# Overview

OpenTURNS is able to perform a complete probabilistic study:

• The first step consists in identifying the numerical model through which one wants to propagate uncertainties, and to define the decision criteria
• The second step consists in quantifying uncertainty sources : OpenTURNS provides tools that can analyze data samples in order to chose the best uncertainty source modeling ;
• The third step of a study allows to propagate the uncertainty in the numerical model ;
• The fourth step of a study gives the possibility to analyze the importance of each parameter (ranking uncertainty sources, sensitivity analysis).

All the OpenTURNS' features (of version 0.12.0) for each step are listed below:

# First step: Quantifying uncertainty sources

• Definition of the numerical model :
• Link with external numerical simulation process (wrapper)
• Analytical formulas
• Python functions
• Meta-model by Taylor Extension
• Meta-model by Least Square method
• Scalar failure criteria

# Second step: Quantifying uncertainty sources

• Empirical cumulative distribution function
• Density by Kernel Smoothing (General Kernel Product)
• Composed distribution (1D marginals and copula)
• Composed copula (copulas linked by the Independent Copula)
• Standard parametric models:
• Normal
• Gumbel
• Logistic
• Student
• Exponential
• Weibull
• Gamma
• Lognormal
• Truncated Normal
• Triangular distribution
• Uniform
• Beta
• Geometric
• Poisson
• Multi-normal
• Non-central Student
• Epanechnikov
• Independent Copula
• Normal Copula
• Clayton copula
• Franck copula
• Gumbel Copula
• Using QQ-plot to compare two samples
• Smirnov's test
• Maximum Likehood Method
• Graphical goodness-of-fit analysis
• Chi-squared goodness-of-fit test
• Kolmogorov Smirnov goodness-of-fit test
• Cramer-Von Mises goodness-of-fit analysis
• Anderson-Darling goodness-of-fit analysis
• Bayesian Information Criterion
• Pearson Correlation Coefficient
• Pearson's correlation test
• Spearman Correlation Coefficient
• Spearman's correlation test
• Chi-squared test for independence
• Linear regression

# Third step: Uncertainty propagation

• Min-Max Approach using Design of Experiments
• Deterministic Min-Max using TNC (Truncated Newton Constrained) algorithm
• Quadratic Combination / Perturbation Method
• Estimating the mean and variance using the MC Method
• Iso-Probabilistic transformation preliminary to FORM-SORM methods
• FORM
• SORM
• Optimization algorithms :
• Cobyla
• AbdoRackwitz
• SQP
• Calculation of Reliability index :
• Hasofer reliability index
• Generalized reliability index
• Generalized Breitung reliability index
• Generalized HohenBichler reliability index
• Generalized Tvedt reliability index
• Design point validation : Strong Maximum Test
• Estimating the probability of an event :
• Monte Carlo Sampling
• Importance Sampling
• Directional Simulation
• Latin Hypercube Sampling
• Controlled Importance Sampling in standard space
• Estimating a quantile by Sampling / Wilk's Method

# Fourth step: Ranking uncertainty sources / sensitivity analysis

• Importance Factors derived from Quadratic Combination Method
• Uncertainty ranking using Pearson's correlation
• Uncertainty ranking using Spearman's correlation
• Uncertainty ranking using Standard Regression Coefficients
• Uncertainty ranking using Pearson's partial correlation coefficients
• Uncertainty ranking using Partial Rank correlation coefficients
• Importance Factors derived from FORM / SORM methods
• Sensitivity Factors from FORM method