Open TURNS User’s Day #5

12 juin 2012

Open TURNS User’s Day #5 Contents

Reception at EDF R&D Clamart

Open TURNS : Identity card

Consortium and Industrial Club

Web site

Open TURNS trainings

New features from the 0.14.0 release Scientific and technologic new features

New modules : FFTW, SVM, Matplotlib, MixMod (modifications)

Open TURNS and the Stochastic Processes « Processus gaussiens et leur simulation » (1h) : F. Poirion (ONERA)

« The Stochastic Processes in Open TURNS v1.0 » (30’) : A. Dutfoy (EDF R&D MRI)

Studies using Open TURNS Study 1 :« Modélisation de variables dépendantes pour l'étude de la dispersion de la signature

infrarouge» (30’) : S. Lefebvre (ONERA) 

Study 2 : « Assistance à l’élaboration de gammes d’assemblage innovantes de structures composites » (30’) : L. Andalfatto (EADS-IW / ENS Cachan) 

Study 3 : « Interfaces Open TURNS  - Java, Matlab, Octave » (30’) : P. Rekapalli (EADS-IW)

Study 4 : « Open TURNS en mécanique » (30’) : A-L. Popelin (EDF)

Study 5 : « Analyse de sensibilité multi-échelle – Open TURNS / YACS » : Y. Caniou 

Next works of the Consortium and Discussion

9h30 – 10h30

10h30 – 12h

14h – 16h

16h15 – 17h

9h – 9h30

12h – 12h30

Open TURNS : identity card

Open TURNS aims at propagating uncertainties through a model until a given variable of interest. It proposes some functionalities in order to quantify uncertainties in order to rank their influence on the variable of interest. Open TURNS also enables to build meta models.

Partners 2005 – 2011 : EDF – R&D / EADS – Innovation Works / PhiMECA : transparency : to make the methodology and the associated software be accepted by certification or control entities

genericity : to benefit from different fields where the same approach is developped (mechanics, hydraulics, ...)

computing performance : to enable the treatment of complex industrial cases (parallélisation, distribution, ...)

Open TURNS : an open source platform dedicated to uncertainty treatment by probabilistic methods Open source Treatment of Uncertainty, Risk ‘N Statistics

sous LGPL licence

www.openturns.org

Uncertainty Methodology

Open TURNS is the software associated to the Global Methodology of Treatment of Uncertainties developped first at EDF R&D in 1990 and then improved by contributions from other companies :

Step A : Study Specification : uncertain input variables, model, variable of interest, analysis criteria (range study, central dispersion, probability to exceed an extrem threshold)

Step B : Uncertainty Quantification : modelisation of the joint distribution of the input uncertain parameters

Step C : Uncertainty Propagation : quantification of the analysis criteria

Step C’ : Uncertainty Ranking

feedback

Step C : Uncertainty Propagation

Step C’ : Sensitivity Analysis, Ranking

ModelG(x,d)

ModelG(x,d)

Input Variables

Incertain : xFixed : d

Input Variables

Incertain : xFixed : d

Variable of interestZ = G(x,d)

Variable of interestZ = G(x,d)

Decision Criteria

Ex: Probability < 10-b

Step A : Study specification

Quantity of interest

ex: variance, probability, ..

Quantity of interest

ex: variance, probability, ..

Step B:

Uncertainty Quantification

joint distribution

Uncertainty Methodology - Open TURNS

Numerous functionalities for each step of the Methodology in particular ....

Code linked to Open TURNS :

Generic Wrapper générique which enables to wrap OT to any code using input and output text files, with authomatic multithreading functionalities (parallélism multiprocessors / multicores)

Interface with python functions, which enables the realisation of complex wrappers without compilation

Standard Interface for the development of wrappers of any complexity (distributed wrapper, binary data) which require the development of an external wrapper

Salome compatibility : OpenTURNS composant of Salome, linked to YACS

Step B : Uncertainty Quantification

Estimation from data :

Distribution fittings (parametric or not),

Validation Tests (quantitative or graphical),

Estimation of the dependence : copula, correlation coefficient

Regression

Modelisation of Joint Distributions of dimension n :

• Combination Marginals + Copula,

• Parametric Distributions of dimension n (normal, student...)

