Estimation of Rare Event Probabilities in Complex Aerospace and Other Systems : A Practical Approach.
Morio, Jerome.
Estimation of Rare Event Probabilities in Complex Aerospace and Other Systems : A Practical Approach. - 1st ed. - 1 online resource (217 pages)
Front Cover -- Estimation of Rare Event Probabilities in Complex Aerospace and Other Systems: A Practical Approach -- Copyright -- Dedication -- Contents -- Preface -- Foreword -- Biography of the external contributors to this book -- Abbreviations -- Chapter 1: Introduction to rare event probability estimation -- 1.1 The book purposes -- 1.2 What are the events of interest considered in this book? -- 1.3 The book organization -- References -- Part One: Essential Background in Mathematics and System Analysis -- Chapter 2: Basics of probability and statistics -- 2.1 Probability theory operators -- 2.1.1 Elements of vocabulary -- 2.1.2 Notion of dependence of random events andconditional probabilities -- 2.1.3 Continuous random variables -- 2.1.3.1 Definitions -- 2.1.3.2 Parameters of continuous random variables -- 2.1.4 Continuous multivariate random variables -- 2.1.4.1 Definitions and theorems -- 2.1.4.2 Dependence of multivariate random variables -- 2.1.5 Point estimation -- 2.2 Random variable modeling -- 2.2.1 Overview of common probability distributions -- 2.2.1.1 Univariate distributions -- Uniform distribution -- Exponential distribution -- Gaussian distribution -- Truncated Gaussian distribution -- Log-normal distribution -- Cauchy distribution -- Chi-squared distribution -- Gamma and beta distributions -- Laplace distribution -- Some properties of univariate distributions -- 2.2.1.2 Multivariate distributions -- Multivariate normal distribution -- 2.2.2 Kernel-based laws -- 2.3 Convergence theorems and sampling algorithms -- 2.3.1 Strong law of large numbers -- 2.3.2 Central limit theorem -- 2.3.3 Simulation of complex laws using the Metropolis-Hastings algorithm -- 2.3.3.1 Markov chain -- 2.3.3.2 Some properties of transition kernels -- 2.3.3.3 The Metropolis-Hastings algorithm -- 2.3.3.4 Transformation of random variables -- References. Chapter 3: The formalism of rare event probability estimation in complex systems -- 3.1 Input-output system -- 3.1.1 Description -- 3.1.2 Formalism -- 3.2 Time-variant system -- 3.2.1 Description -- 3.2.2 Formalism -- 3.3 Characterization of a probability estimation -- References -- Part Two: Practical Overview of the Main Rare Event EstimationTechniques -- Chapter 4: Introduction -- 4.1 Categories of estimation methods -- 4.2 General notations -- 4.3 Description of the toy cases -- 4.3.1 Identity function -- 4.3.2 Polynomial square root function -- 4.3.3 Four-branch system -- 4.3.4 Polynomial product function -- References -- Chapter 5: Simulation techniques -- 5.1 Crude Monte Carlo -- 5.1.1 Principle -- 5.1.2 Application on a toy case -- Four-branch system -- 5.1.3 Conclusion -- 5.2 Simple variance reduction techniques -- 5.2.1 Quasi-Monte Carlo -- 5.2.2 Conditional Monte Carlo -- 5.2.2.1 Principle -- 5.2.2.2 Conclusion -- 5.2.3 Control variates -- 5.2.3.1 Principle -- 5.2.3.2 Application on a toy case -- Four-branch system -- 5.2.3.3 Conclusion -- 5.2.4 Antithetic variates -- 5.2.4.1 Principle -- 5.2.4.2 Application to a toy case -- Identity function -- 5.2.4.3 Conclusion -- 5.3 Importance sampling -- 5.3.1 Principle of importance sampling -- 5.3.2 Nonadaptive importance sampling -- 5.3.2.1 Simple changes of measure -- Principle -- Application to a toy case -- Conclusion -- 5.3.2.2 Exponential twisting -- Principle -- Application to a toy case -- Conclusion -- 5.3.3 Adaptive importance sampling -- 5.3.3.1 Cross-entropy optimization of