Importance analysis for models with correlated variables and its sparse grid solution

Reliability Engineering and System Safety 119 (2013) 207–217

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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress

Importance analysis for models with correlated variables and its sparse grid solution Luyi Li, Zhenzhou Lu n School of Aeronautics, Northwestern Polytechnical University, Xi'an, Shaanxi Province 710072, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 25 June 2012 Received in revised form 21 June 2013 Accepted 26 June 2013 Available online 3 July 2013

For structural models involving correlated input variables, a novel interpretation for variance-based importance measures is proposed based on the contribution of the correlated input variables to the variance of the model output. After the novel interpretation of the variance-based importance measures is compared with the existing ones, two solutions of the variance-based importance measures of the correlated input variables are built on the sparse grid numerical integration (SGI): double-loop nested sparse grid integration (DSGI) method and single loop sparse grid integration (SSGI) method. The DSGI method solves the importance measure by decreasing the dimensionality of the input variables procedurally, while SSGI method performs importance analysis through extending the dimensionality of the inputs. Both of them can make full use of the advantages of the SGI, and are well tailored for different situations. By analyzing the results of several numerical and engineering examples, it is found that the novel proposed interpretation about the importance measures of the correlated input variables is reasonable, and the proposed methods for solving importance measures are efficient and accurate. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Correlated input variables Variance-based importance measure Sparse grid integration Single loop Monte Carlo method

1. Introduction Sensitivity analysis (SA), especially global SA, is widely used in engineering design and probability safety assessment. Global SA, also known as importance analysis, aims at determining which of the input parameters influences output the most in the whole uncertainty range of the inputs. Indicators created for global SA purposes are defined as that uncertainty in the output can be apportioned to different sources of uncertainty in the model input [1]. At present, many importance analysis techniques and indices are available, such as nonparametric techniques [2–4], variance-based importance measure indices [5–8], and momentindependent importance measures [9–11]. Among these methods, variance-based importance measure is known as a versatile and effective tool in uncertainty analysis. Most of the existing importance analysis techniques assume input variables independence for sake of computability. However, in many cases the input variables are correlated with one another and these correlations present among the variables may affect the importance ranking of the inputs dramatically. Therefore, more and more importance analysis techniques are proposed over the past ten years to take the correlation of the variables into consideration, such as the methods proposed in Refs. [12–15].

n

Corresponding author. Tel.: +86 29 88460480. E-mail address: [email protected] (Z. Lu).

0951-8320/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ress.2013.06.036

Nevertheless, these early researches for correlated input variables only provide an overall importance measure of one input variable, which does not distinguish the correlated or uncorrelated contribution of one input variable. To explore the origin of the uncertainty of the output response clearly in case of correlated input variables involved, the contribution of uncertainty to output response by an individual input variable is divided into two parts in Ref. [16]: the uncorrelated contribution and the correlated one. This distinction of contribution for an individual variable can provide engineers a better understanding of the composition of the output uncertainty, and help them to decide whether the uncorrelated part or the correlated part should be focused on. However, the regression-based method proposed in Ref. [16] for decomposing the contribution of the input variables depends on the assumption that the relationship between output response and input variables is approximately linear. Based on covariance decomposition of the unconditional variance of the output, a similar treatment for correlated input variables was proposed in Ref. [17] where the total contribution of an input variable or a subset of input variables to the variance of the output response was decomposed into structural contribution and correlative one. Although this treatment can deal with both linear and nonlinear response function, it relies on the determination of how many components are included in the decomposition. A set of variancebased sensitivity indices is proposed in Ref. [18] to perform importance analysis of models with correlated inputs. The definition of those indices is based on a specific orthogonalisation of the

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L. Li, Z. Lu / Reliability Engineering and System Safety 119 (2013) 207–217

inputs and ANOVA-representations of the model output. They cannot only support nonlinear models and nonlinear dependences, but also reflect the effect of the interaction among variables when decomposing the total variance of the model output. However, the method depends on the order of the inputs in the original set. To obtain the uncorrelated and correlated variance contributions of each input variable, it is essential to transform different sets of circular permutations of the correlated inputs into independent and orthogonal ones, and then calculate the corresponding variance-based importance measures of the transformed variables. Therefore, it is a computationally expensive process to perform analysis with this method. A generalization of the variance-based importance measures for the case of correlated variables is presented in Ref. [19]. The generalized importance measures can perfectly preserve the advantages of the original variance-based importance measures without necessity of determining functional decomposition or orthogonalization of the input factor space and so on. However, it does not consider the differences between the contribution to the output uncertainty by the independent variables and that by the correlated ones, but directly extends the significations of importance measures of the independent variables to those of the correlated ones. Therefore, the interpretation of the importance measures in the case of correlated variables is not reasonable. Furthermore, the solution of the measures in Ref. [19] still relies on numerical simulation method, which usually consumes a relatively heavy computational burden. In this work the variance-based importance measures of the independent variables are reinterpreted in the context of the correlated ones, and the new meanings of these importance measures are proposed. The new interpretation is based directly on the contribution of the correlated inputs to the uncertainty of the output, and decomposes the contribution of the correlated input variables to the variance of the output into main effect of correlated and uncorrelated variations and the total effect of uncorrelated variation. This decomposition cannot only reflect the origin of the output uncertainty, but also interpret the contribution of the correlated input variables clearly and reasonably. In addition, by employing the high efficiency of the sparse grid integration (SGI), two SGI based methods are proposed to perform importance analysis of the correlated inputs in different situations. The proposed methods avoid the sampling procedure, which usually consumes a heavy computational burden, and can be used as effective tools to deal with uncertainty analysis involving correlated inputs. The rest of the paper is organized as follows: in Section 2 the contribution of the correlated input variables to the variance of the output is firstly analyzed, on which the new significations of the variance-based importance measures are proposed. This new interpretation for the variance contribution of the correlated input variables is later discussed and compared with the existing ones. In Section 3, the basic theory of the SGI in case of independent variables is reviewed. Two new proposed methods that incorporate SGI to perform importance analysis for correlated input variables are detailed in Section 4. Several numerical and engineering examples are used to illustrate the new interpretation and the results are discussed to demonstrate the efficiency of the proposed solutions on SGI in Section 5. Finally, the conclusion comes at the end of the paper.