• Non Parametric Distribution of dimension n : kernel fitting (n), Sklar Copula

• Linear combination of pdf

• Linear combination of random variables,

• Random sum of independent discrete variables according to a Poisson process

• …

Uncertainty Methodology - Open TURNS

Open TURNS = repository of innovant or recent algorithms the most recent and efficient algorithms of non uniform distribution generation

• Ziggurat method (2005) for the normal distribution

• sequential reject algorithm (1993) for the binomial distribution,

• Tsang & Marsaglia method (2000) for the gamma distribution,

• Lebrun algorithm (2012) for the MultiNomial distribution,

the most recent algorithms of evaluation of the CDF

• Marsaglia algorithm for the exact statistics of Kolmogorov (2003),

• Benton et Krishnamoorthy algorithm for the distributions non centered Student and non centered Chi2 (2003).

PhD results

• Sparse chaos expansion polynomials : G. Blatman (EDF/R&D/MMC) (2010)

• Accelerated simulation algorithm for the evaluation of low probabilities : M. Munoz (EDF/R&D/MRI) : (in projet)

Innovation : the internal data model is based on the multidimensional cumulative density function which enables :

the sampling approach : estimation of statistical characteristics of the output variable from a large numerical sample

and the analytical approach : exact partial or total résolution of some problems using the probabilistic modelisation and the implementation in Open TURNS of the function R n --> R p algebra until order 2:

• exact determination of the distribution of the sum of random variables thanks to the characteristic functions implemented for each distribution,

• exact representation of the distribution of a random variable such that its pdf is a linear comibnation of pdf,

• exact determination of the distribution of a random variable defined as the random sum according to a Poisson process of iid discrete random variables (for ex, cumulated failure times when failures follow a Poisson process)

• particular manipulations on distributions : extraction of marginals, extraction of dependence structure, ...

Open TURNS in images

Open TURNS in images

Open TURNS in images

Open TURNS in images

Open TURNS : software, doc and Users

Open TURNS is ….

a C++ library giving functionalities for the treatment of uncertainties

a pyhton module offering high level operators

With a detailed documentation :

scientific documentation : Reference Guide,

Users documentation : Use Cases Guide, User Manual, Examples Guide

technical documentation : Architecture Guide, Wrapper Guide, Contribution Guide, Windows port doc.

… and a sympathetic community :

Openturns.org : official web site

a particular page share to communicate about the sofware

the annual Users day

Open TURNS ans its Consortium

The Open TURNS Consortium is organised around two main committees in close collaboration :

Steering Committee : driving the evolution of the scientific content and the architecture, responsible for the releases, organisation of the User’s Day

Technical Committee : responsible for the scientific and technological content, audit of the different contributions, pro active to propose evolutions and partnerships

How to contribute to / interact with Open TURNS ?

Level 1 : the contributor wants to share around the tool by proposing new functionalities (developed under C++ or python with the module mechanism), a script (pre / post processing of data, coupling with other tools, ….)

GO TO share.openturns.org

Level 2 : the contributor wants to contribute to the official version of Open TURNS

Contribution to the C++ library, to the python TUI python

Open TURNS quality criteria to be respected (at the end) : code source + documentation, detailed in the Coding Rules Guide + Contribution Guide

don’t hesitate to contact the Open TURNS Consoritum as soon as possible!

Open TURNS ans its User Club

Current situation :

The Open TURNS user's community is increasing :

• industrial : SNCF, Dassault-Aviation, MBDA, Astrium, Airbus, Michelin, Lafarge, Orange, EDF-SEPTEN, EDF-CIDEN,

• Research centers and universities : LNE, EDF-R&D, EADS-IW, ENS Cachan, IFMA, CEA

• Services : InModelia, IMACS, Phimeca

• projets ANR, européens, ...

Open TURNS is closer to the industrial computational chains :

• EDF : Salome et SalomeMeca, ...

• ERDF : plateform of computation for the safety of electric networks

• Other applications

Open TURNS is interfaced to:

• Matlab,

• SciLab,

• PhimecaSoft, ...

==> increasing needs in the community :

To train the users,

To enrich the tool with specific needs coming from various areas,

To adapt the tool to specific constraints,

To ensure the long-term maintenance and ensure the adequate quality of service, ...

Open TURNS ans its User Club

Since 2012 : creation of the Open TURNS Users Club

Industrial people & Academics

To share best practices around the use of Open TURNS in uncertainty studies :

• discussions

• Scientific presentations around the uncertainty management methodology

• Use-case with the tool

To mutualize the developments to enrich the platform:

• Common requirements shared by several industrial to prototype scientfic developments ( module C++, production of python scripts, ...),

• The actions of developments and maintenance done by partners labelled by the Open TURNS consortium Open TURNS,

To organize training and communication actions

… you are welcome !

