Uncertainty Analysis with High Dimensional Dependence Modelling

"an invaluable reference for researchers, practitioners and graduate students in several areas of statistics, engineering and technology." (Mathematical Reviews, 2007b) "...whether one is a researcher, student, or an industrial practitioner dealing with computer models the book comes highly recommended." (Technometrics, February 2007)

Dorota Kurowicka and Roger M. Cooke are the authors of Uncertainty Analysis with High Dimensional Dependence Modelling, published by Wiley.

Preface ix 1 Introduction 1 1.1 Wags and Bogsats 1 1.2 Uncertainty analysis and decision support: a recent example 4 1.3 Outline of the book 9 2 Assessing Uncertainty on Model Input 13 2.1 Introduction 13 2.2 Structured expert judgment in outline 14 2.3 Assessing distributions of continuous univariate uncertain quantities 15 2.4 Assessing dependencies 16 2.5 Unicorn 20 2.6 Unicorn projects 20 3 Bivariate Dependence 25 3.1 Introduction 25 3.2 Measures of dependence 26 3.2.1 Product moment correlation 26 3.2.2 Rank correlation 30 3.2.3 Kendalls tau 32 3.3 Partial, conditional and multiple correlations 32 3.4 Copulae 34 3.4.1 Frchet copula 36 3.4.2 Diagonal band copula 37 3.4.3 Generalized diagonal band copula 41 3.4.4 Elliptical copula 42 3.4.5 Archimedean copulae 45 3.4.6 Minimum information copula 47 3.4.7 Comparison of copulae 49 3.5 Bivariate normal distribution 50 3.5.1 Basic properties 50 3.6 Multivariate extensions 51 3.6.1 Multivariate dependence measures 51 3.6.2 Multivariate copulae 53 3.6.3 Multivariate normal distribution 53 3.7 Conclusions 54 3.8 Unicorn projects 55 3.9 Exercises 61 3.10 Supplement 67 4 High-dimensional Dependence Modelling 81 4.1 Introduction 81 4.2 Joint normal transform 82 4.3 Dependence trees 86 4.3.1 Trees 86 4.3.2 Dependence trees with copulae 86 4.3.3 Example: Investment 90 4.4 Dependence vines 92 4.4.1 Vines 92 4.4.2 Bivariate- and copula-vine specifications 96 4.4.3 Example: Investment continued 98 4.4.4 Partial correlation vines 99 4.4.5 Normal vines 101 4.4.6 Relationship between conditional rank and partial correlations on a regular vine 101 4.5 Vines and positive definiteness 105 4.5.1 Checking positive definiteness 105 4.5.2 Repairing violations of positive definiteness 107 4.5.3 The completion problem 109 4.6 Conclusions 111 4.7 Unicorn projects 111 4.8 Exercises 115 4.9 Supplement 116 4.9.1 Proofs 116 4.9.2 Results for Section 4.4.6 127 4.9.3 Example of fourvariate correlation matrices 129 4.9.4 Results for Section 4.5.2 130 5 Other Graphical Models 131 5.1 Introduction 131 5.2 Bayesian belief nets 131 5.2.1 Discrete bbns 132 5.2.2 Continuous bbns 133 5.3 Independence graphs 141 5.4 Model inference 142 5.4.1 Inference for bbns 143 5.4.2 Inference for independence graphs 144 5.4.3 Inference for vines 145 5.5 Conclusions 150 5.6 Unicorn projects 150 5.7 Supplement 157 6 Sampling Methods 159 6.1 Introduction 159 6.2 (Pseudo-) random sampling 160 6.3 Reduced variance sampling 161 6.3.1 Quasi-random sampling 161 6.3.2 Stratified sampling 164 6.3.3 Latin hypercube sampling 166 6.4 Sampling trees, vines and continuous bbns 168 6.4.1 Sampling a tree 168 6.4.2 Sampling a regular vine 169 6.4.3 Density approach to sampling regular vine 174 6.4.4 Sampling a continuous bbn 174 6.5 Conclusions 180 6.6 Unicorn projects 180 6.7 Exercise 184 7 Visualization 185 7.1 Introduction 185 7.2 A simple problem 186 7.3 Tornado graphs 186 7.4 Radar graphs 187 7.5 Scatter plots, matrix and overlay scatter plots 188 7.6 Cobweb plots 191 7.7 Cobweb plots local sensitivity: dike ring reliability 195 7.8 Radar plots for importance; internal dosimetry 199 7.9 Conclusions 201 7.10 Unicorn projects 201 7.11 Exercises 203 8 Probabilistic Sensitivity Measures 205 8.1 Introduction 205 8.2 Screening techniques 205 8.2.1 Morris method 205 8.2.2 Design of experiments 208 8.3 Global sensitivity measures 214 8.3.1 Correlation ratio 215 8.3.2 Sobol indices 219 8.4 Local sensitivity measures 222 8.4.1 First order reliability method 222 8.4.2 Local probabilistic sensitivity measure 223 8.4.3 Computing E(X|g o) go 225 8.5 Conclusions 227 8.6 Unicorn projects 228 8.7 Exercises 230 8.8 Supplement 236 8.8.1 Proofs 236 9 Probabilistic Inversion 239 9.1 Introduction 239