Regional moment-independent sensitivity analysis with its applications in engineering

Chinese Journal of Aeronautics, (2017), 30(3): 1031–1042

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

Regional moment-independent sensitivity analysis with its applications in engineering Changcong ZHOU *, Chenghu TANG, Fuchao LIU, Wenxuan WANG, Zhufeng YUE School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, 710129, China Received 22 March 2016; revised 13 December 2016; accepted 2 March 2017 Available online 8 May 2017

KEYWORDS Cumulative distribution function; Moment-independent; Probability density function; Regional importance measure; Sensitivity analysis; Uncertainty

Abstract Traditional Global Sensitivity Analysis (GSA) focuses on ranking inputs according to their contributions to the output uncertainty. However, information about how the specific regions inside an input affect the output is beyond the traditional GSA techniques. To fully address this issue, in this work, two regional moment-independent importance measures, Regional Importance Measure based on Probability Density Function (RIMPDF) and Regional Importance Measure based on Cumulative Distribution Function (RIMCDF), are introduced to find out the contributions of specific regions of an input to the whole output distribution. The two regional importance measures prove to be reasonable supplements of the traditional GSA techniques. The ideas of RIMPDF and RIMCDF are applied in two engineering examples to demonstrate that the regional moment-independent importance analysis can add more information concerning the contributions of model inputs. Ó 2017 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Sensitivity Analysis (SA) is to study ‘‘how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input factors”.1 Generally, SA can be classified into two main categories1–3: Local Sensitivity Analysis (LSA), which is often carried out in the form of derivative of the model output with respect to the input parameters, and Global Sensitivity Analysis (GSA), * Corresponding author. E-mail address: [email protected] (C. ZHOU). Peer review under responsibility of Editorial Committee of CJA.

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which focuses on the output uncertainty over the entire range of the inputs. The limitation of LSA, as a derivative-based approach, lies in that derivatives are only informative at the base points where they are calculated and cannot provide an exploration of the rest of the input space. GSA, on the other hand, explores the whole space of the input factors, and thus is more informative and robust than estimating derivatives at a single point of the input space. Obviously, GSA has a greater potential for engineering applications. Global sensitivity indices are also known as importance measures, and a rapid development in this field has been witnessed in the last several decades.4 The family of importance measures generally includes nonparametric techniques suggested by Saltelli and Marivoet5 and Iman et al.,6 variance-based methods suggested by Sobol7 and further developed by Rabitz and Alis,8 Saltelli et al.,9 and Frey and