(contacts : anne.dutfoy@edf.fr, fabien.mangeant@eads.net, yalamas@phimeca.com)

Open TURNS – www.openturns.org

Quick access to main parts of the website by left and upper menu

Open TURNS – www.openturns.org

Easy acces to the download page

Documentation

Version 1.0 => 168 downloads between the 22th of april and the 1st of june

France : 68 %

Open TURNS – Trainings

EDF

« Uncertainty Management : Open TURNS »

• 3 days to learn the use of Open TURNS : python TUI, wrapping aspects, application of the methodologie from step A to step C & C’

• next session in october 2013, ITECH ref : 4888

contact : Corine Tripet : 01 47 65 58 41

MAI : 26 to 28 novembre 2012, Uncertainty Management in Computational Mechanics

General Methodology, Probabilistic Fatigue Design, etc… & illustrations with OpenTURNS

contact : Marc Berveiller, EDF : 01 60 73 73 66

PHIMECA

Several 2 days sessions on the different steps from the methodology:

Probability & Statistics

Response Surfaces Models

Uncertainty propagation methods for sensitivity & dispersion analysis

Uncertainty propagation methods for reliability evaluation

Introduction to the use of OpenTURNS : 2 days

Introduction to the use of PhimecaSoft : 2 days

New features from the 0.14.0

Python types capabilities

Numpy array / matrix ↔ Open TURNS Matrix conversion Available for all Matrix types : Matrix, SymmetricMatrix, ComplexMatrix, Tensor, … Both ways numpy.array ↔ ot.Matrix: a0 = np.array( ((1.+3j,0.),(3.+7j,4.-9j)) ) m0 = ot.ComplexMatrix( a0 ) a1 = np.array( m0 )

Implicit conversion for Indices, Description, CorrelationMatrix, CovarianceMatrix types No need to specify these types anymore: v0.15 : marginal = distribution.getMarginal( Indices( [ 1, 2, 4 ] ) ) v1.0  : marginal = distribution.getMarginal( [ 1, 2, 4 ] ) Python philosophy : hidden basic types, shorter scripts

More operators for NumericalPoint, NumericalSample, Matrix NumericalPoint : __div__ → x / 3.0 NumericalPoint : __rmul__ → 3.0 * x NumericalSample : __eq__ → if xs == y: Matrix : __rmul__ → 3.0 * m

New features from the 0.14.0

Scientific aspects

Chaos polynomial expansion, sparse chaos :

• Othonormalised polynomials for the discrete distributions : CharlierFactory, MeixnerFactory, KrawtchoukFactory

• Model selection: Kfold

• Projection algorithm: PenalizedLeastSquaresAlgorithmFactory

Additional distributions and factories :

• 1 D continuous distributions : LogUniform, LogUniformFactory

• 1 D discrete distributions : NegativeBinomial

• N D continuous distribution: MinCopula

Numerical models :

• Vectorial functions : AggregatedNumericalMathEvaluationImplementation, AnalyticalNumericalMathEvaluationImplementation, DatabaseNumericalMathEvaluationImplementation

• Process rransformations :DynamicalFunction, TemporalFunction, SpatialFunction,, BoxCoxFactory, BoxCoxTransform, InverseBoxCoxTransform, InverseTrendTransform,, TrendFactory, TrendTransform

• Differentiation: BlendedStep

Last Releases : 0.15.0 in April 2011; 1.0 in April 2012.

New features from the 0.14.0

Scientific aspects

Processes (1.0)

• Families of processes: CompositeProcess, Process, RandomWalk, ARMA, WhiteNoise, SpectralNormalProcess, TemporalNormalProcess

• Second order models : CovarianceModel, CauchyModel, ExponentialCauchy, ExponentialModel, StationaryCovarianceModel, SecondOrderModel, SpectralModel, SpectralModelFactory, UserDefinedSpectralModel

• Statistiocal estimation : WelchFactory, WhittleFactory

• Filtrage: FilteringWindows, Hamming, Hanning

• Event : Domain, EventDomainImplementation, EventProcess

• Data contenairs : ProcessSample, RegularGrid, TimeSeries

Diverse

• Numerical analysis :

• Linear algebra : ComplexMatrix, HermitianMatrix, TriangularComplexMatrix, SquareComplexMatrix

• Fourier transform : FFT, KissFFT

• Design of Experiments : GaussProductExperiment, RandomizedLHS, RandomizedQuasiMonteCarlo, HaselgroveSequence, ReverseHaltonSequence

• Graphical analysis : SimulationSensitivityAnalysis

Some new modules : MixMod & FFTW

See presentation .pdf

Some new modules : Matplotlib based module

New graphing module

Allows to visualize Open TURNS graphs through python matplotlib plotting library

Replaces graphing with R command line, Rpy2, ...