the importance sampling auxiliary density -- Principle -- Application to toy cases -- Conclusion -- 5.3.3.2 Nonparametric adaptive importance sampling -- Principle -- Application to toy cases -- Conclusion -- 5.4 Adaptive splitting technique -- 5.4.1 Description -- 5.4.2 Application to toy cases -- Four-branch system. Polynomial product -- 5.4.3 Conclusion -- References -- Chapter 6: Statistical techniques -- 6.1 Extreme value theory -- 6.1.1 Law of sample maxima -- 6.1.2 Peak over threshold approach -- 6.1.2.1 Principle -- 6.1.2.2 Block maxima versus POT -- 6.1.3 Application to a toy case -- Four-branch system -- 6.1.4 Conclusion -- 6.2 Large deviation theory -- 6.2.1 Conclusion -- References -- Chapter 7: Reliability based approaches -- 7.1 First-order and second-order reliability methods -- 7.1.1 Principle -- 7.1.2 Application to toy cases -- Polynomial square root function -- Four-branch system -- 7.1.3 Conclusion -- 7.2 Line sampling -- 7.2.1 Principle -- 7.2.2 Application to toy cases -- Polynomial square root function -- Four-branch system -- 7.2.3 Conclusion -- 7.3 Directional sampling -- 7.3.1 Principle -- 7.3.2 Application to toy cases -- Four-branch system -- Polynomial product -- 7.3.3 Conclusion -- 7.4 Stratified sampling -- 7.4.1 Principle -- 7.4.2 Monte Carlo method with Latin hypercube sampling -- 7.4.3 Adaptive directional sampling -- 7.4.3.1 Principle -- 7.4.3.2 Application to toy cases -- Polynomial square root function -- Four-branch system -- Polynomial product -- 7.4.4 Conclusion -- 7.5 Geometrical methods -- References -- Chapter 8: Methods for high-dimensional and computationally intensive models -- 8.1 Sensitivity analysis -- 8.1.1 Importance measure-based methods -- 8.1.1.1 Decomposition of the variance -- Sobol indices -- ANOVA by design of experiment -- 8.1.1.2 Standardized regression coefficients -- 8.1.1.3 Correlation coefficients and partial correlation coefficients -- 8.1.2 Screening methods -- 8.1.2.1 One variable at a time -- 8.1.2.2 Morris method -- 8.1.3 General remark about SA for rare event probability estimation -- 8.2 Surrogate models for rare event estimation -- 8.2.1 Introduction -- 8.2.2 Support vector machines. 8.2.2.1 Presentation -- 8.2.2.2 Description -- Support vector machines for classification -- Support vector machines for regression -- 8.2.2.3 Refinement strategies -- Determining the SVM training set by minimizing (φ(x) −T )k -- Adaptive refinement of the SVM -- Subsets by support vector margin algorithm for reliability estimation (2SMART) -- Adaptive refinement of SVM using max-min technique -- Improvement of max-min technique: generalized max-min -- 8.2.3 Kriging -- 8.2.3.1 Presentation -- 8.2.3.2 Description of Kriging -- 8.2.3.3 Refinement strategies -- Direct methods -- One-step look-ahead methods -- K-means clustering strategy for Kriging refinement -- 8.2.4 Conclusion -- References -- Chapter 9: Special developments for time-variant systems -- 9.1 General notations -- 9.2 Toy case -- 9.3 Crude Monte Carlo -- 9.3.1 Principle -- 9.3.2 Application to a toy case -- Brownian bridge -- 9.3.3 Conclusion -- 9.4 Importance sampling -- 9.4.1 Principle -- 9.4.2 Application to a toy case -- Brownian bridge -- 9.4.3 Conclusion -- 9.5 Importance splitting -- 9.5.1 Principle -- 9.5.2 Application to a toy case -- Brownian bridge -- 9.5.3 Conclusion -- 9.6 Weighted importance resampling -- 9.6.1 Principle -- 9.6.2 Application to a toy case -- Brownian bridge -- 9.6.3 Conclusion -- 9.7 Extreme value theory -- 9.7.1 Principle -- 9.7.2 Application to a toy case -- Brownian bridge -- 9.7.3 Conclusion -- References -- Part Three: Benchmark of the Methods to Aerospace Problems -- Chapter 10: Estimation of launch vehicle stage fallout zone -- 10.1 Principle -- 10.2 Simulation description -- 10.3 Analysis of the input space -- 10.4 Estimation results -- 10.4.1 Adaptive splitting technique -- 10.4.2 Importance sampling -- 10.4.2.1 Nonparametric adaptive importance sampling -- 10.4.2.2 Cross-entropy optimization -- 10.4.3 Directional sampling. 