Uncertainty Analysis: With High Dimensional Dependence Modeling

Preface.

1 Introduction.

1.1 Wags and Bogsats.

1.2 Uncertainty analysis and decision support: a recent example.

1.3 Outline of the book.

2 Assessing Uncertainty on Model Input.

2.1 Introduction.

2.2 Structured expert judgment in outline.

2.3 Assessing distributions of continuous univariate uncertain quantities.

2.4 Assessing dependencies.

2.5 Unicorn.

2.6 Unicorn projects.

3 Bivariate Dependence.

3.1 Introduction.

3.2 Measures of dependence.

3.2.1 Product moment correlation.

3.2.2 Rank correlation.

3.2.3 Kendall’s tau.

3.3 Partial, conditional and multiple correlations.

3.4 Copulae.

3.4.1 Fréchet copula.

3.4.2 Diagonal band copula.

3.4.3 Generalized diagonal band copula.

3.4.4 Elliptical copula .

3.4.5 Archimedean copulae.

3.4.6 Minimum information copula.

3.4.7 Comparison of copulae.

3.5 Bivariate normal distribution.

3.5.1 Basic properties.

3.6 Multivariate extensions.

3.6.1 Multivariate dependence measures.

3.6.2 Multivariate copulae.

3.6.3 Multivariate normal distribution.

3.7 Conclusions.

3.8 Unicorn projects.

3.9 Exercises.

3.10 Supplement.

4 High-dimensional Dependence Modelling.

4.1 Introduction.

4.2 Joint normal transform.

4.3 Dependence trees.

4.3.1 Trees.

4.3.2 Dependence trees with copulae.

4.3.3 Example: Investment.

4.4 Dependence vines.

4.4.1 Vines.

4.4.2 Bivariate- and copula-vine specifications.

4.4.3 Example: Investment continued.

4.4.4 Partial correlation vines.

4.4.5 Normal vines.

4.4.6 Relationship between conditional rank and partial correlations on a regular vine.

4.5 Vines and positive definiteness.

4.5.1 Checking positive definiteness.

4.5.2 Repairing violations of positive definiteness.

4.5.3 The completion problem.

4.6 Conclusions.

4.7 Unicorn projects.

4.8 Exercises.

4.9 Supplement.

4.9.1 Proofs.

4.9.2 Results for Section 4.4.6.

4.9.3 Example of fourvariate correlation matrices.

4.9.4 Results for Section 4.5.2.

5 Other Graphical Models.

5.1 Introduction.

5.2 Bayesian belief nets.

5.2.1 Discrete bbn’s.

5.2.2 Continuous bbn’s.

5.3 Independence graphs.

5.4 Model inference.

5.4.1 Inference for bbn’s .

5.4.2 Inference for independence graphs.

5.4.3 Inference for vines.

5.5 Conclusions.

5.6 Unicorn projects.

5.7 Supplement.

6 Sampling Methods.

6.1 Introduction.

6.2 (Pseudo-) random sampling.

6.3 Reduced variance sampling.

6.3.1 Quasi-random sampling.

6.3.2 Stratified sampling.

6.3.3 Latin hypercube sampling.

6.4 Sampling trees, vines and continuous bbn’s.

6.4.1 Sampling a tree.

6.4.2 Sampling a regular vine.

6.4.3 Density approach to sampling regular vine.

6.4.4 Sampling a continuous bbn.

6.5 Conclusions.

6.6 Unicorn projects.

6.7 Exercise.

7 Visualization.

7.1 Introduction.

7.2 A simple problem.

7.3 Tornado graphs.

7.4 Radar graphs.

7.5 Scatter plots, matrix and overlay scatter plots.

7.6 Cobweb plots.

7.7 Cobweb plots local sensitivity: dike ring reliability.

7.8 Radar plots for importance; internal dosimetry.

7.9 Conclusions.

7.10 Unicorn projects.

7.11 Exercises.

8 Probabilistic Sensitivity Measures.

8.1 Introduction.

8.2 Screening techniques.

8.2.1 Morris’ method.

8.2.2 Design of experiments.

8.3 Global sensitivity measures.

8.3.1 Correlation ratio.

8.3.2 Sobol indices.

8.4 Local sensitivity measures.

8.4.1 First order reliability method.

8.4.2 Local probabilistic sensitivity measure.

8.4.3 Computing.

8.5 Conclusions.

8.6 Unicorn projects.

8.7 Exercises.

8.8 Supplement .

8.8.1 Proofs.

9 Probabilistic Inversion.

9.1 Introduction.

9.2 Existing algorithms for probabilistic inversion.

9.2.1 Conditional sampling.

9.2.2 PARFUM.

9.2.3 Hora-Young and PREJUDICE algorithms.

9.3 Iterative algorithms.

9.3.1 Iterative proportional fitting.

9.3.2 Iterative PARFUM.

9.4 Sample re-weighting.

9.4.1 Notation.

9.4.2 Optimization approaches.

9.4.3 IPF and PARFUM for sample re-weighting probabilistic inversion.

9.5 Applications.

9.5.1 Dispersion coefficients.

9.5.2 Chicken processing line.

9.6 Convolution constraints with prescribed margins.

9.7 Conclusions.

9.8 Unicorn projects.

9.9 Supplement.

9.9.1 Proofs.

9.9.2 IPF and PARFUM.

10 Uncertainty and the UN Compensation Commission.

10.1 Introduction.

10.2 Claims based on uncertainty.

10.3 Who pays for uncertainty.

Bibliography.

Index.