Outputs nicer, more customizable plots

Available as an independent python module until next release

view = View( graph )

view.save( 'curve.png' )

view.show()

pyplot.text( 0.15, 0.6, 'Zou!')

New matplotlib based module

Curve

New matplotlib based module

Contour

New matplotlib based module

Histogram

New matplotlib based module

Qqplot

New matplotlib based module

Cloud

New matplotlib based module

CobWeb

New matplotlib based module

Staircase

New matplotlib based module

Convergence graph

Open TURNS : its logo

The Galton Board (1822 - 1911) : british explorer, geograph, inventor, meteorologist, proto-geneticist, psychometrician, statistician

Open TURNS and the Stochastic Processes

« Processus gaussiens et leur simulation   » : F. Poirion (ONERA)

« Les processus stochastiques dans Open TURNS v1.0  » : A. Dutfoy (EDF R&D MRI)

Open TURNS and the Stochastic Processes

Process

getRealization

getTimeGrid

getSample

getMarginalProcess

RegularGrid

Process

TimeSeries

ProcessSample : collection of n TimeSeries

: de dimension m <d

computeTemporalMean

computeMean

computeQuantilePerComponent

NumericalSample

TimeSeries

TimeSeries

computeTemporalMean

drawMarginal(i)

ARMA

RandomWalk SpectralNormalProcess

TemporalNormalProcess

WhiteNoise

CompositeProcess

NumericalPoint

Graph

where

(X(,t)), tR

Open TURNS and the White Noise & Random Walk

WhiteNoise

WhiteNoise(myDistribution, myTimeGrid)

RandomWalk

RandomWalk(myOrigin, myDistribution, myTimeGrid)

Distribution RegularGridDistribution RegularGridNumericalPoint

P(-1) = 0.9; P(10) = 0.1

N(0,1)

N(0,1)

White noise Random walk

Open TURNS and the ARMA processes

ARMA

ARMA models stationnary processes :

ARMA(myARCoef, myMACoef,myWNdist) (a)

ARMA(myARCoef, myMACoef,myWNdist, myCurrentState) (b)

• getRealization()

does not depend on the current state

TimeSeries

==> thermalization step

add Nther instants to the time grid

generate the process on the new time grid

erase the first Nther instants

by default, Nther = f(roots of the AR polynomials)

! authomatically performed with (a) creation

• getSample(n) ProcessSample

• getPossibleFuture(Nit,n)

must take into account the influence of the current state on the first instants

==> NO thermalization step

! creation (b) is mandatory

• getPossibleFuture(Nit) TimeSeries

ProcessSample

! Care : Beyond Nther instants, the future has no link with the present

ARCoefficient

MACoefficient

ARMAStateWhiteNoise

Open TURNS and the ARMA estimation (1d)

Principles : White noise : Gaussian(0,2) Whittle likelihood function which is a ratio between an estimate of the spectral density function of the data and the analytical spectral density function of an ARMA(p,q) process Welch method to estimate the spectral density function of the data using a tappering window (Hamming, Hanning)

Estimation : for a given (p,q) for a range on p and a range on q : (pIndices, qIndices) decision criteria : corrected AIC other information : BIC and AIC criteria

The AIC and AIC c criteria select the best structure - AICc penalizes the high (p,q)

The BIC criteria selects the best prediction model

For n-> , the three selections (p,q) converge

WhittleFactory

• build (TS or PS) :• getHistory()

WhittleFactory(p,q) (a)

WhittleFactory(pIndices,qIndices) (b)

criterion = NumericalPoint()

criterion : in order to get the evaluation of the 3 criteria : AICc, AIC, BIC

• build (TS or PS, criterion) : ARMA WhittleFactoryStateCollection

stores all theARMA(p,q) models of the range ranks the models with respect to the AICc criteria stores the evaluation of the 3 criteria for each ARMA(p,q)

TimeSeries

ProcessSample

NumericalPoint

Indice

Open TURNS and the Normal processes

Normal processes

• if stationnary, defined by its stationnary covariance function

• if stationnary, defined by its spectral density function

ExponentialModel

CauchyModel

UserDefinedModel

TemporalNormalProcess SpectralNormalProcess

==> ==>

SecondOrderModel

ExponentialCauchyModel

in order to insure the coherence between the covariance function and the spectral density function

==>

WechFactory

• may be estimated from data (non parametric Welch estimator)

• defined by its covariance function

Open TURNS and the trend computation

Principles : In general, the time series (Yt)t writes :

(Xt)t stationnaryThe objective here is to identify the trend function f and to remove it from the observed time series in order to analyse the stationary time series (Xt)t .