http://dx.doi.org/10.1016/j.cja.2017.04.006 1000-9361 Ó 2017 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1032 Patil,10 and moment-independent approaches proposed by Park and Ahn,11 Chun et al.,12 Borgonovo,13,14 Liu and Homma,15 and Castaings et al.16 It has been found that nonparametric techniques are insufficient to capture the influences of inputs on the output variability for nonlinear models and also when interactions among inputs emerge.10 As Saltelli underlined,1 an importance measure should satisfy the requirement of being ‘‘global, quantitative, and model free”, and he advocated that the variance-based importance measure was a preferred way of measuring uncertainty importance. However, the use of variance as an uncertainty measure relies on the assumption that ‘‘this moment is sufficient to describe output variability”.13 Borgonovo13 showed that relying on the sole variance as an indicator of uncertainty would sometimes lead a decision maker to noninformative conclusions, since the inputs that influence the variance the most are not necessarily the ones that influence the output uncertainty distribution the most. Borgonovo extended Saltelli’s three requirements by adding ‘‘moment independent”,13,14 and proposed a new importance measure which looks at the influence of the input uncertainty on the entire output distribution without reference to a specific moment of the output. In a similar manner, Liu and Homma15 proposed another moment-independent importance measure, which describes the contribution of the input uncertainty on the Cumulative Distribution Function (CDF) of the model output, while Borgonovo’s importance measure studies the contribution on the Probability Density Function (PDF) of the model output. However, traditional GSA techniques only provide information showing the relative importance of input variables, with no knowledge of which part of a variable is important such as the left or right tail, center region, near center, etc. Such information will be useful for decision makers to identify the important areas inside an input variable, and take corresponding measures. For this purpose, Millwater et al.17 proposed a localized probabilistic sensitivity method to determine the random variable regional importance; however, this method is based on derivatives and suffers the same constraints as those of LSA. In 1993, Sinclair18 introduced the idea of Contribution to Sample Mean (CSM) plot which was further developed by Bolado-Lavin et al.19 As an extension of CSM, Tarantola et al.20 proposed the Contribution to Sample Variance (CSV) plot. The principle behind CSM and CSV is to use a given random sample of input variables, which is generally used for uncertainty analysis, to measure the effects of specific regions of an input variable on the mean and variance of the output. However, the information provided by CSM and CSV will become limited when the mean and variance are insufficient to describe the output distribution, which is possible in both theoretical and real applications. In this work, the idea of CSM and CSV is extended to the domain of moment-independent GSA techniques. Based on the importance measures proposed by Borgonovo13 and Liu and Homma15 respectively, regional moment-independent importance analysis considering the output PDF and CDF is introduced, to find out the contributions of specific regions of an input variable to the whole output distribution. Besides, the regional importance analysis can be performed with the same sample points for computing the traditional momentindependent importance measures, and thus no additional model evaluation is needed. In other words, the regional moment-independent importance analysis can be viewed as a

C. ZHOU et al. byproduct, offering much more information than that provided by the standard importance analysis. The remainder of this paper is organized as follows. The CSM and CSV theories are briefly reviewed in Section 2. Then the regional moment-independent importance analysis and computational strategies are discussed in Section 3. In tion 4, two engineering examples are presented to demonstrate the applicability of the newly proposed concept. Finally, conclusions of this work are highlighted in Section 5. 2. Review of CSM and CSV plots Suppose that the input-output model is denoted by Y ¼ gðXÞ, where Y is the model output and X ¼ ½X1 ; X2 ; . . . ; Xn T (n is the input dimension) is the set of input variables. The uncertainties of the input variables are represented by probability distributions. The joint probability density function PDF of X is denoted as fX ðxÞ, and the marginal PDF of Xi can be formulated as Z Z n Y fXi ðxi Þ ¼ . . . fX ðxÞ dxk ð1Þ k¼1;k–i

Let us recall the definition of CSM for a given input Xi ,18,19 Z þ1 Z þ1 Z F1 ðqÞ Y n Xi 1 CSMXi ðqÞ ¼ ... fXi ðxi Þ EðYÞ 1 1 1 i¼1  gðx1 ; x2 ; . . . ; xn Þdxi dx1 . . . dxi1 dxiþ1 . . . dxn ð2Þ R þ1 R þ1 where q 2 ½0; 1, EðYÞ ¼ 1 . . . 1 gðxÞfX ðxÞdx is the mean of the model output, and F1 Xi ðqÞ is the inverse CDF of Xi at quantile q. The multiple integral in Eq. (2) is taken in the range ð1; þ1Þ for all the input variables except Xi , for which the range is ð1; F1 Xi ðqÞ. In a similar manner, CSV for Xi is defined as20 Z þ1 Z þ1 Z F1 ðqÞ Y n Xi 1 CSVXi ðqÞ ¼ ... fXi ðxi Þ VðYÞ 1 1 1 i¼1  ðgðx1 ;x2 ; .. . ;xn Þ  EðYÞÞ2 dxi dx1 . . .dxi1 dxiþ1 .. . dxn ð3Þ R þ1 R þ1 where VðYÞ ¼ 1 . . . 1 ðgðxÞ  EðYÞÞ2 fX ðxÞdx is the variance of the model output, and the other notations have the same meaning as in the definition of CSM. It is important to note that CSV is defined as a contribution to the output variance with respect to a constant mean EðYÞ over the full range. Both CSM and CSV are plotted in the ½0; 12 space, with q as a point on x-axis representing a fraction of the distribution range of Xi , and CSMXi ðqÞ or CSVXi ðqÞ as a fraction of the output mean or variance corresponding to the values of Xi smaller than or equal to its q quantile. CSM and CSV are meaningful to estimate the contributions of specific ranges of an input variable to the mean and variance of the output. Considering Xi and a specific range of q, say ½0; 0:1 for example, if the CSM or CSV plot is close to the diagonal, it indicates that the contribution to the output mean or variance is almost equal throughout this range of Xi . In addition, the contribution of Xi in the range to the output mean or variance is lower than the average if the CSM and CSV plots are convex downwards; otherwise, the contribution is higher than the average if the plots are convex upwards.

