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Contents

Introduction

Loading the openturns python library

Verbosity level of the Open TURNS platform

Some useful general commands

Link with other python standards

1 Probabilistic input vector modelisation

1.0.1 UC : Creation of the random input vector from a distribution

1.1 Without samples of data

1.1.1 UC : List of usual distributions

1.1.2 UC : Creation of a truncated distribution

1.1.3 UC : Creation of a copula and a composed copula

1.1.4 UC : Creation of nD distribution from (marginals, copula)

1.1.5 UC : Creation of a nD distribution from a Mixture

1.1.6 UC : Manipulation of a distribution

1.2 With samples of data : manipulation of data

1.2.1 UC : Import / Export data from a file at format CSV (Comma Separated Value)

1.2.2 UC : Drawing Empirical CDF, Histogram, Clouds / PDF or superposition of two clouds from data

1.2.3 UC : Do two samples have the same distribution : QQ-plot visual test, Smirnov numerical test

1.2.4 UC : Are two scalar samples independent : ChiSquared test, Pearson test, Spearman test

1.2.5 UC : Particular manipulations of the Pearson and Spearman tests, when the first sample is of dimension superior to 1.

1.2.6 UC : Regression test between two scalar numerical samples

1.2.7 UC : Distribution fitting tests, numerical and visual validation tests : Chi-squared test, Kolmogorov test, QQ-plot graph

1.2.8 UC : Normal distribution fitting test, visual validation tests (Henry line) and numerical validation tests in extreme zones (Anderson Darling test and Cramer Von Mises test)

1.2.9 UC : Making a choice between multiple fitted distributions : Kolmogorov ranking, ChiSquared ranking and BIC ranking

1.2.10 UC : PDF fitting by kernel smoothing and graphical validation : superposition of the empirical and kernel smoothing CDF

1.2.11 UC : Estimating a Copula from a sample

1.2.12 UC : Validating a Copula with the Kendall Plot Test

1.2.13 UC : Building and validating a linear model from two samples

1.2.14 UC : Statistical manipulations on data : min, max, covariance, skewness, kurtosis, quantile, empirical CDF, Pearson, Kendall and Spearman correlation matrixes and rank/sort functionnalities

1.2.15 UC : Drawing one cloud

1.2.16 UC : Maximum likelihood of a given probability density function

1.3 Bayesian modeling

1.3.1 UC : Creation of a random vector with random parameters

2 Creation of the limit state function and the output variable of interest

2.1 Creation of the limit state function

2.1.1 UC : From an external wrapper with gradient and hessian implementations

2.1.2 UC : From an analytical formula declared inline

2.1.3 UC : From a fonction defined in the script python

2.1.4 UC : Some particular functions : linear combination, agregation, composition

2.1.5 UC : Introducing some deterministic variables, using a LinearNumericalMathFunction

2.1.6 UC : Introducing some deterministic variables, optimizing memory and CPU time

2.1.7 UC : Manipulation of a NumericalMathFunction

2.1.8 UC : Creation of a Dynamical Function

2.2 Creation of the output variable of interest from the limit state function and the input random vector

2.2.1 UC : Creation of the ouput random vector

2.2.2 UC : Extraction of a random subvector from a random vector

2.3 Creation of the output variable of interest defined as an affine combination of input random variables

2.3.1 UC : Creation of a Random Mixture

2.4 Creation of the output variable of interest from the result of a polynomial chaos expansion

2.4.1 UC : Creation of the output variable of interest from the result of a polynomial chaos expansion

2.4.2 UC : Creation of a specialized random vector for the global sensitivity analysis using a polynomial chaos expansion

