Modelling Under Risk and Uncertainty

Transcription

1 Modelling Under Risk and Uncertainty An Introduction to Statistical, Phenomenological and Computational Methods Etienne de Rocquigny Ecole Centrale Paris, Universite Paris-Saclay, France WILEY A John Wiley & Sons, Ltd., Publication

2 Contents Preface Acknowledgements Introduction and reading guide Notation Acronyms and abbreviations xv xvii xix xxxiii xxxvii 1 Applications and practices of modelling, risk and uncertainty Protection against natural risk The popular 'initiator/frequency approach' Recent developments towards an 'extended frequency approach' Engineering design, safety and structural reliability analysis (SRA) The domain of structural reliability Deterministic safety margins and partial safety factors Probabilistic structural reliability analysis Links and differences with natural risk studies Industrial safety, system reliability and probabilistic risk assessment (PRA) The context of systems analysis Links and differences with structural reliability analysis, The case of elaborate PRA (multi-state, dynamic) Integrated probabilistic risk assessment (IPRA) Modelling under uncertainty in metrology, environmental/sanitary assessment and numerical analysis Uncertainty and sensitivity analysis (UASA) Specificities in metrology/industrial quality control Specificities in environmental/health impact assessment Numerical code qualification (NCQ), calibration and data assimilation Forecast and time-based modelling in weather, operations research, economics or finance Conclusion: The scope for generic modelling under risk and uncertainty Similar and dissimilar features in modelling, risk and uncertainty studies Limitations and challenges motivating a unified framework 30 References 31

3 viii CONTENTS 2 A generic modelling framework The system under uncertainty Decisional quantities and goals of modelling under risk and uncertainty The key concept of risk measure or quantity of interest Salient goals of risk/uncertainty studies and decision-making Modelling under uncertainty: Building separate system and uncertainty models The need to go beyond direct statistics Basic system models Building a direct uncertainty model on variable inputs Developing the underlying epistemic/aleatory structure Summary Modelling under uncertainty - the general case Phenomenological models under uncertainty and residual model error The model building process Combining system and uncertainty models into an integrated statistical estimation problem The combination of system and uncertainty models: A key information choice The predictive model combining system and uncertainty components Combining probabilistic and deterministic settings Preliminary comments about the interpretations of probabilistic uncertainty models Mixed deterministic-probabilistic contexts Computing an appropriate risk measure or quantity of interest and associated sensitivity indices Standard risk measures or q.i. (single-probabilistic) A fundamental case: The conditional expected utility Relationship between risk measures, uncertainty model and actions Double probabilistic risk measures The delicate issue of propagation/numerical uncertainty Importance ranking and sensitivity analysis Summary: Main steps of the studies and later issues 73 Exercises 74 References 75 3 A generic tutorial example: Natural risk in an industrial installation Phenomenology and motivation of the example The hydro component The system's reliability component The economic component Uncertain inputs, data and expertise available A short introduction to gradual illustrative modelling steps Step one: Natural risk standard statistics Step two: Mixing statistics and a QRA model 89

4 CONTENTS ix Step three: Uncertainty treatment of a physical/engineering model (SRA) Step four: Mixing SRA and QRA Step five: Level-2 uncertainty study on mixed SRA-QRA model Step six: Calibration of the hydro component and updating of risk measure Step seven: Economic assessment and optimisation under risk and/or uncertainty " Summary of the example 99 Exercises 101 References ; 101 Understanding natures of uncertainty, risk margins and time bases for probabilistic decision-making Natures of uncertainty: Theoretical debates and practical implementation Defining uncertainty - ambiguity about the reference Risk vs. uncertainty - an impractical distinction The aleatory/epistemic distinction and the issue of reducibility Variability or uncertainty - the need for careful system specification Other distinctions Understanding the impact on margins of deterministic vs. probabilistic formulations Understanding probabilistic averaging, dependence issues and deterministic maximisation and in the linear case Understanding safety factors and quantiles in the monotonous case Probability limitations, paradoxes of the maximal entropy principle Deterministic settings and interval computation - uses and limitations Conclusive comments on the use of probabilistic and deterministic risk measures Handling time-cumulated risk measures through frequencies and probabilities The underlying time basis of the state of the system Understanding frequency vs. probability Fundamental risk measures defined over a period of interest Handling a time process and associated simplifications Modelling rare events through extreme value theory Choosing an adequate risk measure - decision-theory aspects The salient goal involved Theoretical debate and interpretations about the risk measure when selecting between risky alternatives (or controlling compliance with a risk target) The choice of financial risk measures 137

5 x CONTENTS The challenges associated with using double-probabilistic or conditional probabilistic risk measures Summary recommendations 140 Exercises 140 References Direct statistical estimation techniques The general issue Introducing estimation techniques on independent samples Estimation basics Goodness-of-fit and model selection techniques A non-parametric method: Kernel modelling Estimating physical variables in the flood example Discrete events and time-based statistical models (frequencies, reliability models, time series) Encoding phenomenological knowledge and physical constraints inside the choice of input distributions Modelling dependence Linear correlations Rank correlations Copula model Multi-dimensional non-parametric modelling Physical dependence modelling and concluding comments Controlling epistemic uncertainty through classical or Bayesian estimators Epistemic uncertainty in the classical approach Classical approach for Gaussian uncertainty models (small samples) Asymptotic covariance for large samples Bootstrap and resampling techniques Bayesian-physical settings (small samples with expert judgement) Understanding rare probabilities and extreme value statistical modelling The issue of extrapolating beyond data - advantages and limitations of the extreme value theory The significance of extremely low probabilities 201 Exercises 203 References ' Combined model estimation through inverse techniques Introducing inverse techniques Handling calibration data Motivations for inverse modelling and associated literature Key distinctions between the algorithms: The representation of time and uncertainty, One-dimensional introduction of the gradual inverse algorithms Direct least square calibration with two alternative interpretations Bayesian updating, identification and calibration An alternative identification model with intrinsic uncertainty Comparison of the algorithms Illustrations in the flood example 229

