Analysis of covariance (ANCOVA) using polynomial chaos expansions B. SUDRET ETH Zurich,¨ Institute of Structural Mechanics, Chair of Risk, Safety & Uncertainty Quantification Wolfgang-Pauli-Strasse 15, CH-8093 Zurich¨ Y. CANIOU Phimeca Engineering, Centre d’Affaires du Zenith,´ 34 Rue de Sarlieve,` 63800 Cournon, France ABSTRACT: Global sensitivity analysis aims at quantifying the uncertainty of the output of a computer model that may be attributed to each input parameter or combination thereof. Variance decomposition tech- niques that lead to the well-known Sobol’ indices are now well established. However this classical framework only holds for independent input parameters. Moreover, when the computational model under consideration is costly to evaluate, it is not possible to resort to crude Monte Carlo simulation to evaluate sensitivity indices. In this paper we extend the polynomial chaos-based Sobol’ indices derived by Sudret (2006, 2008) to the case of dependent input parameters using the covariance decomposition recently proposed by Li et al. The functional decomposition which is natively given by the polynomial chaos expansion is taken advantage of, and the pro- posed approach is consistent with the classical Sobol’ indices when the input variables are independent. The methodology is illustrated on a tolerance analysis problem. 1 INTRODUCTION that the input parameters of the computational model are statistically independent. Computer simulation has become an inescapable tool In engineering problems though, many situations of modern engineering. Computational models of exhibit correlations between model parameters, e.g. ever increasing complexity are built by engineers and between loads applied onto a structural system or scientists in order to describe in the most refined between material parameters describing a constitu- way complex natural or man-made systems. However tive law. Deriving sensitivity indices that are able computer models always represent an idealized vision to account for the input correlation structure is thus of the real world. In this context it is important to as- of practical importance. Until recently not much at- sess the uncertainty in the model predictions that are tention has been devoted to this topic. Distribution- caused by the natural variability (aleatory uncertainty) based sensitivity indices have been introduced by and lack of knowledge (epistemic uncertainty) in the Borgonovo (2007), Borgonovo et al. (2011), which model input parameters. are equally defined when the model input parame- In the field of uncertainty quantification, sensitiv- ters are dependent or independent. The interpretation ity analysis aims at determining what are the in- of the resulting δ-indices is not easy though since put parameters or combinations thereof that have the they do not sum up to 1 as in the case of Sobol’ largest impact onto the model predictions, i.e. those indices. In contrast several approaches to generalize which explain at best the response variability. A re- the Sobol’ indices to the case of dependent parame- view of classical sensitivity analysis can be found in ters have been recently proposed, see Xu and Gertner Saltelli et al. (2004). In this paper we focus on so- (2008), Da Veiga et al. (2009), Mara and Tarantola called variance-based sensitivity indices originally in- (2012), Kucherenko et al. (2012). However no con- troduced by Sobol’ (1993). This approach is based on sensus has been attained so far. the decomposition of the variance of the model re- In this paper the covariance decomposition origi- sponse into a sum of contributions that can be related nally proposed by Li et al. (2010) is exploited. In the to each single model input parameter, pairs of param- original paper the authors first build a so-called High- eters, triplets, etc. The main drawback of these so- Dimensional Model Representation (HDMR) of the called Sobol’ indices is that there definition assumes model. In the present paper we propose to make use of polynomial chaos expansions (Ghanem & Spanos For instance, the truncation scheme may corre- 2003) which have already proven efficient for sensi- spond to all the multivariate polynomials whose tivity analysis in the case of independent input param- PM total degree (computed as jjαjj1 = i=1 αi) is eters, see Sudret (2008), Blatman and Sudret (2010). smaller than or equal to a prescribed p, namely As a summary the goal of the paper is to provide n M o A = α : P α ≤ p . Note that other trunca- a computationally efficient framework for sensitiv- M;p i=1 i ity analysis in case of dependent parameters which tion schemes such as the hyperbolic sets may be more is based on a covariance decomposition of the model appropriate for large dimensional problems, see Blat- output and polynomial chaos expansions. The paper is man & Sudret (2011). organized as follows: in Section 2 the basics of poly- The set of PC coefficients fyα; α 2 Ag in Eq.