Reference Guide
OpenTURNS 1.7
Documentation built from package openturns-doc-16.03
2016/03/17 15:59:39

Contents

1 Introduction

1.1 Presentation of the flood example

2 Global methodology of an uncertainty study

2.1 Step A: specification of the case-study

2.1.1 Variables of interest, model and input variables

2.1.1.1 Illustration on the flood example

2.1.2 Criteria of the uncertainty study

2.1.2.1 Deterministic criteria

2.1.2.2 A/ Criteria for random vectors

2.1.2.3 Probabilistic criteria: probability of exceeding a threshold / failure probability, and quantile

2.1.2.4 Probabilistic criteria: central dispersion

2.1.2.5 Illustration on the flood example

2.1.2.6 B/ Criteria for stochastic processes

2.1.2.7 Same deterministic and probabilistic criteria as for random vectors

2.1.2.8 Specific criteria for stochastic processes

2.1.2.9 Illustration on the flood example

2.2 Step B: quantification of the uncertainty sources

2.2.0.10 Deterministic criteria

2.2.0.11 Probabilistic criteria

2.2.0.12 Illustration on the flood example

2.2.0.13 B/ Criteria for stochastic processes

2.2.0.14 Remarks

2.2.0.15 Illustration on the flood example

2.3 Step C: uncertainty propagation

2.3.0.16 Deterministic criteria

2.3.0.17 Probabilistic criteria

2.3.0.18 Illustration on the flood example

2.4 Step C': Ranking uncertainty sources / Sensitivity analysis (only for random vectors)

2.4.0.19 Illustration on the flood example

3 OpenTURNS' methods for Step B: quantification of the uncertainty sources

3.1 Probabilistic models proposed in OpenTURNS

3.1.1 Non-parametric models

3.1.2 Parametric models

3.2 Classical statistical tools for uncertainty quantification

3.2.1 Aggregation of two samples

3.2.2 Estimation of a parametric models

3.2.3 Analysis of the goodness of fit of a parametric model

3.2.4 Detection and quantification of dependencies among uncertainty sources

3.3 Methods description

3.3.1 Step B  – Empirical cumulative distribution function

3.3.2 Step B  – Kernel Smoothing

3.3.3 Step B  – Standard parametric models

3.3.4 Step B  – Copula

3.3.5 Step B  – Random Mixture : affine combination of independent univariate distributions

3.3.6 Step B  – Using QQ-plot to compare two samples

3.3.7 Step B  – Comparison of two samples using Smirnov test

3.3.8 Step B  – Maximum Likelihood Principle

3.3.9 Step B  – Bayesian Calibration

3.3.10 Step B  – The Metropolis-Hastings Algorithm

3.3.11 Step B  – Parametric Estimation

3.3.12 Step B  – Graphical goodness-of-fit tests : QQ-plot, Kendall Plot and Henry line

3.3.13 Step B  – Chi-squared goodness of fit test

3.3.14 Step B  – Kolmogorov-Smirnov goodness-of-fit test

3.3.15 Step B  – Cramer-Von Mises goodness-of-fit test

3.3.16 Step B  – Anderson-Darling goodness-of-fit test

3.3.17 Step B  – Bayesian Information Criterion (BIC)

3.3.18 Step B  – Pearson Correlation Coefficient

3.3.19 Step B  – Pearson's correlation test

3.3.20 Step B  – Spearman correlation coefficient

3.3.21 Step B  – Spearman correlation test

3.3.22 Step B  – Chi-squared test for independence

3.3.23 Step B  – Linear regression

4 OpenTURNS' methods for Step C: uncertainty propagation

4.1 Min-max criterion

4.2 Probabilistic criterion

4.2.1 Central dispersion

4.2.2 Probability of exceeding a threshold / failure probability / probability of an event

4.2.3 Quantile of a variable of interest

4.3 Methods description

4.3.1 Step C  – Uniform Random Generator

4.3.2 Step C  – Distribution realizations

4.3.3 Step C  – Low Discrepancy Sequence

4.3.4 Step C  – Min-Max Approach

4.3.5 Step C  – Design of Experiments

4.3.6 Step C  – Optimization Algorithms

4.3.7 Step C  – Taylor variance decomposition / Perturbation Method

4.3.8 Step C  – Estimating the mean and variance using the Monte Carlo Method

4.3.9 Step C  – Isoprobabilistic transformation preliminary to FORM-SORM methods

4.3.10 Step C  – Generalized Nataf Transformation

4.3.11 Step C  – Rosenblatt Transformation

4.3.12 Step C  – FORM

4.3.13 Step C  – SORM

4.3.14 Step C  – Reliability Index

4.3.15 Step C  – Sphere sampling method

4.3.16 Step C  – Strong Maximum Test : a design point validation

4.3.17 Step C  – Monte Carlo Simulation

4.3.18 Step C  – Importance Simulation

4.3.19 Step C  – Directional Simulation

4.3.20 Step C  – Latin Hypercube Simulation

4.3.21 Step C  – Quasi Monte Carlo

4.3.22 Step C  – Estimating a quantile by Sampling / Wilks' Method

5 OpenTURNS' methods for Step C': ranking uncertainty sources / sensitivity analysis

5.1 Probabilistic criteria

5.1.1 Central dispersion probabilistic criterion

5.1.2 Probability of exceeding a threshold / failure probability

5.2 Methods description

5.2.1 Step C'  – Importance Factors derived from Taylor Variance Decomposition Method

5.2.2 Step C'  – Uncertainty ranking using Pearson's correlation

5.2.3 Step C'  – Uncertainty ranking using Spearman's correlation

5.2.4 Step C'  – Uncertainty Ranking using Standard Regression Coefficients

5.2.5 Step C'  – Uncertainty Ranking using Pearson's Partial Correlation Coefficients

5.2.6 Step C'  – Uncertainty Ranking using Partial Rank Correlation Coefficients

5.2.7 Step C'  – Sensivity analysis using Sobol indices

5.2.8 Step C'  – Sensivity analysis for models with correlated inputs

5.2.9 Step C'  – Sensivity analysis by Fourier decomposition

5.2.10 Step C'  – Importance Factors from FORM-SORM methods

5.2.11 Step C'  – Sensitivity Factors from FORM method

6 OpenTURNS' methods for the construction of response surfaces

6.1 Deterministic response surfaces

6.1.1 Polynomial response surfaces

6.1.2 Error estimation based on cross-validation

6.1.3 Kriging

6.2 Stochastic response surfaces

6.2.1 Functional chaos expansions

6.2.2 Polynomial chaos basis

6.2.3 Truncation schemes for the polynomial chaos expansions

6.2.4 Computation of the polynomial chaos coefficients

6.3 Methods description

6.3.1 Step Resp. Surf.  – Linear and Quadratic Taylor Expansions

6.3.2 Step Resp. Surf.  – Numerical methods to solve least squares problems

6.3.3 Step Resp. Surf.  – Polynomial response surface based on least squares

6.3.4 Step Resp. Surf.  – Polynomial response surface based on sparse least squares

6.3.5 Step Resp. Surf.  – Kriging

6.3.6 Step Resp. Surf.  – Assessment of the polynomial approximations by cross validation

6.3.7 Step Resp. Surf.  – Functional Chaos Expansion

6.3.8 Step Resp. Surf.  – Orthogonal polynomials

6.3.9 Step Resp. Surf.  – Polynomial chaos basis

6.3.10 Step Resp. Surf.  – Strategies for enumerating the polynomial chaos basis

Index