2. Variance-based importance measures for correlated input variables 2.1. Importance measures for independent input variables Considering a mathematical or computational model of the form y ¼ gðxÞ, where x ¼ ðx1 ; x2 ; …; xd ÞT is the input vector with d-dimensional variables. The variance-based importance measures

of these input variables are related to a decomposition of the function gðxÞ itself into terms of increasing dimensionality (High Dimensional Model Representation, HDMR) [5], i.e. gðxÞ ¼ g 0 þ ∑ g i ðxi Þ þ ∑ ∑ g ij ðxi ; xj Þ þ ⋯ þ g 12…d ðx1 ; …; xd Þ

ð1Þ

i j4i

i

where the various terms in Eq. (1) are defined as g 0 ¼ EðyÞ, g i ðxi Þ ¼ Eðyjxi Þg 0 , and g ij ðxi ; xj Þ ¼ Eðyjxi ; xj ÞEðyjxi ÞEðyjxj Þ þ g 0 and so on. For the independent input variables, a decomposition scheme of the total unconditional variance VðyÞ, equivalent to HDMR (1), is derived by Sobol etc. as follows, VðyÞ ¼ ∑ V i þ ∑ ∑ V Iij þ ⋯ þ V I12…d

ð2Þ

i j4i

i

where V i ¼ Var½g i ðxi Þ ¼ VðEðyjxi ÞÞ V Iij ¼ Var½g ij ðxi ; xj Þ ¼ VðEðyjxi ; xj ÞÞV i V j …

ð3Þ

here Var½ represents the variance of , VðÞ and EðÞ are the simplified variance operator and expectation operator, respectively. It can be seen from decomposition (1) that while V i is the contribution of xi to the variance of y due to its unique variation, V Iij is the contribution of xi and xj caused by their interaction in the response function. Normalizing the terms in decomposition (3) by the total unconditional variance VðyÞ, the variance-based importance measures are obtained: Si ¼ V i =VðyÞ; SIij ¼ V Iij =VðyÞ; …

ð4Þ

At present, the two key measures used extensively are the main effect index Si ¼

V ðEðyjxi ÞÞ VðyÞ

ð5Þ

and the total effect index STi ¼

VðyÞVðEðyjxi ÞÞ VðyÞ

ð6Þ

where xi denotes the vector of all input variables except xi . While the main effect measures the unique contribution of the input variable xi to the variance of y, the total effect measures the overall contribution of xi on y, including the unique contribution of xi as well as the contributions by all the interaction of xi with other input variables. 2.2. New interpretation for correlated input variables The variance decomposition (2) is proposed under the assumption of input independence, thus it may not hold when the input variables are correlated. However, for the correlated input variables, the various variance terms VðEðyjxi ÞÞ, VðEðyjxi ; xj ÞÞ and so on in Eq. (3) are still able to reflect the variance contribution of the input variables, yet the difference from the independent case lies in the connotation. In this subsection, we will also concentrate on the two key measures and propose a new interpretation for them in case of correlated input variables. When the input variables x of the computational model y ¼ gðxÞ are correlated, it also can be decomposed into terms of increasing dimensionality as follows [20]: gðxÞ ¼ g 0 þ ∑ g i ðxi Þ þ ∑ ∑ g ij ðxi ; xj Þ þ ⋯ þ g 12…d ðx1 ; …; xd Þ i

where Z g0 ¼

gðxÞf X ðxÞdx

i j4i

ð7Þ

L. Li, Z. Lu / Reliability Engineering and System Safety 119 (2013) 207–217

Z g i ðxi Þ ¼

gðxÞf Xi ðxi Þdxi g 0 Z

g ij ðxi ; xj Þ ¼

gðxÞf Xij ðxij Þdxij g i ðxi Þg j ðxj Þg 0 :::

ð8Þ

here xij denotes the vector of all input variables except xi and xj . f X ðxÞ is the joint probability density function (PDF) for x, and f Xi ðxi Þ and f Xij ðxij Þ are marginal PDF's obtained by integrating the distribution over xi and xi ,xj , respectively. Eq. (7) is the same as Eq. (1) in terms of the form of expression, yet the difference from the independent case lies in that HDMR component functions in this case are only hierarchically, but not mutually orthogonal. In the case of correlated input variables, when xi is fixed, the independent part of xi and all the parts of xi which are correlated with xi are also fixed. Then, Eðyjxi Þ ¼ g 0 þ g i ðxi Þ þ ∑ g^ k ðxi Þ þ ∑ ∑ g^ kl ðxi Þ þ ⋯ þ g^ 12…d ðxi Þ

ð9Þ

k l4k

k≠i

where g^ u ðxi Þðu D f1; 2; ::; dg; u≠figÞ are functions obtained from integration of the corresponding functions g u ðxu Þðu D f1; 2; ::; dg; u≠figÞ in Eq. (7) with respect to xi under the conditional PDF f Xi jX i ðxi Þ, and fig denotes the subset only containing element i. Since the above resultant functions are not mutually orthogonal, we have 2 3 VðEðyjxi ÞÞ ¼ Var4g 0 þ g i ðxi Þ þ ∑ g^ k ðxi Þ þ ∑ ∑ g^ kl ðxi Þ þ ⋯ þ g^ 12…d ðxi Þ5 k l4k

k≠i

2

of xi in this paper. Consequently, the importance measure in Eq. (5) is actually the main effect index of correlated and uncorrelated variations of xi in case of correlated input variables, which is denoted by STc i in the sequel to differ from the importance measure of the independent variables. Similarly, when xi is fixed, all xi and the part of xi which is correlated with xi are also fixed. Then, Eðyjxi Þ ¼ g 0 þ g^ i ðxi Þ þ ∑ g k ðxk Þ k≠i

þ∑ ∑ g kl ðxk ; xl Þ þ ∑ g^ ik ðxi Þ þ ⋯ þ g^ 12…d ðxi Þ k l 4k k; l≠i

VðEðyjxi ÞÞ ¼ Var g 0 þ g^ i ðxi Þ þ ∑ g k ðxk Þ k≠i

k l 4k k; l≠i

" k≠i

#

þ∑ ∑ g kl ðxk ; xl Þ þ ∑ g^ ik ðxi Þ þ ⋯ þ g^ 12…d ðxi Þ k l 4k k; l≠i

k≠i

¼ Var½g^ i ðxi Þ þ ∑ Var½g k ðxk Þ k≠i

k l4k

    þVar g^ 12…d ðxi Þ þ 2∑Cov qr ðxi Þ; qs ðxi Þ

k≠i

¼ Var g^ i ðxi Þ þ ∑ g k ðxk Þ

k l4k

k≠i

#

þ∑ ∑ g kl ðxk ; xl Þ þ ∑ g^ ik ðxi Þ þ ⋯ þ g^ 12…d ðxi Þ

¼ Var4g i ðxi Þ þ ∑ g^ k ðxi Þ þ ∑ ∑ g^ kl ðxi Þ þ ⋯ þ g^ 12…d ðxi Þ5       ¼ Var g i ðxi Þ þ ∑ Var g^ k ðxi Þ þ ∑ ∑ Var g^ kl ðxi Þ þ ⋯

ð11Þ

k≠i

where g^ u ðxi Þðfig D u D f1; 2; ::; dgÞ are functions obtained from integration of the corresponding functions g u ðxu Þðfig D u D f1; 2; ::; dgÞ in Eq. (7) with respect to xi under the conditional PDF f X i jXi ðxi Þ. Taking variance for both sides of Eq. (11), we have "

