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Eprints ID: 9178

To cite this document: Dubreuil, Sylvain and Berveiller, Marc and Petitjean, Frank and

Salaün, Michel Determination of Bootstrap confidence intervals on sensitivity indices

obtained by polynomial chaos expansion. (2012) In: JFMS12 - Journées Fiabilité des Matériaux et des Structures, 04-06 Jun 2012, Chambéry, France.

Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@inp-toulouse.fr Determination of Bootstrap confidence intervals on sensitivity indices obtained by polynomial chaos expansion Sylvain Dubreuil 1, Marc Berveiller 2, Frank Petitjean 3, Michel Salaun 4

1. ISAE - Institut Clément Ader (ICA) F-31055 Toulouse, France {sylvain.dubreuil,michel.salaun}@isae.fr

2. EDF R&D - Département MMC Site des Renardières F-77818 Moret-sur-Loing, France marc.berveiller@edf.fr

3. Institut Catholique d’Arts et Métiers (ICAM) F-31300 Toulouse, France frank.petitjean@icam.fr ABSTRACT. Sensitivity analysis aims at quantify the influence of the dispersion of input parameters on the dispersion of the outputs of a mechanical model. Among all approximation methods, polynomial chaos expansion is one of the most efficient in order to calculate sensitivity indices because they are computed analytically from the coefficients of the expansion (Sudret (2008)).

Indices are approximated and it is difficult to evaluate the error due to approximation. In order to evaluate the reliability of these indices we propose the construction of confidence intervals by Bootstrap re-sampling (Efron, Tibshirani (1993)) on the experimental design used to build the polynomial chaos approximation. Since the evaluation of the sensitivity indices is obtained with confidence intervals, it is possible to find an optimal design of experiments allowing the computation of sensitivity indices with a given accuracy.

RÉSUMÉ. L’analyse de sensibilité a pour but d’évaluer l’influence de la variabilité d’un ou plusieurs paramètres d’entrée d’un modèle sur la variabilité d’une ou plusieurs réponses.

Parmi toutes les méthodes d’approximations, le développement sur une base de chaos polynômial est une des plus efficace pour le calcul des indices de sensibilité, car ils sont obtenus analytiquement grâce aux coefficients de la décomposition (Sudret (2008)). Les indices sont donc approximés et il est difficile d’évaluer l’erreur due à cette approximation. Afin d’évaluer la conance que l’on peut leur accorder nous proposons de construire des intervalles de confiance par ré-échantillonnage Bootstrap (Efron, Tibshirani (1993)) sur le plan d’expérience utilisé pour construire l’approximation par chaos polynômial. L’utilisation de ces intervalles de confiance permet de trouver un plan d’expérience optimal garantissant le calcul des indices de sensibilité avec une précision donnée.

KEYWORDS: Sensitivity analysis, polynomial chaos expansion, Bootstrap re-sampling MOTS-CLÉS : Analyse de sensibilité, chaos polynômial, ré-échantillonnage Bootstrap

1. Introduction Performing global sensitivity analysis is often a major step in uncertainties propagation. It helps to understand how uncertainties of a mechanical response could be explained and reduced. Sensitivity indices coming from variance decomposition are relevant information but asked for estimation of many partial variances. These calculations could be done by polynomial chaos expansion approximation of the quantity of interest. In order to know if this approximation is efficient enough (to estimated partial variance) we propose to construct confidence interval by Bootstrap re-sampling.

The first part of this paper recalls some important features about polynomial chaos expansion and determination of sensitivity indices. One important point deals with the method used to construct the polynomial basis. In our study, two different methods are compared (OpenTURNS (2011) and Blatman (2009)). The second part presents application of Bootstrap re-sampling (Efron, Tibshirani (1993)) to sensitivity indices when they are calculated by polynomial chaos expansion (PCE). It recalls some results about the determination of confidence intervals and presents the algorithm which is set up to build a design of experiment allowing to obtain sensitivity indices with a given degree of confidence. Finally this methodology is tested on an academic case (Ishigami function) and used for sensitivity analysis on a finite elements model of satellite (TARANIS designed by Centre National d’Etudes Spatiales).

2. Determination of sensitivity indices by polynomial chaos expansion

2.1. Polynomial chaos expansion

–  –  –

Determination of the coefficients Several methods can be used in order to find the best coefficients Ci of (2). Here, we choose a regression method which is quite efficient and simple (Berveiller (2005)).

Coefficients Ci are determined by minimizing the error εy = Y−ΦC in quadratic norm between some exact values y(X) estimated at k different points (experimental design of size k) concatenated into vector Y, and their estimation by the truncated polynomial expansion, concatenated into vector ΦC where C is the vector of unknown coefficients Ci of (2), and Φ the matrix of regressors. The least square minimization criterion leads to, C = (Φt Φ)−1 Φt Y (3) Construction of the basis About the construction of the polynomial basis, it is shown (Xiu, Karniadakis (2002)) that for usual distributions (see table 1), classical univariate polynomial basis should be used. Then the orthogonal multivariate polynomials basis is obtained from the product of each univariate polynomial. Here, this approach is chosen because only usual distributions are used.

–  –  –

Table 1. Univariate orthogonal polynomials for usual random variables This multivariate polynomial basis is composed of an infinity of terms.