Regional moment-independent sensitivity analysis

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Furthermore, considering another range ½q1 ; q2  of Xi , the values of CSMXi ðq2 Þ  CSMXi ðq1 Þ or CSVXi ðq2 Þ  CSVXi ðq1 Þ can measure the contribution of this range to the output mean or variance. Clearly, CSM and CSV provide much more information than solely performing the standard GSA techniques, along with the convenience that the plots can be depicted without additional model evaluations. Readers are referred to Refs.18–20 for more detailed proofs and interpretations of CSM and CSV. 3. Regional moment-independent importance analysis 3.1. Two moment-independent importance measures Instead of evaluating the output uncertainty only by the mean or variance, the moment-independent importance analysis estimates the contributions of input variables on the whole output distribution. In this work, we extend the idea of regional sensitivity analysis to two moment-independent importance measures considering the PDF and CDF of the output. The unconditional PDF and CDF of the model output Y are denoted as fY ðyÞ and FY ðyÞ, respectively. The conditional PDF and CDF of Y are denoted as fYjXi ðyÞ and FYjXi ðyÞ, which can be obtained by fixing the input of interest Xi at its realization xi generated by its distribution. The conditional and unconditional distribution functions of the output are depicted in Fig. 1. Borgonovo13 proposed his importance measure by considering the contributions of inputs to the PDF of the model output. The shift between fY ðyÞ and fYjXi ðyÞ is measured by the area sðXi Þ closed by the two PDFs, which is given as Z sðXi Þ ¼ jfY ðyÞ  fYjXi ðyÞjdy ð4Þ The expected shift can be obtained by varying the value of Xi over its distribution range, i.e., Z EXi ½sðXi Þ ¼ fXi ðxi ÞsðXi Þdxi ð5Þ where fXi ðxi Þ is the marginal PDF of Xi . Then Borgonovo’s importance measure is defined as 1 di ¼ EXi ½sðXi Þ 2

In a similar manner, Liu and Homma15 proposed a new moment-independent importance measure considering the CDF of the model output. The deviation of FYjXi ðyÞ from FY ðyÞ is measured by using the area AðXi Þ closed by the two CDFs, and can be calculated as Z ð7Þ AðXi Þ ¼ jFY ðyÞ  FYjXi ðyÞjdy The expected deviation of FYjXi ðyÞ from FY ðyÞ can be obtained as Z EXi ½AðXi Þ ¼ fXi ðxi ÞAðXi Þdxi ð8Þ EXi ½AðXi Þ can be utilized to illustrate the influence of the input Xi on the output distribution. Liu and Homma15 defined this measure as the CDF-based sensitivity indicator and denoted ðCDFÞ it as Si . Furthermore, for a general model of which the unconditional expectation of the output EðYÞ is not equal to ðCDFÞ 0, Si can be normalized as ðCDFÞ