3 k≠i

209

þ∑ ∑ Var½g kl ðxk ; xl Þ þ ∑ Var½g^ ik ðxi Þ þ ⋯

ð10Þ

k l 4k k; l≠i

r;s

where Cov½;  represents the covariance of ½; , and ½qr ðxi Þ; qs ðxi Þ is any pair of the non-constant functions in Eq. (9). Analyzing the origin of each term in Eq. (10), it can be seen that   Var g i ðxi Þ is the variance contribution by the variation of xi , which can be seen later consisting of the independent contribution by xi itself and the contribution associated with the correlation of xi  with the other variables. Var½g^ u ðxi Þði∉u D f1; 2; ::; dgÞ and Cov qr ðxi Þ; qs ðxi Þ, where ½qr ðxi Þ; qs ðxi Þ is any pair of the functions in ðg i ðxi Þ; g^ u ðxi Þði∉u D f1; 2; ::; dgÞÞ, are the contributions due to pure correlation between xi and xi , considering that g^ u ðxi Þði∉u D f1; 2; ::; dgÞ are functions resulting only from the correlation of xi and xi . Similarly, since the cross terms g^ u ðxi Þðfig⊂u D f1; 2; …; dgÞ are produced by both the interaction and correlation of xi and xi (this is kind of interaction “carried over” by correlation), the variance terms and covariance terms in Eq. (10) associated with these functions are the contributions by both the interaction and correlation of xi and xi . It is necessary to be pointed out that the variance contribution of the correlated part of an individual variable (or a set of variables), is different from the interaction contribution of this variable (or these variables) to the output variance. These two contributions are absolutely different concepts. The former is coming from the correlation among the input variables, while the latter is produced by variable interaction in the form of the model function. Considering that the independent contribution by xi itself contained in Var½g i ðxi Þ can be seen as self-correlated contribution, all the components of VðEðyjxi ÞÞ are associated with the correlation of xi , so VðEðyjxi ÞÞ can be seen as the total correlated contribution of xi . To describe the components contained in VðEðyjxi ÞÞ clearly, it is termed as main effect of correlated and uncorrelated variations

k≠i

þVar½g^ 12…d ðxi Þ þ 2∑Cov½qr ðxi Þ; qs ðxi Þ

ð12Þ

r;s

where ½qr ðxi Þ; qs ðxi Þ is any pair of the non-constant components in Eq. (11). Analogous to Eq. (10), Eq. (12) shows that VðEðyjxi ÞÞ represents the main effect of correlated and uncorrelated variations of xi , which also includes the variance purely contributed by each variable in xi correlated with xi e.g. Var½g^ i ðxi Þ, as well as the variance contributed by both the interaction and   correlation between each variable in xi and xi , e.g. Var g^ ik ðxi Þ . In case of correlated input variables, the variance of the output can be computed as " # VðyÞ ¼ Var g 0 þ ∑ g k ðxk Þ þ ∑ ∑ g kl ðxk ; xl Þ þ ⋯ þ g 12…d ðx1 ; …; xd Þ k l4k

k

"

#

¼ Var ∑ g k ðxk Þ þ ∑ ∑ g kl ðxk ; xl Þ þ ⋯ þ g 12…d ðx1 ; …; xd Þ k l4k

k

¼ Var½g i ðxi Þ þ ∑ Var½g k ðxk Þ k≠i

þ∑ ∑ Var½g kl ðxk ; xl Þ þ ∑ Var½g ik ðxi ; xk Þ þ ⋯ k l 4k k; l≠i

k≠i

þVar½g 12…d ðxÞ þ 2∑Cov½qr ðxr Þ; qs ðxs Þ

ð13Þ

r;s

where ½qr ðxr Þ; qs ðxs Þ is any pair of the non-constant components in Eq. (7). Then VðyÞVðEðyjxi ÞÞ ¼ Var½g i ðxi ÞVar½g^ i ðxi Þ

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L. Li, Z. Lu / Reliability Engineering and System Safety 119 (2013) 207–217

þ∑ fVar½g ik ðxi ; xk ÞVar½g^ ik ðxi Þg þ ⋯ k≠i

þfVar½g 12…d ðxÞVar½g^ 12…d ðxi Þg þ2 ∑ fCov½qr ðxr Þ; qs ðxs ÞCov½qr ðxi Þ; qs ðxi Þg

ð14Þ

r;s∈Φ

where Φ ¼ fr; sjfig Dr∪sg represents that at least one of the functions qr ðxr Þ and qs ðxs Þ is related to xi . Analyzing the terms in Eq. (14), we can find that Var½g i ðxi ÞVar½g^ i ðxi Þ is the variance contribution of xi minus the variance contribution associated with the correlation between xi and xi . Thus, it gives the independent contribution by xi itself to the output variance (i.e. the aforementioned independent contribution of xi contained in VðEðyjxi ÞÞ). The rest variance and covariance terms in Eq. (14) are all involving the interactions between xi and xi , and by analyzing each pair of variance and covariance terms in the braces “{}”, it can be seen that they are the contributions by interactions between xi and xi minus the corresponding contributions by the interaction associated with correlation between xi and xi . Therefore, they are the independent part of the contributions by interactions between xi and xi (i. e. the variance contribution is not associated with any correlation relationship, only produced by interactions). Taking the variance pair fVar½g ik ðxi ; xk ÞVar½g^ ik ðxi Þg as an illustration, it is the variance contribution by the interaction between xi and xk minus the corresponding variance contribution by the interaction between xi and xk associated with the correlation of xi and xi , including the correlation of xi and xk . Thus, it gives the independent part of the contribution by interaction between xi and xk . Similar interpretation can be derived for other variance and covariance pairs in the braces “{}”. In brief, VðyÞVðEðyjxi ÞÞ includes the independent contribution by xi itself and the independent part of the contributions by interactions between xi and xi . All components of VðyÞVðEðyjxi ÞÞ are independent of the correlations of xi , so it is the total uncorrelated contribution of xi , and termed as the total effect of uncorrelated variation of xi in this paper. As a result, when the correlation is present among the input variables, the variance ratio in Eq. (6) should represent the ratio of the total uncorrelated variance contribution of xi to the unconditional variance of the output. That is to say that the variance ratio in Eq. (6) is the total effect index of uncorrelated variation of xi in the presence of correlation, and it is denoted as STu i to differ from the importance measure of the independent variables. Tu Analogously, STc I ¼ VðEðyjx I ÞÞ=VðyÞ and SI ¼ ½VðyÞVðEðyjx I ÞÞ =VðyÞ are the main effect index of correlated and uncorrelated variations and the total effect index of uncorrelated variation of a set of correlated input variables xI , respectively, where I represents a set of subscript variables I ¼ fi1 ; …; ig g; 1 ≤i1 ≤⋯ ≤ ig ≤ d.

3. Review on sparse grid integration (SGI) The sparse grid method based on Smolyak algorithm has become more and more popular since it was introduced. In this method, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. In this way, the number of function evaluations and the numerical accuracy become independent of the dimension of the problem up to logarithmic factors [21]. In view of this, it has been widely used in numerical integration [21,22], interpolation [23,24], solving differential equations with random inputs [25], uncertainty propagation [26] and so on, and has been demonstrated as an efficient discretization scheme that is especially wellsuited for high dimensional scenarios. In this paper, we mainly extend the sparse grid method associated with numerical integration to the importance analysis of the correlated input variables. The basic concepts of SGI are reviewed in this section. i

i

Let U 1j and w1j represent the quadrature points and weights of the jth variable in the one-dimension space, which can be obtained by Gaussian quadrature, Clenshaw–Curtis rules, etc. [21]. Then the quadrature points Ukd in d-dimension space with k-level (k≥0) accuracy generated by the sparse grid technique can be constructed by the Smolyak algorithm [25] as follows: Ukd ¼

∪

U i11 ⊗U i12 ⊗…⊗U i1d

ð16Þ

kþ1 ≤ jij ≤ q

where ⊗ represents tensor product. q ¼ k þ d, and jij ¼ i1 þ; :::; þid is the sum of the multi-indices. According to Smolyak algorithm, the weight ωl corresponding to the lth quadrature point ξl ¼ ½ξij1 ; :::; ξijd ∈Ukd can be obtained by the following equation: i1 id ! d1 qjij ωl ¼ ð1Þ ð17Þ ðωij1 …ωijd Þ i1 id qjij

2.3. Comparison with the existing importance measures of the correlated input variables Starting from the total variance equation VðyÞ ¼ V ðEðyjxi ÞÞ þ EðVðyjxi ÞÞ;

measures of the correlated input variables in this paper is more reasonable, which also can be proved later by the example. In addition, the total effect index of uncorrelated variation of xi ,STu i , defined in this paper is consistent with that defined in Ref. [18], which is derived by independent-orthogonalisation of the original correlated inputs. This manifests again the rationality of the importance measures of the correlated input variables proposed in this paper. However, different from the importance measures in Ref. [18], the estimation of the proposed importance measures does not need to transform each order of the original inputs into independent and orthogonal variables, which avoids the tremendous computational cost associated with this process. Therefore, the new interpretation is more concise in form and convenient in computation. In the following sections, we intend to establish an efficient solution for the importance measures of the correlated inputs by sparse grid integration.