As it is shown in (2), this basis is truncated to a finite number of polynomials, say P. In the following, these polynomials are ranked by order (first polynomials are univariate of degree one, then multivariate using two variables at degree one, then the univariate at degree two, etc.). The simplest way to truncate the basis is then to choose the P first polynomials. For example, number P of polynomials necessary to reach order p is P = (N + p)!/(N!p!) where N is the number of random variables. This strategy is efficient for problem of small dimensions and responses that can be approached by low degree polynomials. When it is not the case the number of terms becomes too important and leads to calculation problems and pollution of the approximation. That is why different kinds of selection strategies allowing sparse basis were introduced

and are now briefly presented:

1. Cleaning Strategy (OpenTURNS (2011)). The main idea is to remove non significant polynomials from the reference basis of size P and to select a significant

basis of size B. From a complete basis of B terms, non significant terms are removed :a

term Ci φi (X1, · · ·, XN ) is considered as non significant if Ci 10−3 max{Ck, k ∈ [1, B]}.

The basis is then constructed iteratively, replacing the removed terms, which number is j, by next terms of the basis (terms B + j). This method is efficient to clean the polynomial basis from useless terms which pollute the meta-model and lead to error, especially in sensitivity analysis. Nevertheless, as the calculation of all terms needs to be done, its numerical coast is higher than the construction of a full basis of degree p. Parameters of this method are essentially the size B of the basis to construct (how many polynomials have to be kept) and size P.

2. LAR and Leave One Out selection (Blatman (2009); OpenTURNS (2011)).

This method aims at construct sparse polynomial basis (as the Cleaning Strategy) selecting the most relevant terms iteratively. Selection criterion is based on the Least Angle Regression method. LAR algorithm return a family of sparser and sparser PC approximations then, the one which minimizes the leave one out error is chosen. Finally this method is the most efficient and allows fast construction of a sparse basis.

As for the Cleaning Strategy, the main parameter of this method is the size P (chosen for the degree of the polynomials it allows to reach).

For our application, Cleaning Strategy and LAR methods are both used because the construction of a sparse basis is more relevant for sensitivity analysis.

2.2. Computation of sensitivity indices

–  –  –

with α = (α1, · · ·, αN ), where αk means that the variable k is used with order αk by the polynomial φα.

This formula means that sensitivity indices are given by the coefficients of the polynomials acting on the variables of interest. For example, the first order sensitivity indice of a variable Xi is computed with the coefficient of all the univariate polynomials acting on Xi. The major advantage of polynomial chaos expansion for calculation of sensitivity indices is that it does not need more calculation than the one necessary for determination of the coefficients. This method is much more efficient than the method based on Monte Carlo sampling, in terms of numerical cost due to the number of model evaluations. Nevertheless, the quality of the sensitivity indices is directly connected to the quality of the approximation by the meta-model. So the problem is how to choose the construction parameters of the polynomial chaos expansion in order to get sensitivity indices with enough confidence.

3. Bootstrap re-sampling applied to polynomial chaos expansion

3.1. Construction of confidence intervals by Bootstrap re-sampling

The Bootstrap (Efron, Tibshirani (1993)) is a re-sampling method which aims at determining confidence interval on a statistic variable with only one design of experiment. The main idea is to create several new designs of experiment (B) by drawing with replacement in the first one (source design) and then to use these new designs to get an empirical distribution of the statistic variable calculated on these designs. This methodology has been applied on several surrogate models as in (Janon et al. (2010)) and (Gayton et al. (2003)).

Here, we are interested in sensitivity indices (denoted as collection T ). PCE of the response of interest gives an estimator of these indices (denoted as collection T ).

For every new design Pk, k = 1 · · · B, the methodology of construction of polynomial chaos expansion (Cleaning strategy or LAR) is used, leading to the sensitivity indices Tk for the design Pk (it should be noted that to avoid ill conditioning of regression matrix Φ defined in (3), the size of the design of experiments is three times higher than the size of the polynomial basis, see Roussouly (2011)). After the computation of the B sensitivity indices collections (one per re-sampled design), several confidence intervals are built. Let us denote T ∗ and VarT the estimator of the mean and variance of the empirical distribution of the B collection of indices Tk.

1. Direct quantiles estimation: Confidence interval is such that

T[α/2] T T[1−α/2]

where T[α/2] and T[1−α/2] are the α/2 et 1 − α/2 empirical quantiles.

This interval does not need hypothesis on the distribution of T but needs more resampling B (higher than 500, see (Notin (2010))) in order to approximate these quantiles with a sufficient precision.

2. Empirical quantiles corrected: The idea is to take into account bias between estimator of the mean T ∗ and the value of indices estimated on the first design of experiment (source design). This bias is then used to correct the evaluation of empirical quantiles. Details can be found on Notin (2010) and Efron, Tibshirani (1993).

This study essentially use direct empirical quantiles estimation as it is faster and easier to set up, but example of section 4 uses empirical quantiles corrected method for comparison.

3.2. Sequential construction of an optimized design of experiments

This part presents application of this method to optimize the number of points in design of experiment in order to get the sensitivity indices from polynomial chaos with a fixed level of confidence. The algorithm described here is inspired by a previous work, using Bootstrap re-sampling in a similar way on reliability indices (Notin (2010)). The iterative methodology presented figure 1 is split in five

main steps:

1. An experimental design of size N is used to construct a polynomial chaos metamodel. The size of the polynomial basis is less equal to N/3 for reason of conditioning described previously.

2. B new bootstrapped samples are used to construct the 95% confidence intervals (CI) with one of the methods presented in the previous section. Number B depends on the complexity of the response and of the kind of chosen interval. In our application, we mainly used the direct quantiles estimation with B = 700 (in order to fix B, we increase this number until the convergence on the size of the CI).

3. Convergence criterion: Here, we choose to stop the algorithm at iteration for which all CI sizes are less than 10% of the maximum mean value of the sensitivity indices (let us recall that if there is no interaction, the sum of the first order sensitivity indices is equal to one). Of course if many variables have an important influence or if variables act mainly in interaction, this criterion is going to be too restrictive because the maximum mean value of the sensitivity indices will be small (compared to one).

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