Si

¼

EXi ½AðXi Þ jEðYÞj

ð9Þ

ðCDFÞ

Both di and Si evaluate the influence of the entire distribution of the input Xi on that of the model output Y, except that the former focuses on the PDF of the output while the latter focuses on the CDF. For both measures, the effect of Xi on the output uncertainty is estimated by varying all the other inputs over their variation ranges. The two momentindependent importance measures represent the expected shift in a decision maker’s view on the output uncertainty provoked ðCDFÞ will be assoby Xi . The inputs with larger values of di or Si ciated with a higher importance, indicating that these inputs should be paid more attention to if we want to control the model performance. In fact, the two measures work regardless of whether the model is linear or nonlinear, additive or nonadditive, and can be easily extended to a group of inputs, for which more detailed information can be found in the works of Borgonovo13,14 and Liu and Homma.15 ðCDFÞ

3.2. Regional importance analysis based on di and Si ðCDFÞ

ð6Þ

The two measures, di and Si , add new insights into the GSA. However, just like the other GSA techniques, both measures can only estimate the contribution of the entire input distribution to the output uncertainty. In this respect, we consider ðCDFÞ , and proto extend the idea of CSM and CSV to di and Si pose a new concept of regional moment-independent importance analysis. Just like the way of defining CSM and CSV, instead of integrating sðXi Þ at ð1; þ1Þ in Eq. (5), we cut the integration range down to ð1; F1 Xi ðqÞ for investigating the contribution of Xi within this range to the output PDF. We now put forward the Regional Importance Measure based on PDF (RIMPDF), and present its formulation as  Z F1 ðqÞ Z þ1 Xi 1 RIMPDFXi ðqÞ ¼ jfY ðyÞ  fYjXi ðyÞjdy fXi ðxi Þdxi 2di 1 1 ð10Þ

Fig. 1 Unconditional and conditional distribution functions of output.

RIMPDFXi ðqÞ can be plotted in the ½0; 12 space. Compared to di which can rank the inputs according to their

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contributions, RIMPDFXi ðqÞ is able to further investigate the effects that specific areas of an input have on the output PDF. In fact, a similar idea of considering the regional effects of the inputs on di was also discussed by Wei et al.21; however, in this work, it is discussed in a different and more comprehensive way. Similarly, the Regional Importance Measure based on CDF (RIMCDF), can be established as Z F1 ðqÞ Z þ1 Xi 1 RIMCDFXi ðqÞ ¼ ðCDFÞ jFY ðyÞ jEðYÞj 1 Si 1  ð11Þ  FYjXi ðyÞjdy fXi ðxi Þdxi RIMCDFXi ðqÞ is also plotted in the ½0; 12 space, and it investigates the contributions of specific areas of an input variable to the output CDF. Obviously, RIMPDFXi ðqÞ and RIMCDFXi ðqÞ can be viewed as regional versions of the two moment-independent ðCDFÞ measures di and Si . However, it is very meaningful to perform such extensions, as RIMPDFXi ðqÞ and RIMCDFXi ðqÞ ðCDFÞ can act as good supplements to di and Si , and enrich the available information by moment-independent importance analysis. Besides, they can be easily estimated by using the ðCDFÞ same samples for evaluating di and Si , and thus no additional model evaluation is needed, which will be discussed later in this work. 3.3. Interpretations of RIMPDFXi ðqÞ and RIMCDFXi ðqÞ

ðCDFÞ½q1 ;q2 

Si

ðCDFÞ Si

¼

RIMCDFXi ðq2 Þ  RIMCDFXi ðq1 Þ q2  q1

ð13Þ

Important information can be provided by Property III: given the quantiles q1 and q2 , reduction of the two momentindependent importance measures by reducing the range of Xi from ð1; þ1Þ to ½xi;1 ; xi;2  can be estimated by Eqs. (12) and (13). Since q1 and q2 can be any value from the interval ½0; 1 as long as satisfying q1 < q2 , a 3D plot can be created indicating the region where the reduction of the input contribution to the whole output distribution can be most effectively achieved. Obviously, such information will be missed if only performing the traditional importance analysis reviewed in Section 3.1. Properties I and II can be easily derived by the definitions of regional importance measures, and here we only give the proof of Property III. Proof. Let us consider the input variable Xi with its PDF fXi ðxi Þ over ð1; þ1Þ. Now the range of Xi is reduced to ½xi;1 ; xi;2  (corresponding to the quantile interval ½q1 ; q2 ), and according to the principles of truncated distribution, the PDF of Xi after the reduction can be established as20 fXi ðxi Þ f ðxi Þdxi xi;1 Xi

fXi ðxi Þ ¼ R xi;2

ð14Þ

Then Borgonovo’s and Liu’s importance measures of Xi after the distribution reduction can be computed as