ð15Þ

two importance measures of xi in the forms of Eqs. (5) and (6) are also derived in Ref. [19]. However, it does not analyze the actual connotations of the two importance measures in the case of correlated input variables involved, but defines VðEðyjxi ÞÞ=VðyÞ and ½V ðyÞVðEðyjxi ÞÞ=VðyÞ as the first order effect index and the total effect index of xi , respectively. It is a direct extending of the meanings of the importance measures of the independent input variables. This definition is obviously inappropriate from the analysis in Section 2.2, since VðyÞV ðEðyjxi ÞÞ also excludes the variance contributions which is related to the correlation of xi with xi , the remainder cannot represent the total contribution of xi . Therefore, compared with the definitions in Ref [19], the explanation of the importance

Then, the integration by sparse grid method for response function gðxÞ with d-dimension input variables x ¼ ðx1 ; x2 ; ⋯; xd ÞT can be calculated by Eq. (18) with up to ð2k þ 1Þ-order polynomial accuracy [21] Z

N kd

N kd

l¼1

l¼1

gðxÞf X ðxÞdx≈ ∑ ωl gðT 1 ðξl ÞÞ ¼ ∑ ωl gðxl Þ

ð18Þ

where f X ðxÞ is the joint PDF of the input variables x. Nkd is the number of all the quadrature points in Ukd . T 1 ðξl Þ is the inverse transformation of Tðxl Þ which transforms x space into ξ space, and the corresponding input variable for ξl is xl . If x are normally distributed random variables with the mean vector μx and standard deviation vector sx , then xl ¼ T 1 ðξl Þ ¼ μx þ ξl C rx with Gaussian

L. Li, Z. Lu / Reliability Engineering and System Safety 119 (2013) 207–217

type integration rule, where C rx is d  d dimension diagonal matrix with the elements of sx on the main diagonal. Note that the general expression for the function gðT 1 ðξl ÞÞ is not necessary, and that the inverse transformation is only required at the quadrature points. Based on Eq. (18), the first two moments of the computational model y ¼ gðxÞ estimated by the SGI can be expressed as follows [26]: Z

N kd

EðyÞ ¼

gðxÞf X ðxÞdx≈ ∑ ωl gðxl Þ

ð19Þ

l¼1

Z

N kd

ðgðxÞEðyÞÞ2 f X ðxÞdx≈ ∑ ωl ðgðxl ÞEðyÞÞ2

VðyÞ ¼

211

worthwhile to mention that the SGI method reviewed in Section 3 relies on the assumption of input independence, thus cannot be directly employed for cases with correlated inputs. When dealing with correlated input variables, it requires a transformation between the correlation space and the independence one. Fortunately, in the case of normal distributions this transformation is analytical [27,28], and in the general case Nataf [29] or Copula [30] transformation can be employed to reduce the problem to the case of the correlated normal distribution. Therefore, we take the case of correlated normal distribution as example to introduce the proposed two methods in the sequel.

ð20Þ

l¼1

It can be seen that by the restrictive condition k þ 1 ≤ jij ≤ q, the sparse grid method can intelligently exclude large number of points from full grids that contribute less to the improvement of the required integration accuracy. Thus it can overcome the “curse of dimension” meaning that the computational cost grows exponentially with the dimension of the problem to a certain extent, which accounts for its well applicability for the high dimensional integration. Furthermore, the implementation of the method is simple and flexible, different kinds of one-dimensional formula can be employed to treat problems with different input distribution types, and the integration accuracy can be easily controlled by adjusting the level k.

4.1.1. Transformation between the correlated normal space and the independent one The joint PDF of the d-dimensional correlated normal input vector x ¼ ðx1 ; x2 ; ⋯; xd ÞT is defined as    1=2 1   f X ðxÞ ¼ ð2πÞd=2 C x  exp  ðxμx ÞT C1 x ðxμx Þ 2 where 8 s2x1 > > > > < Covðx2 ; x1 Þ Cx ¼ > ⋮ > > > : Covðx ; x Þ d 1

μx ¼ ðμx1 ; μx2 ; ⋯; μxd ÞT Covðx1 ; x2 Þ

⋯

s2x2 ⋮

⋯ ⋱

Covðxd ; x2 Þ

⋯

9 Covðx1 ; xd Þ > > > > Covðx2 ; x Þ = d

⋮ s2xd

> > > > ;

ð27Þ and

are the mean

vector and the covariance matrix of x, respectively. jCx j is deter4. Two new methods based on SGI for the importance analysis of correlated input variables 4.1. Double-loop nested SGI (DSGI) Tu Expressions of STc i and Si , i.e.

STc i ¼

VðEðyjxi ÞÞ VðyÞ

ð21Þ

and STu i

VðyÞV ðEðyjxi ÞÞ ; ¼ VðyÞ

ð22Þ

show that the key point in calculating the importance measures of the correlated input variables is to estimate the variance contributions VðEðyjxi ÞÞ and V ðEðyjxi ÞÞ. Both of them are double integrations, which can be seen as the nested integration of expectation and variance operator. Taking VðEðyjxi ÞÞ as an example, it can be seen as the nested integration of Z Eðyjxi Þ ¼ g Xi jX i ðxi Þf X i jX i ðxi Þdxi ð23Þ Rd1

and

Z

VðEðyjxi ÞÞ ¼

R1

ðEðyjxi ÞEðyÞÞ2 f X i ðxi Þdxi

ð24Þ

where f X i ðxi Þ and f Xi jX i ðxi Þ are the marginal PDF of xi and conditional PDF of xi conditional on xi , respectively. g Xi jX i ðxi Þ is the ðd1Þ-dimensional response function after fixing xi . Z gðxÞf X ðxÞdx ð25Þ EðyÞ ¼ Rd

and

Z

VðyÞ ¼ R

d

ðgðxÞEðyÞÞ2 f X ðxÞdx

ð26Þ

are the unconditional expectation and variance of the output response, respectively. In view of the analysis above, this work tries to introduce the SGI into variance analysis for input correlation cases. However, it is

minant value of the covariance matrix Cx , and C1 x is the reverse matrix of Cx . According to the basic theory in linear algebra [27,28], there is an orthogonal matrix A that links x with a random variable vector y ¼ ðy1 ; y2 ; ⋯; yd ÞT by the following equation: ! 1 d y2 f X ðAy þ μx Þ ¼ ð2πÞd=2 ðλ1 λ2 ⋯λd Þ1=2 exp  ∑ i ð28Þ 2 i ¼ 1 λi where λ1 ; λ2 ; …; λn are the eigenvalues of Cx . Based on this conclusion, d-dimensional correlated input variables x ¼ ðx1 ; x2 ; ⋯; xd ÞT are transformed to independent normal variables y ¼ ðy1 ; y2 ; ⋯; yd ÞT , and the transformation formula is y ¼ AT ðxμx Þ