10.4.4 Adaptive directional sampling -- 10.4.5 FORM/SORM -- 10.4.6 Line sampling -- 10.5 Conclusion -- Chapter 11: Estimation of collision probability between space debris and satellites -- 11.1 Principle -- 11.2 Simulation description -- 11.3 Analysis of the input space -- 11.4 Estimation results -- 11.4.1 Adaptive splitting technique -- 11.4.2 Importance sampling -- 11.4.2.1 Nonparametric adaptive importance sampling -- 11.4.2.2 Cross-entropy optimization -- 11.4.3 Directional sampling -- 11.4.4 Adaptive directional sampling -- 11.4.5 FORM/SORM -- 11.4.6 Line sampling -- 11.5 Conclusion -- References -- Chapter 12: Analysis of extreme aircraft wake vortex circulations -- 12.1 Principle -- 12.2 Simulation description -- 12.3 Estimation results -- 12.4 Conclusion -- References -- Chapter 13: Estimation of conflict probability between aircraft -- 13.1 Principle -- 13.2 Simulation description -- 13.3 Estimation results -- 13.3.1 Importance splitting -- 13.3.2 Weighted importance resampling -- 13.4 Conclusion -- References -- Part Four: Practical Guidelines of Rare Event Probability Estimation -- Chapter 14: Synthesis of rare event probability estimation methods for input-output systems -- 14.1 Synthesis -- 14.2 Some remarks for a successful practical rare event probability estimation -- Chapter 15: Synthesis for time-variant systems -- 15.1 Synthesis -- 15.2 Some remarks for a successful practical rareevent probability estimation -- Index -- Back Cover.
9780081001110
Reliability (Engineering)--Graphic methods.
Electronic books.
TL545
629.1
Estimation of Rare Event Probabilities in Complex Aerospace and Other Systems : A Practical Approach. - 1st ed. - 1 online resource (217 pages)
Front Cover -- Estimation of Rare Event Probabilities in Complex Aerospace and Other Systems: A Practical Approach -- Copyright -- Dedication -- Contents -- Preface -- Foreword -- Biography of the external contributors to this book -- Abbreviations -- Chapter 1: Introduction to rare event probability estimation -- 1.1 The book purposes -- 1.2 What are the events of interest considered in this book? -- 1.3 The book organization -- References -- Part One: Essential Background in Mathematics and System Analysis -- Chapter 2: Basics of probability and statistics -- 2.1 Probability theory operators -- 2.1.1 Elements of vocabulary -- 2.1.2 Notion of dependence of random events andconditional probabilities -- 2.1.3 Continuous random variables -- 2.1.3.1 Definitions -- 2.1.3.2 Parameters of continuous random variables -- 2.1.4 Continuous multivariate random variables -- 2.1.4.1 Definitions and theorems -- 2.1.4.2 Dependence of multivariate random variables -- 2.1.5 Point estimation -- 2.2 Random variable modeling -- 2.2.1 Overview of common probability distributions -- 2.2.1.1 Univariate distributions -- Uniform distribution -- Exponential distribution -- Gaussian distribution -- Truncated Gaussian distribution -- Log-normal distribution -- Cauchy distribution -- Chi-squared distribution -- Gamma and beta distributions -- Laplace distribution -- Some properties of univariate distributions -- 2.2.1.2 Multivariate distributions -- Multivariate normal distribution -- 2.2.2 Kernel-based laws -- 2.3 Convergence theorems and sampling algorithms -- 2.3.1 Strong law of large numbers -- 2.3.2 Central limit theorem -- 2.3.3 Simulation of