The newly proposed regional measures, RIMPDFXi ðqÞ and RIMCDFXi ðqÞ, can be interpreted similarly as CSM and CSV, except that the former two consider the whole output distribution while the latter only consider the first two moments of the output. If the plot of RIMPDFXi ðqÞ or RIMCDFXi ðqÞ is close to the diagonal, it means that the contribution of the input variable to the output PDF or CDF is nearly unvarying in its current range. If the plot is convex upwards, it means that in this range, the input variable contributes more than the average to the output PDF or CDF; otherwise, the contribution is lower than the average if the plot exhibits an opposite trend. Furthermore, some interesting properties of these two regional moment-independent importance measures are noticed and listed as follows. Property I. 0 6 RIMPDFXi ðqÞ 6 1 and 0 6 RIMCDFXi ðqÞ 6 1. Property II. RIMPDFXi ð0Þ ¼ RIMCDFXi ð0Þ ¼ 0 RIMPDFXi ð1Þ ¼ RIMCDFXi ð1Þ ¼ 1.

and

and

½xi;1 ;xi;2 

di

¼

1 2

Z

xi;2

Z

þ1

1

xi;1

 jfY ðyÞ  fYjXi ðyÞjdy fXi ðxi Þdxi

1 ¼ R xi;2 2 xi;1 fXi ðxi Þdxi

Z

xi;2

Z

þ1 1

xi;1

 jfY ðyÞ  fYjXi ðyÞjdy fXi ðxi Þdxi ð15Þ

and ðCDFÞ½xi;1 ;xi;2 

Si

¼ ¼

1 jEðYÞj

Z

xi;2 xi;1

Z

þ1 1

 jFY ðyÞ  FYjXi ðyÞjdy fXi ðxi Þdxi

1

jEðYÞj

R xi;2 xi;1

fXi ðxi Þdxi

Z

xi;2 xi;1

Z

þ1 1

 jFY ðyÞ  FYjXi ðyÞjdy fXi ðxi Þdxi ð16Þ

Meanwhile, the following equations hold by Eqs. (10) and (11): RIMPDFXi ðq2 Þ  RIMPDFXi ðq1 Þ  Z F1 ðq2 Þ Z þ1 Xi 1 ¼ jfY ðyÞ  fYjXi ðyÞjdy fXi ðxi Þdxi 2di F1 1 ðq1 Þ X

ð17Þ

i

Property III. For any given quantiles q1 and q2 , which satisfy the condition 0 < q1 < q2 < 1, the following relationships can 1 be obtained by noting that F1 Xi ðq1 Þ ¼ xi;1 and FXi ðq2 Þ ¼ xi;2 : ½q1 ;q2 

di

di

¼

RIMPDFXi ðq2 Þ  RIMPDFXi ðq1 Þ q2  q1

ð12Þ

and RIMCDFXi ðq2 Þ  RIMCDFXi ðq1 Þ  Z F1 ðq2 Þ Z þ1 Xi 1 ¼ ðCDFÞ jfY ðyÞ  fYjXi ðyÞjdy fXi ðxi Þdxi jEðYÞj F1 Si 1 ðq1 Þ Xi ð18Þ