ð29Þ

where the column vectors of A are the orthogonal eigenvectors of Cx . The covariance matrix Cy and mean vector μy of y are 2 3 λ1 0 ⋯ 0 6 ⋮ λ2 ⋯ 0 7 6 7 ð30Þ C y ¼ AT C x A ¼ 6 7 40 ⋮ ⋱ ⋮5 0 0 ⋯ λn and μy ¼ f0; 0; ⋯; 0g

ð31Þ

respectively. Using Eqs. (30) and (31), y can be further transformed into standard normal variables ξ ¼ ðξ1 ; :::; ξd ÞT , i.e. qffiffiffiffiffiffi ð32Þ ξ ¼ ð C y Þ1 y Substitute the inverse transformation of Eq. (32) into (29), and take inverse transformation for the results, the transformation between the original correlated variables x and the independent standard normal one ξ can be obtained: x ¼ Að

qffiffiffiffiffiffi C y ξÞ þ μx

ð33Þ

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4.1.2. Basic procedure of DSGI method for the importance measures of correlated input variables For the normal variables, we choose to use SGI formulas that are based on one-dimensional Gaussian–Hermite quadrature nodes and weights in this paper, which can be calculated by an appropriate method (such as the moment-matching method) or table according to the numbers of quadrature nodes. The basic procedure of DSGI method for the importance measure STc i is as follows: (1) Estimate the unconditional expectation EðyÞ and variance VðyÞ (1.1) Specify the desired accuracy level k, generate quadrature points

ξl ðl ¼ 1; …; N kd Þ

and

corresponding

weights

wl ðl ¼ 1; …; Nkd Þ for d-dimension random space with Eqs. (16) and (17), respectively, by the sparse grid technique. (1.2) Based on the mean vector μx and the covariance matrix Cx of x, transform ξl ðl ¼ 1; …; N kd Þ into the original correlated normal space and obtain xl ðl ¼ 1; …; N kd Þ using the transformation in Eq. (33). (1.3) Calculate EðyÞ and VðyÞ according to xl ðl ¼ 1; …; N kd Þ and wl ðl ¼ 1; …; Nkd Þ by Eqs. (19) and (20), respectively. (2) Estimate Eðyjxi Þ (2.1) Specify the desired accuracy level k, generate quadrature points ξl ðl ¼ 1; …; N kd1 Þ and corresponding weights wl ðl ¼ 1; …; N kd1 Þ for ðd1Þ-dimension random space with Eqs. (16) and (17), respectively, by the sparse grid technique. (2.2) It can be seen from Eq. (23) that to calculate Eðyjxi Þ with SGI, ξl ðl ¼ 1; …; N kd1 Þ need to be transformed into ðd1Þ-dimensional variables xi;l ðl ¼ 1; …; N kd1 Þ with conditional PDF f Xi jX i ðxi Þ. This also can be done by the transformation in Eq. (33), yet the difference from the transformation in step (1.2) lies in that the mean vector μxi jxi and covariance matrix Cxi jxi of f Xi jX i ðxi Þ should be used in the transformation process. For the multivariate normal distribution, μxi jxi and Cxi jxi can be obtained analytically as follows: Since the components xi , xi of the d-dimensional normal random x are also normally distributed with mean vectors μxi , μxi and covariance matrices Cxi ,Cxi correspondingly, the mean vector μx and the covariance matrix Cx of x can " # Cxi Cxi xi T be partitioned as μx ¼ ½μxi μxi  and Cx ¼ , Cxi xi Cxi respectively. With the knowledge of statistics [31], the conditional distribution of xi on xi fixed is still of normal type, which can be expressed as xi jxi  Nðμxi jxi ; Cxi jxi Þ. And the conditional mean and covariance matrix can be obtained by the following formulas: μxi jxi ¼ μxi þ C xi xi C 1 xi ðxi μxi Þ

ð34Þ

C xi jxi ¼ C xi C xi xi C 1 xi C xi xi

ð35Þ

(2.3) Estimate Eðyjxi Þ by Eq. (19) using xi;l ðl ¼ 1; …; N kd1 Þ and wl ðl ¼ 1; …; N kd1 Þ. (3) Estimate VðEðyjxi ÞÞ The estimated result of Eðyjxi Þ in step (2) should be a univariate function of xi , which is denoted by ψðxi Þ in the sequel for concision, thus its expectation Eðψðxi ÞÞ and variance Vðψðxi ÞÞ can be easily computed by the SGI, with the mean and variance of xi already known. The detail process is as follows: (3.1) Specify the desired accuracy level k, for univariate function

(3.2) Transform ξl ðl ¼ 1; …; N k1 Þ into the original correlated normal space to obtain xi;l ðl ¼ 1; …; N k1 Þ according to xi;l ¼ ξl sxi þ μxi . (3.3) Estimate the expectation Eðψðxi ÞÞ and variance Vðψ ðxi ÞÞ of ψðxi Þusing Eqs. (19) and (20) based on xi;l ðl ¼ 1; …; N k1 Þ and wl ðl ¼ 1; …; N k1 Þ. (4) Substituting Vðψðxi ÞÞ and VðyÞ into Eq. (21), we can get the importance measure STc i . For the importance measure STu i , as mentioned above that the key job is to find the estimation of VðEðyjxi ÞÞ, which also can be seen as the nested integration of Eðyjxi Þ and VðEðyjxi ÞÞ. Therefore, it can be obtained by the same procedure above, while the major difference from the estimation of STc i lies in steps (2) and (3). To get the estimation of Eðyjxi Þ in step (2), one-dimensional quadrature points are used, which are then transformed into original correlated normal space according to the conditional mean vector μxi jxi and covariance matrix Cxi jxi of xi on xi fixed. And then, when coming to calculate V ðEðyjxi ÞÞ, since Eðyjxi Þ is a function of ðd1Þ-dimensional input variables xi , ðd1Þ-dimensional quadrature points are firstly generated in step (3) by the sparse grid technique. These quadrature points are then transformed into the original correlated normal space based on the mean vector μxi and covariance matrix Cxi of xi . After V ðEðyjxi ÞÞ is obtained, the estimation of STu is straightforward according to i Eq. (22). 4.2. Single loop SGI (SSGI) Substitute Eq. (23) into (24), the full expression of V ðEðyjxi ÞÞ can be obtained: 2 Z Z g Xi jX i ðxi Þf X i jX i ðxi Þdxi f X i ðxi Þdxi E2 ðyÞ VðEðyjxi ÞÞ ¼ R1

Rd1

ð36Þ Borrowing the idea of the single loop Monte Carlo method [32], which was created for the importance analysis of independent input variables, the expression (36) can be written as V ðEðyjxi ÞÞ i R R hR ¼ R1 Rd1 g X i jX i ðxi Þf X i jX i ðxi Þdxi  Rd1 g X i jX i ðx′i Þf X i jX i ðx′i Þdx′i f X i ðxi Þdxi E2 ðyÞ