complex laws using the Metropolis-Hastings algorithm -- 2.3.3.1 Markov chain -- 2.3.3.2 Some properties of transition kernels -- 2.3.3.3 The Metropolis-Hastings algorithm -- 2.3.3.4 Transformation of random variables -- References. Chapter 3: The formalism of rare event probability estimation in complex systems -- 3.1 Input-output system -- 3.1.1 Description -- 3.1.2 Formalism -- 3.2 Time-variant system -- 3.2.1 Description -- 3.2.2 Formalism -- 3.3 Characterization of a probability estimation -- References -- Part Two: Practical Overview of the Main Rare Event EstimationTechniques -- Chapter 4: Introduction -- 4.1 Categories of estimation methods -- 4.2 General notations -- 4.3 Description of the toy cases -- 4.3.1 Identity function -- 4.3.2 Polynomial square root function -- 4.3.3 Four-branch system -- 4.3.4 Polynomial product function -- References -- Chapter 5: Simulation techniques -- 5.1 Crude Monte Carlo -- 5.1.1 Principle -- 5.1.2 Application on a toy case -- Four-branch system -- 5.1.3 Conclusion -- 5.2 Simple variance reduction techniques -- 5.2.1 Quasi-Monte Carlo -- 5.2.2 Conditional Monte Carlo -- 5.2.2.1 Principle -- 5.2.2.2 Conclusion -- 5.2.3 Control variates -- 5.2.3.1 Principle -- 5.2.3.2 Application on a toy case -- Four-branch system -- 5.2.3.3 Conclusion -- 5.2.4 Antithetic variates -- 5.2.4.1 Principle -- 5.2.4.2 Application to a toy case -- Identity function -- 5.2.4.3 Conclusion -- 5.3 Importance sampling -- 5.3.1 Principle of importance sampling -- 5.3.2 Nonadaptive importance sampling -- 5.3.2.1 Simple changes of measure -- Principle -- Application to a toy case -- Conclusion -- 5.3.2.2 Exponential twisting -- Principle -- Application to a toy case -- Conclusion -- 5.3.3 Adaptive importance sampling -- 5.3.3.1 Cross-entropy optimization of the importance sampling auxiliary density -- Principle -- Application to toy cases -- Conclusion -- 5.3.3.2 Nonparametric adaptive importance sampling -- Principle -- Application to toy cases -- Conclusion -- 5.4 Adaptive splitting technique -- 5.4.1 Description -- 5.4.2 Application to toy cases -- Four-branch system. Polynomial product -- 5.4.3 Conclusion -- References -- Chapter 6: Statistical techniques -- 6.1 Extreme value theory -- 6.1.1 Law of sample maxima -- 6.1.2 Peak over threshold approach -- 6.1.2.1 Principle -- 6.1.2.2 Block maxima versus POT -- 6.1.3 Application to a toy case -- Four-branch system -- 6.1.4 Conclusion -- 6.2 Large deviation theory -- 6.2.1 Conclusion -- References -- Chapter 7: Reliability based approaches -- 7.1 First-order and second-order reliability methods -- 7.1.1 Principle -- 7.1.2 Application to toy cases -- Polynomial square root function -- Four-branch system -- 7.1.3 Conclusion -- 7.2 Line sampling -- 7.2.1 Principle -- 7.2.2 Application to toy cases -- Polynomial square root function -- Four-branch system -- 7.2.3 Conclusion -- 7.3 Directional sampling -- 7.3.1 Principle -- 7.3.2 Application to toy cases -- Four-branch system -- Polynomial product -- 7.3.3 Conclusion -- 7.4 Stratified sampling -- 7.4.1 Principle -- 7.4.2 Monte Carlo method with Latin hypercube sampling -- 7.4.3 Adaptive directional sampling -- 7.4.3.1 Principle -- 7.4.3.2 Application to toy cases -- Polynomial square root function -- Four-branch system -- Polynomial product -- 7.4.4 Conclusion -- 7.5 Geometrical methods -- References -- Chapter 8: Methods for high-dimensional and computationally intensive models -- 8.1 Sensitivity analysis -- 8.1.1 Importance measure-based methods -- 8.1.1.1 Decomposition of the variance -- Sobol indices -- ANOVA by design of experiment -- 8.1.1.2 Standardized regression coefficients -- 8.1.1.3 Correlation coefficients and partial correlation coefficients -- 8.1.2 Screening methods -- 8.1.2.1 One variable at a time -- 8.1.2.2 Morris method -- 8.1.3 General remark about SA for rare event probability estimation -- 8.2 Surrogate models for rare event estimation -- 8.2.1 Introduction -- 8.2.2 Support vector machines. 