelling of uncertainty 378\u003c\/p\u003e \u003cp\u003e10.1.1 Modelling through a random variable 378\u003c\/p\u003e \u003cp\u003e10.1.2 The impact of data and the estimation uncertainty 380\u003c\/p\u003e \u003cp\u003e10.1.3 Continuous probabilistic distributions 382\u003c\/p\u003e \u003cp\u003e10.1.4 Dependence and stationarity 382\u003c\/p\u003e \u003cp\u003e10.1.5 Non-statistical approach of probabilistic modelling 384\u003c\/p\u003e \u003cp\u003e10.2 Annex 2 – comments about the probabilistic foundations of the uncertainty models 386\u003c\/p\u003e \u003cp\u003e10.2.1 The overall space of system states and the output space 386\u003c\/p\u003e \u003cp\u003e10.2.2 Correspondence to the Kaplan\/Garrick risk analysis triplets 389\u003c\/p\u003e \u003cp\u003e10.2.3 The model and model input space 389\u003c\/p\u003e \u003cp\u003e10.2.4 Estimating the uncertainty model through direct data 391\u003c\/p\u003e \u003cp\u003e10.2.5 Model calibration and estimation through indirect data and inversion techniques 393\u003c\/p\u003e \u003cp\u003e10.3 Annex 3 – introductory reflections on the sources of macroscopic uncertainty 394\u003c\/p\u003e \u003cp\u003e10.4 Annex 4 – details about the pedagogical example 397\u003c\/p\u003e \u003cp\u003e10.4.1 Data samples 397\u003c\/p\u003e \u003cp\u003e10.4.2 Reference probabilistic model for the hydro component 399\u003c\/p\u003e \u003cp\u003e10.4.3 Systems reliability component – expert information on elementary failure probabilities 399\u003c\/p\u003e \u003cp\u003e10.4.4 Economic component – cost functions and probabilistic model 403\u003c\/p\u003e \u003cp\u003e10.4.5 Detailed results on various steps 404\u003c\/p\u003e \u003cp\u003e10.5 Annex 5 – detailed mathematical demonstrations 414\u003c\/p\u003e \u003cp\u003e10.5.1 Basic results about vector random variables and matrices 414\u003c\/p\u003e \u003cp\u003e10.5.2 Differentiation results and solutions of quadratic likelihood maximisation 415\u003c\/p\u003e \u003cp\u003e10.5.3 Proof of the Wilks formula 419\u003c\/p\u003e \u003cp\u003e10.5.4 Complements on the definition and chaining of monotony 420\u003c\/p\u003e \u003cp\u003e10.5.5 Proofs on level-2 quantiles of monotonous system models 422\u003c\/p\u003e \u003cp\u003e10.5.6 Proofs on the estimator of adaptive Monte-Carlo under monotony (section 7.4.3) 423\u003c\/p\u003e \u003cp\u003eReferences 426\u003c\/p\u003e \u003cp\u003eEpilogue 427\u003c\/p\u003e \u003cp\u003eIndex 429\u003c\/p\u003e","brand":"John Wiley 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