Z ¼ R2d1

gðxÞg X i jX i ðx′i Þf X ðxÞf X i jX i ðx′i Þdxdx′i E2 ðyÞ ¼ EðGÞE2 ðyÞ ð37Þ

where xi and x′i are two different random vectors generated from the condition PDF f Xi jX i ðxi Þ, and GðxÞ ¼ gðxÞg Xi jX i ðx′i Þ. Eq. (37) allows for reducing the double-loop nested integration in Eq. (36) into a single integration by increasing the dimensionality of the integral from d to 2d1. Application of Eq. (37) only requires estimating the expectation of GðxÞ, i.e. EðGÞ, and the unconditional expectation EðyÞ, which will be done easily by the SGI method. Analogously, the double-loop nested integration of VðEðyjxi ÞÞ, i.e. Z 2 Z g X i jX i ðxi Þf X i jX i ðxi Þdxi f X i ðxi Þdxi E2 ðyÞ VðEðyjxi ÞÞ ¼ Rd1

R1

ð38Þ can be reduced into Z VðEðyjxi ÞÞ ¼

Rdþ1

gðxÞg X i jXi ðx′i Þf X ðxÞf X i jX i ðx′i Þdxdx′i E2 ðyÞ ¼ EðG′ÞE2 ðyÞ

ð39Þ

ψðxi Þ, one-dimensional quadrature points ξl ðl ¼ 1; …; N k1 Þ

Then, the key point in the application of Eq. (39) only involves calculating the expectation EðG′Þ of the ðd þ 1Þ-dimensional function G′ðxÞ ¼ gðxÞg X i jXi ðx′i Þ. The basic procedure of SSGI method for

and corresponding weights wl ðl ¼ 1; …; N k1 Þ are used.

Tu the estimation of STc i and Si is based on Eqs. (37) and (39), which

L. Li, Z. Lu / Reliability Engineering and System Safety 119 (2013) 207–217

consists of the following steps: (1) Specify the desired accuracy level k, generate quadrature points ξl ðl ¼ 1; …; N k2d Þ and corresponding weights wl ðl ¼ 1; …; N k2d Þ for 2d-dimension random space with Eqs. (16) and (17), respectively, by the sparse grid technique. And then split ξl ðl ¼ 1; …; N k2d Þ into three parts: the initial (1:d) low-dimensional nodes are denoted as ξdl ðl ¼ 1; …; N k2d Þ, the middle (d+1:2d 1) ðl ¼ 1; …; N k2d Þ, and the dimensional nodes are denoted as ξd1 l (2)

(3) (4)

(5) (6) (7)

(8) (9)

last one dimensional nodes are denoted as ξ1l ðl ¼ 1; …; N k2d Þ. Based on the mean vector μx and the covariance matrix Cx of x, transform ξdl ðl ¼ 1; …; N k2d Þ into the original correlated normal space to obtain xl ðl ¼ 1; …; N k2d Þ using the transformation in Eq. (33), and then split xl into two components xi;l and xi;l . Estimate EðyÞ and V ðyÞ using xl ðl ¼ 1; …; Nk2d Þ and wl ðl ¼ 1; …; N k2d Þ by Eqs. (19) and (20), respectively. Transform ξd1 ðl ¼ 1; …; Nk2d Þ into ðd1Þ-dimensional variables l x′i;l ðl ¼ 1; …; N k2d Þ with conditional mean vector μxi jxi and covariance matrix Cxi jxi by the transformation (33). In this step μxi jxi and Cxi jxi in Eqs. (34) and (35) are computed using xi;l generated in step (2). Estimate EðGÞ of GðxÞ ¼ gðxÞg Xi jX i ðx′i Þ by Eq. (19) using xl ðl ¼ 1; …; N k2d Þ, x′i;l ðl ¼ 1; …; Nk2d Þ and wl ðl ¼ 1; …; Nk2d Þ. Substitute EðGÞ and EðyÞ into Eq. (37) to get VðEðyjxi ÞÞ, and then substitute VðEðyjxi ÞÞ and VðyÞ into Eq. (21), STc i can be obtained. Transform ξ1l ðl ¼ 1; …; N k2d Þ into one-dimensional variables x′i;l ðl ¼ 1; …; N k2d Þ with conditional mean vector μxi jxi and covariance matrix Cxi jxi by the transformation (33). In this step μxi jxi and Cxi jxi are computed using xi;l generated in step (2). Estimate EðG′Þ of G′ðxÞ ¼ gðxÞg X i jXi ðx′i Þ by Eq. (19) based on xl ðl ¼ 1; …; N k2d Þ, x′i;l ðl ¼ 1; …; Nk2d Þ and wl ðl ¼ 1; …; Nk2d Þ. Substitute EðG′Þ and EðyÞ into Eq. (39), and then substitute the obtained VðEðyjxi ÞÞ and VðyÞ into Eq. (22), STu i can be obtained.

4.3. Discussions on the proposed two SGI methods From the basic theories and procedures of the proposed two SGI methods above, it can be seen that DSGI method can make full use of the advantages of the SGI by viewing the importance measures of the correlated input variables as nested integration of expectation and variance operators, while SSGI can take full advantages of the fact that SGI is especially well-suited for high dimensional integration based on the theory of the single loop Monte Carlo method. It seems at first sight that SSGI should be more efficient than DSGI. Although this is actually true in many cases, for the importance measures of the correlated input variables, DSGI can be more efficient than SSGI sometimes, e.g. when calculating the importance measure of an individual input variable. The reason is that the number of quadrature points of SGI

213

relies on the dimensionality of the input variables. To calculate the importance measure of a set of input variables xI ðI ¼ i1 … Tu is ; 1 ≤i1 ; …; is ≤ dÞ, ðSTc I ; SI Þ, while the total number of the model evaluation of DSGI method is 2N ks N kds þ Nkd , varying with the subscript variable s, the total computational cost of SSGI always equal to 3Nk2d . When s or ds is very small, the corresponding N ks or Nkds is also a small number, leading to a smaller computational cost of DSGI. For different dimensionality d and level k, Table 1 lists the amount of quadrature points needed in the two methods when calculating the importance measures of the correlated input variables. It can be found that while SSGI has the advantage of calculating the importance measure of a set of input variaTu bles, DSGI is always more efficient in estimating ðSTc i ; Si Þ, the importance measure of an individual correlated input variable. They are tailored to satisfy different engineering purpose. Furthermore, since the sparse grid technique is basically under the assumption of input independence, when dealing with correlated input variables in this paper, transformation between the correlation space and the independence one is necessary. The procedures of the two proposed methods described above show that by obtaining the conditional distribution of xi with xi fixed or that of xi with xi fixed in the process, the correlation between xi and xi can be kept completely in the transformation process.

