Siu-Kui Au, University of Liverpool, UK Yu Wang, City University of Hong Kong, China

• About the Authors xiiiPreface xvAcknowledgements xviiNomenclature xix1 Introduction 11.1 Formulation 21.2 Context 51.3 Extreme Value Theory 51.4 Exclusion 61.5 Organization of this Book 71.6 Remarks on the Use of Risk Analysis 71.7 Conventions 8References 82 A Line of Thought 92.1 Numerical Integration 102.2 Perturbation 102.3 Gaussian Approximation 122.3.1 Single Design Point 122.3.2 Multiple Design Points 142.4 First/Second-Order Reliability Method 142.4.1 Context 152.4.2 Design Point 162.4.3 FORM 172.4.4 SORM 182.4.5 Connection with Gaussian Approximation 222.5 Direct Monte Carlo 242.5.1 Unbiasedness 252.5.2 Mean-Square Convergence 252.5.3 Asymptotic Distribution (Central Limit Theorem) 282.5.4 Almost Sure Convergence (Strong Law of Large Numbers) 312.5.5 Failure Probability Estimation 322.5.6 CCDF Perspective 342.5.7 Rare Event Problems 382.5.8 Variance Reduction by Conditioning 412.6 Importance Sampling 442.6.1 Optimal Sampling Density 452.6.2 Failure Probability Estimation 452.6.3 Shifting Distribution 462.6.4 Benefits and Side-Effects 482.6.5 Bias 502.6.6 Curse of Dimension 532.6.7 CCDF Perspective 562.7 Subset Simulation 582.8 Remarks on Reliability Methods 602A.1 Appendix: Laplace Type Integrals 61References 623 Simulation of Standard Random Variable and Process 653.1 Pseudo-Random Number 653.2 Inversion Principle 663.2.1 Continuous Random Variable 673.2.2 Discrete Random Variables 673.3 Mixing Principle 683.4 Rejection Principle 693.4.1 Acceptance Probability 713.5 Samples of Standard Distribution 723.6 Dependent Gaussian Variables 783.6.1 Cholesky Factorization 783.6.2 Eigenvector Factorization 813.7 Dependent Non-Gaussian Variables 833.7.1 Nataf Transformation 833.7.2 Copula 873.8 Correlation through Constraint 893.8.1 Uniform in Sphere 893.8.2 Gaussian on Hyper-plane 923.9 Stationary Gaussian Process 953.9.1 Autocorrelation Function and Power Spectral Density 953.9.2 Discrete-Time Process 993.9.3 Sample Autocorrelation Function and Periodogram 1003.9.4 Time Domain Representation 1013.9.5 The ARMA Process 1033.9.6 Frequency Domain Representation 1083.9.7 Remarks 1153A.1 Appendix: Variance of Linear System Driven by White Noise 1153A.2 Appendix: Verification of Spectral Formula 117References 1184 Markov Chain Monte Carlo 1194.1 Problem Context 1194.2 Metropolis Algorithm 1224.2.1 Proposal PDF 1234.2.2 Statistical Properties 1234.2.3 Detailed Balance 1284.2.4 Biased Rejection 1324.2.5 Reversible Chain 1344.3 Metropolis–Hastings Algorithm 1344.3.1 Detailed Balance 1354.3.2 Independent Proposal and Importance Sampling 1354.4 Statistical Estimation 1374.4.1 Properties of Estimator 1374.4.2 Chain Correlation 1394.4.3 Ergodicity 1434.5 Generation of Conditional Samples 1484.5.1 Curse of Dimension 1494.5.2 Independent Component MCMC 152References 1555 Subset Simulation 1575.1 Standard Algorithm 1575.1.1 Simulation Level 0 (Direct Monte Carlo) 1585.1.2 Simulation Level i = 1,…,m − 1 (MCMC) 1595.2 Understanding the Algorithm 1605.2.1 Direct Monte Carlo