6 CONTENTS xi 6.3 The general structure of inverse algorithms: Residuals, identifiability, estimators, sensitivity and epistemic uncertainty The general estimation problem Relationship between observational data and predictive outputs for decision-making Common features to the distributions and estimation problems associated to the general structure Handling residuals and the issue bf model uncertainty Additional comments on the model-building process Identifiability Importance factors and estimation accuracy Specificities for parameter identification/calibration or data assimilation algorithms The BLUE algorithm for linear Gaussian parameter identification An extension with unknown variance: Multidimensional model calibration Generalisations to non-linear calibration Bayesian multidimensional model updating Dynamic data assimilation Intrinsic variability identification A general formulation Linearised Gaussian case Non-linear Gaussian extensions Moment methods Recent algorithms and research fields Conclusion: The modelling process and open statistical and computing challenges 267 Exercises 267 References Computational methods for risk and uncertainty propagation Classifying the risk measure computational issues ' Risk measures in relation to conditional and combined uncertainty distributions Expectation-based single probabilistic risk measures Simplified integration of sub-parts with discrete inputs Non-expectation based single probabilistic risk measures Other risk measures (double probabilistic, mixed deterministic-probabilistic) The generic Monte-Carlo simulation method and associated error control Undertaking Monte-Carlo simulation on a computer Dual interpretation and probabilistic properties of Monte-Carlo simulation Control of propagation uncertainty: Asymptotic results Control of propagation uncertainty: Robust results for quantiles (Wilks formula) Sampling double-probabilistic risk measures Sampling mixed deterministic-probabilistic measures 299

7 xii CONTENTS 7.3 Classical alternatives to direct Monte-Carlo sampling Overview of the computation alternatives to MCS Taylor approximation (linear or polynomial system models) Numerical integration Accelerated sampling (or variance reduction) Reliability methods (FORM-SORM and derived methods) Polynomial chaos and stochastic developments Response surface or meta-models ' Monotony, regularity and robust risk measure computation Simple examples of monotonous behaviours Direct consequences of monotony for computing the risk measure. " ~ Robust computation of exceedance probability in the monotonous case Use of other forms of system model regularity Sensitivity analysis and importance ranking Elementary indices and importance measures and their equivalence in linear system models Sobol sensitivity indices Specificities of Boolean input/output events - importance measures in risk assessment Concluding remarks and further research Numerical challenges, distributed computing and use of direct or adjoint differentiation of codes 342 Exercises 342 References Optimising under uncertainty: Economics and computational challenges Getting the costs inside risk modelling - from engineering economics to financial modelling Moving to costs as output variables of interest - elementary engineering economics Costs of uncertainty and the value of information The expected utility approach for risk aversion Non-linear transformations Robust design and alternatives mixing cost expectation and variance inside the optimisation procedure The role of time - cash flows and associated risk measures Costs over a time period - the cash flow model The issue of discounting Valuing time flexibility of decision-making and stochastic optimisation, Computational challenges associated to optimisation Static optimisation (utility-based) Stochastic dynamic programming Computation and robustness challenges The promise of high performance computing The computational load of risk and uncertainty modelling 369

8 CONTENTS xiii The potential of high-performance computing 371 Exercises 372 References Conclusion: Perspectives of modelling in the context of risk and uncertainty and further research Open scientific challenges Challenges involved by the dissemination of advanced modelling in the context of risk and uncertainty 377 References 377 v 10 Annexes Annex 1 - refresher on probabilities and statistical modelling of uncertainty Modelling through a random variable The impact of data and the estimation uncertainty Continuous probabilistic distributions Dependence and stationarity Non-statistical approach of probabilistic modelling Annex 2 - comments about the probabilistic foundations of the uncertainty models The overall space of system states and the output space Correspondence to the Kaplan/Garrick risk analysis triplets The model and model input space Estimating the uncertainty model through direct data Model calibration and estimation through indirect data and inversion techniques Annex 3 - introductory reflections on the sources of macroscopic uncertainty Annex 4 - details about the pedagogical example Data samples Reference probabilistic model for the hydro component Systems reliability component - expert information on elementary failure probabilities Economic component - cost functions and probabilistic model Detailed results on various steps Annex 5 - detailed mathematical demonstrations Basic results about vector random variables and matrices Differentiation results and solutions of quadratic likelihood maximisation Proof of the Wilks formula Complements on the definition and chaining of monotony Proofs on level-2 quantiles of monotonous system models Proofs on the estimator of adaptive Monte-Carlo under monotony (section 7.4.3) 423 References 426 Epilogue 427 Index 429