5. Examples Example 1. Consider the linear model Y ¼ gðxÞ ¼ x1 þ x2 þ x3 in Ref. [19], where all the input variables are normally distributed 0 1 1 0 0 B 0 1 2ρ C with zero mean and covariance matrix C x ¼ @ A. The 0 2ρ 22 computational results of the importance measures by the two proposed SGI methods are listed in Table 2. Additionally, the results of the quasi Monte Carlo (QMC) method as well as the analytical (ANA) results in Ref. [19] are also presented for comparison. Example 2. Consider the nonlinear model Y ¼ gðxÞ ¼ x1 x3 þ x2 x4 in Ref. [19], where ðx1 ; x2 ; x3 ; x4 Þ∼Nðμx ; Cx Þ, with μx ¼ ð0; 0; 250; 400Þ 0 1 16 2:4 0 0 B 2:4 4 C 0 0 B C and Cx ¼ B C. Table 3 shows the @ 0 1:8⋅104 A 0 4⋅104 0 0 1:8⋅104 9⋅104 numerical estimates of the importance measures by the two proposed methods. Once again the results of QMC and analytical analysis are cited from Ref. [19] for comparison. Tables 2 and 3 show that for the linear and nonlinear model in Examples 1 and 2, both the computational results of the proposed two SGI methods can agree with the analytical values completely. Compared with the QMC method employed in Ref. [19], the

Table 1 Tu Computational costs of the two SGI methods for a set ðSTc I ; SI Þ. Tu ðSTc i ; Si Þ

Measures

Tu ðSTc ijl ; Sijl Þ

Tu ðSTc ij ; Sij Þ

k¼1

k¼2

k¼1

k¼2

k¼1

k¼2

d

DSGI

SSGI

DSGI

SSGI

DSGI

SSGI

DSGI

SSGI

DSGI

SSGI

DSGI

SSGI

4 8 10 13

37 77 97 127

51 99 123 159

191 823 1307 2243

435 1635 2523 4215

59 147 191 257

51 99 123 159

379 2355 3991 7255

435 1635 2523 4215

37 171 231 321

51 99 123 159

191 3195 5871 11415

435 1635 2523 4215

214

L. Li, Z. Lu / Reliability Engineering and System Safety 119 (2013) 207–217

Table 2 Computational results of the importance measures of the correlated input variables in Example 1. STc i

Measures

STu i

ρ

Methods

DSGI

SSGI

QMC

ANA

DSGI

SSGI

0

x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3

0.167 0.167 0.667 0.125 0.500 0.781 0.250 0.000 0.563 0.109 0.735 0.852 0.357 0.129 0.514

0.167 0.167 0.667 0.125 0.500 0.781 0.250 0.000 0.563 0.109 0.735 0.852 0.357 0.129 0.514

0.167 0.168 0.669 0.126 0.502 0.784 0.251 0.000 0.564 0.109 0.737 0.855 0.357 0.129 0.515

0.167 0.167 0.667 0.125 0.5 0.781 0.250 0.000 0.563 0.109 0.735 0.852 0.357 0.129 0.514

0.167 0.167 0.667 0.125 0.094 0.375 0.250 0.188 0.750 0.109 0.039 0.157 0.357 0.129 0.514 67

0.167 0.167 0.667 0.125 0.094 0.375 0.250 0.188 0.750 0.109 0.039 0.157 0.357 0.129 0.514 91

0.5

 0.5

0.8

 0.8

Tu Total number of model evaluations for a set ðSTc i ; Si Þði ¼ 1; 2; 3Þ

QMC

ANA

0.167 0.167 0.669 0.125 0.094 0.376 0.251 0.188 0.752 0.109 0.039 0.157 0.358 0.129 0.515

0.167 0.167 0.667 0.125 0.094 0.375 0.250 0.188 0.750 0.109 0.039 0.157 0.357 0.129 0.514

216

Table 3 Computational results of the importance measures of the correlated input variables in Example 2. Measures

STc i

STu i

Methods

DSGI

SSGI

QMC

ANA

DSGI

SSGI

QMC

ANA

x1 x2 x3 x4

0.507 0.399 0.000 0.000

0.507 0.399 0.000 0.000

0.503 0.400 0.009 0.007

0.507 0.399 0.000 0.000

0.492 0.300 0.192 0.108 641

0.492 0.300 0.192 0.108 1305

0.495 0.298 0.185 0.108 15000

0.492 0.300 0.192 0.108

Tu Total number of model evaluations for a set ðSTc i ; Si Þði ¼ 1; …; 4Þ

presented two SGI methods not only have higher accuracy, but also can improve the computational efficiency of the importance measures dramatically. This testifies the accuracy and efficiency of the proposed two methods. Example 3. Ishigami function Y ¼ gðxÞ ¼ sin ðx1 Þ þ 7 sin 2 ðx2 Þ þ 0:1x43 sin ðx1 Þ with strong non-linearity and non-monotonicity is also considered in this paper, where all the input variables are uniformly distributed in the interval ½π; π. As assumed in Ref. [19], x1 and x3 are correlated with the correlation coefficient ρ13 ranging from  0.9 to 0.9. Fig. 1 shows the variation of the importance measures at different values of ρ13 obtained by the proposed two methods as well as the QMC method. For the uniformly distributed input variables in this example, we choose to use the zeroes of the Gauss-legendre polynomials as the one dimensional quadrature points. Gaussian copula is used to transform the original problem to the case of the correlated normal one. To obtain a smooth curve of the importance measure with enough accuracy, 105 samples are used in the QMC method, and a high level of k ¼ 7 is specified for the proposed two SGI methods, representing fifteenth order polynomial exactness. Fig. 1 shows that the computational results of the proposed two SGI methods are in good agreement with the QMC method, which suggests the applicability of the proposed methods to the strong non-linear and non-monotone function. However, for each value of ρ13 , 7  105 model evaluations Tu are needed in the QMC method for a set ðSTc i ; Si Þði ¼ 1; 2; 3Þ, whereas the total numbers of model evaluations of the DSGI and SSGI methods are 10881 and 232351, respectively, indicating that both of the proposed methods are more efficient than QMC even for function

with strong non-linearity and non-monotonicity. This shows the accuracy and efficiency of the proposed two methods once again. Furthermore, it can be seen from Fig. 1 that for the input variable Tu x2 , STc 2 is always equal to S2 . This agrees with the variable settings, since x2 is involved neither in correlation nor in interaction with Tu only contain its uncorrelated other variables, both STc 2 and S2 contribution. For the input variable x3 , both the importance measure Tu STc 3 and S3 are influenced strongly by the correlation, yet they show Tu totally distinct variation: when ρ13 ¼ 0, STc 3 ¼ 0 while S3 reaches its Tu maximum. As jρ13 j-1, STc 3 increases to its maximum, whereas S3 goes to zero. This is easy to understand considering the significances Tu of STc 3 and S3 . Since there is no first order term of x3 in the response function, the independent contribution by x3 itself is zero. As a result,

STc 3 only contains the variance contribution related to the correlated variation of x3 , whereas STu 3 includes the independent contribution by interaction between x3 and other variables. When x1 and x3 are totally independent (ρ13 ¼ 0), the correlated contribution of x3 is zero, and the independent contribution by interaction between x3 Tu and the others reaches its maximum. As a result, STc 3 ¼ 0 and S3 reaches maximum. As the correlation of x1 and x3 increasing (jρ13 j-1), the correlated contribution of x3 increases accordingly while the independent contribution by interaction between x3 and Tu the others goes to zero, which leads to a maximum STc 3 and a zero S3 . However, the situation for x1 is different. For the input variable x1 ,

STu 1 decreases obviously with correlation, showing a similar pattern as Tc STu 3 , whereas S1 is only weakly affected by the correlation. This is also

L. Li, Z. Lu / Reliability Engineering and System Safety 119 (2013) 207–217

215

Fig. 1. Variation of the importance measures of the correlated input variables in Example 3 at different values of ρ13 . Solid lines refer to STc ; broken lines refer to STu .