Indispensible 1605.2.2 Rare Regime Explored by MCMC 1615.2.3 Stationary Markov Chain from the Start 1615.2.4 Multiple Chains 1615.2.5 Seeds Discarded 1625.2.6 CCDF Perspective 1625.2.7 Repeated Samples 1625.2.8 Uniform Conditional Probabilities 1635.3 Error Assessment in a Single Run 1665.3.1 Heuristic Argument 1675.3.2 Efficiency Over Direct Monte Carlo 1695.4 Implementation Issues 1735.4.1 Proposal Distribution 1735.4.2 Ergodicity 1735.4.3 Generalizations 1745.4.4 Level Probability 1755.5 Analysis of Statistical Properties 1795.5.1 Random Intervals 1805.5.2 Random CCDF Values 1815.5.3 Summary of Results 1825.5.4 Expectation 1835.5.5 Variance 1855.6 Auxiliary Response 1905.6.1 Statistical Properties 1925.6.2 Design of Driving Response 1945.7 Black Swan Events 1955.7.1 Diagnosis 1975.8 Applications 1995.9 Variants 201References 2026 Analysis Using Conditional Failure Samples 2056.1 Probabilistic Failure Analysis 2066.2 Uncertain Parameter Sensitivity 2076.3 Conditional Samples from Direct Monte Carlo 2086.3.1 Conditional Expectation 2086.3.2 Parameter Sensitivity 2106.4 Conditional Samples from Subset Simulation 2166.4.1 Sample Partitioning 2176.4.2 Conditioning Structure 2196.4.3 Conditional Expectation 2206.4.4 Parameter Sensitivity 224References 2317 Spreadsheet Implementation 2337.1 Microsoft Excel and VBA 2337.1.1 Excel Spreadsheet 2347.1.2 Illustrative Example – Polynomial Function 2367.1.3 Visual Basic for Applications (VBA) 2427.1.4 VBA User-Defined Functions 2457.1.5 VBA Subroutines 2477.1.6 Macro Recorder 2517.2 Software Package UPSS 2557.2.1 Installation in Excel 2003 2557.2.2 Installation in Excel 2010 2587.2.3 Software Context 2607.2.4 Deterministic System Modeling 2617.2.5 Uncertainty Modeling 2627.2.6 Uncertainty Propagation 2627.2.7 Pre-Processing Tools 2657.2.8 Post-Processing Tools 2687.3 Tutorial Example – Polynomial Function 2697.3.1 Deterministic System Modeling 2707.3.2 Uncertainty Modeling 2707.3.3 Uncertainty Propagation 2727.3.4 Direct Monte Carlo 2747.3.5 Subset Simulation 2757.4 Tutorial Example – Slope Stability 2787.4.1 Problem Context* 2787.4.2 Deterministic System Modeling 2797.4.3 Uncertainty Modeling 2797.4.4 Histogram Tool 2817.4.5 Uncertainty Propagation 2827.4.6 CCDF of Driving Variable 2867.4.7 Auxiliary Variable 2867.5 Tutorial Example – Portal Frame 2887.5.1 Problem Context* 2897.5.2 Deterministic System Modeling 2907.5.3 Uncertainty Modeling 2917.5.4 Uncertainty Propagation 2947.5.5 Transforming Standard Normal Random Variables 2957.5.6 Introducing Correlation 299References 302A Appendix: Mathematical Tools 303A.1 Calculus 303A.1.1 Lagrange Multiplier Method 303A.1.2 Asymptotics 304A.2 Linear Algebra 304A.2.1 Linear Independence, Span, Basis 304A.2.2 Orthogonality and Norm 305A.2.3 Gram–Schmidt Procedure 306A.2.4 Eigenvalue Problem 307A.2.5 Real Symmetric Matrices 307A.2.6 Function of Real Symmetric Matrices 308A.3 Probability Theory 309A.3.1 Conditional Expectation 309A.3.2 Conditional Variance Formula 310A.3.3 Chebyshev’s Inequality 310A.3.4 Jensen’s Inequality 310A.3.5 Modes of Stochastic Convergence 311Index 313