(4) nomial chaos expansions is summarized. In Section 3 are interpreted as the coordinates of random variable the tools of variance-based sensitivity analysis are re- Y in the orthonormal basis made of the Ψα’s. Many called. In Section 4 the proposed polynomial chaos / computational methods have been proposed for their covariance-based sensitivity indiced are derived. The evaluation. In practical engineering applications, so- approach is illustrated by two application examples in called non intrusive methods are most suited, in the Section 5. sense that they only use a set of runs of the origi- nal model M using selected points in the domain of definition of the input parameters, namely a so-called 2 POLYNOMIAL CHAOS EXPANSIONS experimental design denoted by X = fx1;:::; xN g. Such non intrusive methods include stochatic colloca- Let us consider a computational model M : x 2 M tion approaches based on sparse grids (Xiu 2007), re- DX ⊂ R 7! y = M(x) 2 R. The uncertain input gression (Berveiller, Sudret, & Lemaire 2006) or vari- parameters are modelled by a random vector X with able selection techniques such as Least Angle Regres- DX through its probability density function (PDF) sion (Blatman & Sudret 2011). In this paper we focus denoted by fX . We assume that the model output on classical least-square regression for the sake of il- has a finite variance, i.e. Var [M(X)] < +1. In this lustration. case Y = M(X) belongs to the Hilbert space of sec- The idea behind least-square regression is to con- ond order random variables. Thus Y may be repre- sider the infinite series in Eq.(1) as a sum of the sented by its coordinates onto a countably infinite ba- truncated series in Eq.(4) and a residual. The coef- sis (Soize & Ghanem 2004): def ficients y = fyα ; α 2 Ag are obtained by mini- X MS Y = M(X) = yα Ψα(X) (1) mizing the empirical mean-square error between the α2 M original model response and its approximation by PC N expansion over the selected experimental design: M where the construction of the basis Ψα ; α 2 N is now described. Assume now that the input random N " #2 vector X has independent components, i.e. fX (x) = X (i) X (i) M y = arg min M(x ) − yα Ψα(x ) Q MS A i=1 fXi (xi) where fXi is the marginal distribution y2 card R i=1 α2A of the i-th component Xi. It is possible to define an Hilbertian basis made of multivariate polynomials in the input random vector. More precisely, for any M- (5) tuple α = fα1; : : : ; αM g, one defines: Typical choices of experimental designs are Latin Hy- percube designs M or low-discrepancy sequences, and Y the size of the experimental design is selected by the Ψ (X) = P (i)(X ) (2) α αi i thumb rule N = 2 − 3 cardA. Note that robust error i=1 estimates based on cross-validation techniques may n (i) o be derived, see Blatman & Sudret (2010). where P ; n 2 is the set of orthonormal poly- n N As a conclusion, based on an experimental de- nomials with respect to the probability measure sign X of appropriate size and the model responses P (dxi) = fXi (xi) dxi. By construction the multivari- fM(xi) i = 1;:::;Ng, a surrogate model of the ate basis is orthonormal (δαβ is the Kronecker sym- original model is obtained, which is nothing but a bol): polynomial response surface with specific orthogo- nality properties. E [Ψα(X)Ψβ(X)] = δαβ (3) For computational purpose, the series is truncated 3 SENSITIVITY ANALYSIS by selected a truncation set A, which lead to the poly- nomial chaos approximation: 3.1 Sobol’ indices def X Considering a computational model M with inde- YA = MA(X) = yα Ψα(X) (4) α2A pendent uncertain input parameters gathered in X, the Sobol’ decomposition (also known as functional proposed, see e.g. Janon et al. (2012) for recent devel- ANOVA decomposition) reads: opments including asymptotic properties. Whatever the selected estimator, the computational cost in or- M X der to get an accurate result is still large due to the M(x) = M0 + Mi(xi) sampling procedure. i=1 In contrast polynomial chaos-based Sobol’ indices (6) provide an exact result for the sensitivity indices once X + Mij(xi; xj) + ··· + M12:::M (x) the PC expansion is available. As originally shown in 1≤i<j≤M Sudret (2006), Sudret (2008), the truncated expansion in Eq.(4) may be rearranged so as to reflect the de- In this equation, M0 is the mean value of the re- composition into summands of increasing order. For sponse, i.e. M0 = E [M(X)] and the other terms any non-empty set u ⊂ f1;:::;Mg and any finite are summands of increasing order. Introducing the truncation set A ⊂ NM let us define: def generic index set u = fi1; : : : ; ikg ⊂ f1;:::;Mg and denoting by xu the subvector of x obtained by ex- A = fα 2 A : k 2 u , α 6= 0; k = 1;:::;Mg tracting the components labelled by the indices in u, u k the above equation concisely rewrites: (12) X M(x) = M0 + Mu(xu) (7) In other words Au contains all the multi-indices u⊂{1; ::: ;Mg within the truncation set A which have non zero com- u6=; ponents αk 6= 0 if and only if k 2 u.