Tu very lucid in view of the meanings of STc 1 and S1 proposed in this paper. Since x1 is correlated to x3 in the parameter setting and has

interaction with x3 in the response function, STc 1 contains both the independent contribution by x1 itself and the contribution by the correlation between x1 and x3 , and STu 1 consists of the independent contribution by x1 itself and the independent contribution by interaction between x1 and other variables. When x1 and x3 are totally independent (ρ13 ¼ 0), the contribution by the correlation between x1 and x3 is zero, and the independent contribution by x1 itself and independent contribution by interaction between x1 and the others Tc have their maxima. This leads to a maximum STu 1 and a nonzero S1 due to the nonzero independent contribution by x1 itself. As the correlation of x1 and x3 increasing (jρ13 j-1), the contribution by the correlation between x1 and x3 increases accordingly while both the independent contribution by x1 itself and independent contribution by interaction between x1 and the others go to zero. As a result,

STu 1

STc 1

tends to zero. The unobvious variation of resulting from the equivalently opposite changing rates of the independent and the correlated contribution of x1 , which suggests that the contribution by the uncorrelated and the correlated variation of x1 to the output variance are basically the same. The analysis above shows that the new interpretation of the variance-based importance measures of correlated input variables can reflect the uncorrected and correlated contribution of an individual variable to the output variance clearly and reasonably. Tu Nevertheless, STc i and Si are defined as the first order effect index ηi and the total effect index ηTi of xi in Ref. [19]. This definition is obviously inappropriate from the computational results shown in Fig. 1. According to the definition (6) of the total effect in the Sobol's importance measure, ηTi should include the first order effect ηi of xi as well as all the interactive effect of xi with other variable, thus there must be ηTi ≥ηi . However, in this example as jρ13 j-1, the total effects of x1 and x3 go to zero while their first order effects tend to maxima, which is a clear contravention of the meanings of the total effect. Therefore, when correlation is present among the input variables, the same importance measure will have totally different meanings from that of the independent one. A direct extension of the meanings of the importance measures in the case of independent input variables to the correlated case is unreasonable. Example 4. In this example, we apply the presented importance measures and the established methods to the importance analysis of a nine-box wing in engineering. The results will provide useful information for reliability design and optimization of the nine-box

Fig. 2. Diagram of the nine-box wing.

Table 4 Distribution parameters of the input variables of nine-box wing. Random variables

L

A

E

P

TH

Mean value Variation coefficient

0.2 m 0.1

0.0001 m2 0.1

71 GPa 0.1

1500 N 0.1

0.003 m 0.1

wing in the case of correlated input variables. The diagram of the nine-box wing is shown in Fig. 2. It consists of 64 bars and 42 plates. The sectional area of all the bars is denoted as A, while the length of the bars in the y, x and z directions are L, 2L and 3L, respectively. The thickness of all the plates is denoted as TH. E is the elastic modulus of all bars and plates, and P is the external load. We assume L, A，TH，E and P are normally distributed random variables with parameters shown in Table 4. A and L are correlated with correlation coefficient ρAL ¼ 0:5. Taking the displacement of y direction not exceeding 0.0111 m as the constrain condition, the following reliability model can be constructed, Y ¼ g ¼ 0:0111jΔyj, where Δy ¼ ΔðL; A; E; P; THÞ is an implicit function of the input random variables, and determined by the finite element method. For the implicit model in this example, k ¼ 3 is specified to obtain a result with up to seventh order polynomial exactness. Table 5 shows the importance measures estimated by the proposed two SGI methods, indicating the applicability of the proposed methods to the implicit finite element model. Furthermore, for the input variables E，P and TH, as they are not involved in correlation with other variables, their importance measures STc i should only consist of independent contribution by

216

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Table 5 Computational results of the importance measures for the wing structure. Measures

STc i

STu i

Methods

DSGI

SSGI

L A E P TH

0.0028 0.0020 0.3732 0.3503 0.2576

0.0027 0.0018 0.3733 0.3504 0.2577

Tu Total number of model evaluations for a set ðSTc i ; Si Þ

the variable itself, while their importance measures STu contains i both the independent contribution by the variable itself and the independent contribution by the interaction among variables. However, it can be seen from Table 5 that STc i are nearly equal to for these variables. This suggests that the contributions of STu i these variables to the output variance mainly come from the unique variations of these variables, the interactions of these variables are all very weak in the model function. For the input Tc variables L and A, STu i 4 Si , indicating that their contributions to the output variance mostly originate from the uncorrelated variations, and they have interaction with each other in the reliability model, (otherwise, STu i will contain only independent contribution by the variable itself, which will lead to a smaller STu i ). Therefore, the application of the proposed importance measures cannot only clearly reflect the origin of the variance contributions of the correlated inputs, but provide useful information for identifying the structure of the model function. The results will provide important guidance for engineering optimization and design of the nine-box wing. Finally, it is worth pointing out that although application of STc i and STu can provide useful information about the origin of the i variance contributions of the correlated inputs as well as the structure of model function, it is inappropriate to directly use STc i and STu i as measures to rank the correlated input variables. This is Tu contain covariance terms because the expressions of STc i and Si which can be negative depending on correlation patterns. Due to different signs the terms contained in the expressions of STc i and may cancel one another. For ranking the correlated input STu i variables, the importance measures defined in Ref. [17], based on covariance decomposition of the unconditional variance of the output, may be more appropriate if the components included in the decomposition are correctly determined.

6. Conclusions A novel interpretation for the variance-based importance measures in case of correlated input variables is presented. The interpretation is directly based on the contribution of the correlated input variables to the output variance, and divides the contribution of the correlated input variables into main effect of correlated and uncorrelated variations and the total effect of uncorrelated variation. Therefore, compared with the generalization-based measures, it cannot only reflect the uncorrelated and correlated contribution of the input variables to the output variance clearly, but be more reasonable for interpreting the variance-based measures of the correlated input variables. Additionally, the new interpretation can obtain the same importance measure as the independentorthogonalisation based method does, while does not need to perform

DSGI

SSGI

0.0077 0.0070 0.3794 0.3564 0.2628 5721

0.0078 0.0069 0.3798 0.3567 0.2632 17391

circular orthogonalization of the original correlated inputs. Thus, it is more convenient for computation. By viewing the variance based importance measures as nested expressions of expectation and variance operators, and performing transformation between the correlation space and the independence one, SGI is successfully introduced into variance based importance analysis for model with correlated input variables, on which two SGI methods are proposed to deal with different situations. The basic theories and detailed procedures of the two proposed methods are given. Compared with the traditional methods based on sampling, e.g. QMC, the proposed two methods do not need a large number of samples, thus can act as a competitive tool for importance analysis. At last, the application of the proposed importance measures and methods to the engineering example provides guidance for the engineers to decrease the variability of the output and identify the model function of the nine-box wing.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. NSFC 51175425), the Doctorate Foundation of Northwestern Polytechnical University (Grant No. CX201205), the Ministry of Education Fund for Doctoral Students Newcomer Awards of China and the Excellent Doctorate Foundation of Northwestern Polytechnical University (Grant No. DJ201301). Additionally, the authors would like to thank the anonymous reviewers for their valuable comments.

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