Regional moment-independent sensitivity analysis Remember that F1 F1 Xi ðq1 Þ ¼ xi;1 , Xi ðq2 Þ ¼ xi;2 , R xi;2 f ðx Þdx ¼ q  q , so we can further obtain i i 2 1 xi;1 Xi RIMPDFXi ðq2 Þ  RIMPDFXi ðq1 Þ Z xi;2 ½x ;x  ½x ;x  d i;1 i;2 d i;1 i;2 ¼ fXi ðxi Þdxi i ¼ ðq2  q1 Þ i di di xi;1

1035 and

ð19Þ

and RIMCDFXi ðq2 Þ  RIMCDFXi ðq1 Þ Z xi;2 ðCDFÞ½xi;1 ;xi;2  ðCDFÞ½xi;1 ;xi;2  S S ¼ fXi ðxi Þdxi i ðCDFÞ ¼ ðq2  q1 Þ i ðCDFÞ Si Si xi;1

ð20Þ

Property III is readily obtained based on Eqs. (19) and (20). h 3.4. Computational strategies The moment-independent importance measures add necessary tools to the domain of sensitivity analysis for quantifying the contributions of inputs to the model output. Basically, it needs a double-loop sampling procedure to estimate di and ðCDFÞ 13,15,22 Si . Now we describe the computational method proposed by Borgonovo13 to obtain di . Firstly, the unconditional PDF of the output, fY ðyÞ, is obtained by uncertainty propagation, varying the inputs over their variation ranges. Secondly, one generates a value for Xi , namely x1i sampled by fXi ðxi Þ. With this value, the conditional PDF of the output, fYjXi ðyÞ, is obtained by propagating uncertainty in the model fixing Xi ¼ x1i . Then sðx1i Þ is computed from the numerical integration of the absolute value of the difference between fYjXi ðyÞ and fY ðyÞ. Repeat the procedure for a certain number of times, ðCDFÞ and di can be finally estimated. In fact, the calculation of Si can be implemented in a similar way, only needing to substitute the PDFs in Borgonovo’s method with CDFs. As we have noted before, the regional importance measures, RIMPDFXi ðqÞ and RIMCDFXi ðqÞ, can be obtained ðCDFÞ simultaneously when estimating di and Si , with the same sample points. Now we describe the generalized steps for estimating regional importance measures. Step 1. Generate N samples xt ¼ ½x1t ; x2t ; . . . ; xnt ðt ¼ 1; 2; . . . ; N Þ of the input variables randomly according to the joint distribution, and obtain the corresponding model output values y t . Step 2. Sort the samples of the ith input variable ð1Þ ð2Þ ðN Þ X i ðxi ; xi ; . . . ; xi Þ in an ascending order, and compute the corresponding values of sðX i Þ and AðX i Þ in the way we just described above, denoted as ðsðX i Þð1Þ ; sðX i Þð2Þ ; . . . ; sðX i ÞðN Þ Þ and ðAðX i Þð1Þ ; AðX i Þð2Þ ; . . . ; AðX i ÞðN Þ Þ. Step 3. Estimate the regional moment-independent measures for X i as PjqNj ð jÞ j¼1 sðXi Þ ð21Þ RIMPDFXi ðqÞ ¼ PN ð jÞ j¼1 sðXi Þ PjqNj j¼1

RIMCDFXi ðqÞ ¼ PN

AðXi Þð jÞ

j¼1 AðXi Þ

ð jÞ

where jqNj is the largest integer not greater than qN.