3 Uncertainty propagation and Uncertainty sources ranking

3.1 UC : Parametrisation of the Random Generator

3.2 UC : Generation of low discrepancy sequences

3.3 Min/Max approach

3.3.1 UC : Creation of a deterministic design of experiments : Axial, Box, Composite, Factorial patterns

3.3.2 UC : Creation of a random design of experiments : Monte Carlo, LHS patterns

3.3.3 UC : Re-use of a specified numerical sample as design of experiments

3.3.4 UC : Creation of a mixed deterministic / random design of experiments

3.3.5 UC : Drawing an design of experiments in dimension 2

3.3.6 UC : Min/Max research from an design of experiments and sensitivity analysis

3.3.7 UC : Min/Max research with an optimization algorithm

3.4 Random approach : central uncertainty

3.4.1 UC : Sensitivity analysis : Sobol indices

3.4.2 UC : Sensitivity analysis : Cobweb graph

3.4.3 UC : Moments evaluation from the Taylor variance decomposition method and evaluation of the importance factors associated

3.4.4 UC : Moments evaluation of a random sample of the output variable of interest

3.4.5 UC : Correlation analysis on samples : Pearson and Spearman coefficients, PCC, PRCC, SRC, SRRC coefficients

3.4.6 UC : Quantile estimations : Wilks and empirical estimators

3.5 Random approach : threshold exceedance

3.5.1 UC : Creation of an event in the physical and the standard spaces

3.5.2 UC : Manipulation of a StandardEvent

3.5.3 UC : Creation of an analytical algorithm : FORM/SORM

3.5.4 UC : Run and results exploitation of a FORM/SORM algorithm : probability estimation, importance factors, reliability indexes, sensitivity factors

3.5.5 UC : Validate the design point with the Strong Maximum Test

3.5.6 UC : Creation of a Monte Carlo / LHS / Quasi Monte Carlo / Importance Sampling simulation algorithm

3.5.7 UC : Creation of a Directional Sampling simulation algorithm

3.5.8 UC : Parametrisation of a simulation algorithm

3.5.9 UC : Run and results exploitation of a simulation algorithm : probability estimation, estimator variance, confidence interval, convergence graph, stored samples

3.5.10 UC : Probability evaluation from an analytical method (FORM/SORM) followed by a simulation method centered on the design point

4 Construction of a response surface

4.1 Taylor approximations

4.1.1 UC : Linear and Quadratic Taylor approximations

4.2 Least Squares approximation

4.2.1 UC : Linear Least Squares approximation from a sample of the input vector and the real function

4.2.2 UC : Linear Least Squares approximation from a sample of the input vector and a sample of the output vector

4.3 Polynomial chaos expansion

4.3.1 UC : Creation of a polynomial chaos algorithm

4.3.2 UC : Run and results exploitation of a polynomial chaos algorithm : coefficients, polynomial model, multivariate basis, truncated multivariate basis, ...

4.3.3 UC : Draw some usefull graphs associated to a polynomial chaos algorithm : polynomial graphs, comparison graph between numerical samples from the model and the meta-model, ...

5 Stochastic process

5.1 UC : Creation of a time grid

5.2 UC : Manipulation of a process

5.3 UC : Manipulation of a time series

5.4 UC : Manipulation of a process sample

5.5 UC : Manipulation of a White Noise

5.6 UC : Manipulation of a Random Walk

5.7 ARMA stochastic process

5.7.1 UC : Creation of an ARMA model

5.7.2 UC : Manipulation of an ARMA model

5.7.3 UC : Estimation of an ARMA model using the Whittle likelihood function

5.7.4 UC : Trend computation : identification and removal

5.7.5 UC : Box Cox method : identification and transformation

5.8 Normal process

5.8.1 UC : Creation of a stationary normal process from its covariance function

5.8.2 UC : Creation of a stationary normal process from its spectral density function

5.8.3 UC : Creation of stationary normal process from temporal and spectral views

5.8.4 UC : Manipulation of a stationary normal process

5.8.5 UC : Creation of a spectral model

5.8.6 UC : Estimation of a spectral model

5.9 Output stochastic process of interest

5.9.1 UC : Creation of a Composite Process

5.10 Event based on process

5.10.1 UC : Creation and manipulation of a process event

5.10.2 UC : Probability of a process event

6 How to save and load a study ?

6.1 UC : How to save a study ?

6.2 UC : How to load a study ?

Index

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