==> We decompose f over the functional basis B :

Coefficients identification : algorithms which combine cross validation techniques and advanced regression strategies, in order to provide the estimation of the coefficients associated to the best model among the basis functions (sparse model).

==> Construction of the ‘Trend Factory’ :

TrendFactory TrendFactory(myBasisSequenceFact, myFittingAlgo)

BasisSequenceFactory FittingAlgo

ex : LAR CorrectedLeaveOneOut

KFold==> generates a sub basis of B

==> estimates the empirical error

ex :

Open TURNS and the trend computation

This ‘Trend Factory’ is used to create a trend transform : • with a particular functional basis B• and a particluar time serie

• build(myTimeSeries, myFunctionBasis) TrendTransformTrendFactory

TimeSeries NumericalMathFunctionCollection

• add the trend to (Xt)t :

>> myYTimeSeries = myTrendTranform(Xt )

• remove the trend from (Yt)t :

>> myXTimeSeries = myTrendTranform . getInverse( )(Yt )

• evaluate the trend at t :

>> trend_t = myTrendTranform.getFunction ()(t)

Open TURNS and the Box Cox transform

Principles : the Box Cox transform h : (Yt)t -> (h(Yt))t with a stabilized variance (if it exists) : such that Var[h(Y t)_] is constant with respect to the time for each i.For each scalar marginal process :

In general, the Box Cox transformation writes (where is scalar) :

• may be estimated from the data or given by the User

==> For ex, if then :

==> For ex : if then :

BoxCoxFactory • BoxCoxFactory()

if (Yt)t has negative values, translate it of alpha : (alpha + Yt)t is a positive time series

• BoxCoxFactory(alpha)

==> More generally : if then :

Open TURNS and the Box Cox transform

This ‘Box Cox Factory’ is used to create a Box Cox transform with a particular time series where the factor is estimated from the data

• build(myTimeSeries) BoxCoxTransformBoxCoxFactory

TimeSeries

• transform (Yt)t into (h(Yt))t :

>> myStabilizedTS = myBoxCoxTranform(myYt )

• transform (h(Yt))t into (Yt)t

>> myYt = myBoxCoxTranform.getInverse()(myStabilizedTS )

or :

or :

Open TURNS and the Box Cox and Trend transforms

We illustrate the full process of analysis

Box Cox transform : stabilization of the variance = 0.046

Trend suppression : LAR & cross validationbasis : polynomials of degree 4linear trend

ARMA(0,1)

Open TURNS and the propagation of processes

Dynamical function :

TemporalFunction SpatialFunction

DynamicalFunction

Event :

Monte Carlo estimation :

Composite Process :

TemporalFunction(myFunctiong)

NumericalMathFunction

SpatialFunction(myFunctiong)

NumericalMathFunction

DynamicalFunction(myTemporalFunction)

myYt = CompositeProcess(f,myXt)

DynamicalFunction(mySpatialFunction)

Studies with Open TURNS

Study 1 : « Modélisation de variables dépendantes pour l'étude de la dispersion de la signature infrarouge  » - S. Lefebvre - ONERA

Study 2 : « Assistance à l’élaboration de gammes d’assemblage innovantes de structures composites   » : L. Champaney » – L. Andalfatto EADS-IW / ENS-CACHAN

Study 3 : « Interfaces Open TURNS  - Java, Matlab, Octave  » - P. Rekapalli  – EADS-IW

Study 4 : « Open TURNS in Mecanics   » – AL Popelin - EDF R&D

Study 5 : « Analyse de sensibilité multi-échelle – Open TURNS / YACS   » – Y. Caniou - PhiMECA

Perspectives

Processus stochastiques : new developments to come

Dickey-Fuller stationarity test

Multidimensional ARMA estimation

Estimation of the covariance function stationnary : FFT algorithm, non stationnary : crude sample autocovariance function

Bayesian modelisation

Bayesian updating of « a priori » distribution in conjunction with physical models

Use Case discussion meeting : 19 juin 2012

OpenTURNS in its environment…

Python friendly

Automatic conversion (e.g. array/matrix)

Use of MatPlotLib library for the visualisation

Other ideas ?

Use of the module mecanism in order to extend the OpenTURNS capabilities

Discussion

Some ideas ...

points on which you would like some improvements ?

additional functionnalities ?

how to improve the Open TURNS diffusion ?

It’s up to you !

The end ....

Thanks for your participation ... and see you next year !