8.2.2.1 Presentation -- 8.2.2.2 Description -- Support vector machines for classification -- Support vector machines for regression -- 8.2.2.3 Refinement strategies -- Determining the SVM training set by minimizing (φ(x) −T )k -- Adaptive refinement of the SVM -- Subsets by support vector margin algorithm for reliability estimation (2SMART) -- Adaptive refinement of SVM using max-min technique -- Improvement of max-min technique: generalized max-min -- 8.2.3 Kriging -- 8.2.3.1 Presentation -- 8.2.3.2 Description of Kriging -- 8.2.3.3 Refinement strategies -- Direct methods -- One-step look-ahead methods -- K-means clustering strategy for Kriging refinement -- 8.2.4 Conclusion -- References -- Chapter 9: Special developments for time-variant systems -- 9.1 General notations -- 9.2 Toy case -- 9.3 Crude Monte Carlo -- 9.3.1 Principle -- 9.3.2 Application to a toy case -- Brownian bridge -- 9.3.3 Conclusion -- 9.4 Importance sampling -- 9.4.1 Principle -- 9.4.2 Application to a toy case -- Brownian bridge -- 9.4.3 Conclusion -- 9.5 Importance splitting -- 9.5.1 Principle -- 9.5.2 Application to a toy case -- Brownian bridge -- 9.5.3 Conclusion -- 9.6 Weighted importance resampling -- 9.6.1 Principle -- 9.6.2 Application to a toy case -- Brownian bridge -- 9.6.3 Conclusion -- 9.7 Extreme value theory -- 9.7.1 Principle -- 9.7.2 Application to a toy case -- Brownian bridge -- 9.7.3 Conclusion -- References -- Part Three: Benchmark of the Methods to Aerospace Problems -- Chapter 10: Estimation of launch vehicle stage fallout zone -- 10.1 Principle -- 10.2 Simulation description -- 10.3 Analysis of the input space -- 10.4 Estimation results -- 10.4.1 Adaptive splitting technique -- 10.4.2 Importance sampling -- 10.4.2.1 Nonparametric adaptive importance sampling -- 10.4.2.2 Cross-entropy optimization -- 10.4.3 Directional sampling. 10.4.4 Adaptive directional sampling -- 10.4.5 FORM/SORM -- 10.4.6 Line sampling -- 10.5 Conclusion -- Chapter 11: Estimation of collision probability between space debris and satellites -- 11.1 Principle -- 11.2 Simulation description -- 11.3 Analysis of the input space -- 11.4 Estimation results -- 11.4.1 Adaptive splitting technique -- 11.4.2 Importance sampling -- 11.4.2.1 Nonparametric adaptive importance sampling -- 11.4.2.2 Cross-entropy optimization -- 11.4.3 Directional sampling -- 11.4.4 Adaptive directional sampling -- 11.4.5 FORM/SORM -- 11.4.6 Line sampling -- 11.5 Conclusion -- References -- Chapter 12: Analysis of extreme aircraft wake vortex circulations -- 12.1 Principle -- 12.2 Simulation description -- 12.3 Estimation results -- 12.4 Conclusion -- References -- Chapter 13: Estimation of conflict probability between aircraft -- 13.1 Principle -- 13.2 Simulation description -- 13.3 Estimation results -- 13.3.1 Importance splitting -- 13.3.2 Weighted importance resampling -- 13.4 Conclusion -- References -- Part Four: Practical Guidelines of Rare Event Probability Estimation -- Chapter 14: Synthesis of rare event probability estimation methods for input-output systems -- 14.1 Synthesis -- 14.2 Some remarks for a successful practical rare event probability estimation -- Chapter 15: Synthesis for time-variant systems -- 15.1 Synthesis -- 15.2 Some remarks for a successful practical rareevent probability estimation -- Index -- Back Cover.
9780081001110
Reliability (Engineering)--Graphic methods.
Electronic books.
TL545
629.1