ð22Þ

A few comments need to be added here. In the second step, when computing the values of sðXi Þ and AðXi Þ with Xi fixed at one sample point, estimation of PDF and CDF is involved. In this work, a new adaptive Kernel Density Estimator (KDE) proposed by Botev et al.23 is adopted to estimate PDF and CDF. According to the steps described above, a total of nN2 times of model evaluations is needed to perform the regional moment-independent importance analysis. Obviously, the computational burden may become unbearable in some applications if the model evaluation is rather time-consuming. Now let us consider if there is any more efficient way to perform such analysis. Recently, Wei et al.24 has developed a sing-loop Monte Carlo method to estimate Borgonovo’s importance measure, di . This method proves to be rather efficient with acceptable accuracy. Noting that fYjXi ðyÞ ¼ fY;Xi ðy; xi Þ=fXi ðxi Þ, the following transformation is proposed by Wei et al.24:  Z Z   1  fY ðyÞ  fYjX ðyÞdy fX ðxi Þdxi di ¼ i i 2   ZZ   1  fY ðyÞ  fY;Xi ðy; xi ÞfX ðxi Þdxi dy ¼  fXi ðxi Þ  i 2 ZZ   1  fY ðyÞfX ðxi Þ  fY;X ðy; xi Þdxi dy ¼ ð23Þ i i 2  ZZ   1 fY ðyÞfXi ðxi Þ  1fY;X ðy; xi Þdxi dy ¼   i 2 fY;Xi ðy; xi Þ   f ðyÞfXi ðxi Þ  1  1 ¼ EY;Xi  Y 2 fY;Xi ðy; xi Þ where fY;Xi ðy; xi Þ is the joint PDF of the model output Y and the input variable Xi . By such a transformation, estimation of di turns to be a single-loop procedure, which will cut down the computational efforts remarkably. The robustness of this method has been demonstrated by Wei et al. 24 It can be seen that the final formulation in Eq. (23) can be easily calculated by the Monte Carlo simulation method except that the marginal PDF fY ðyÞ and the joint PDF fY;Xi ðy; xi Þ need to be estimated. ðCDFÞ Similarly, we can do a similar transformation to Si as ðCDFÞ

Si

Z 1 jEðYÞj Z 1 ¼ jEðYÞj Z 1 ¼ jEðYÞj Z 1 ¼ jEðYÞj ¼

Z

   FY ðyÞ  FYjX ðyÞdy fX ðxi Þdxi i

i

  Z Z y   dy fX ðxi Þdxi  ðf ðuÞ  f ðuÞÞdu Y YjX   i i 1   Z Z y   dy dxi  ðf ðuÞf ðx Þ  f ðuÞf ðx ÞÞdu i i Y X YjX X   i i i 1   Z Z y    ðfY ðuÞfXi ðxi Þ  fY;Xi ðu; xi ÞÞdudy dxi  1

ð24Þ ðCDFÞ

After fY ðyÞ and fY;Xi ðy; xi Þ are estimated, Si can be obtained by numerical integration in Eq. (24). It is admitted that in some cases, the numerical integration may be difficult to converge to produce a precise result. Nevertheless, the transformation in Eq. (24) provides an optional way to estimate ðCDFÞ Si in a single-loop procedure. In fact, the single-loop Monte Carlo method for di and ðCDFÞ Si can be extended to the regional moment-independent importance analysis. The general procedure is summarized as follows:

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Step 1. Generate N samples xt ¼ ½x1t ; x2t ; . . . ; xnt ðt ¼ 1; 2; . . . ; N Þ of the input variables randomly according to the joint distribution, and obtain the corresponding model output values y t . Step 2. Estimate f Y ðyÞ and f Y ;X i ðy; xi Þ using the KDE technique proposed by Botev et al.23 Sort the sample of X i in an ð1Þ ð2Þ ðN Þ ascending order as ðxi ; xi ; . . . ; xi Þ, and then the output values are correspondingly sorted as ðy ð1Þ ; y ð2Þ ; . . . ; y ðN Þ Þ. Calculate the corresponding values of the PDFs at these sample points: 8 ð jÞ ð jÞ > > < fY ¼ fY ðy Þ ð jÞ ð jÞ ð25Þ fY;Xi ¼ fY;Xi ðyð jÞ ; xi Þ j ¼ 1; 2; . . . ; N > > : ð jÞ ð jÞ fXi ¼ fXi ðxi Þ Step 3. For RIMPDF, it can be estimated as    PjqNj fYð jÞ fXð jÞi   1 j¼1  f ð jÞ    ð jÞY;XðijÞ RIMPDFXi ðqÞ ¼   PN fY fXi  j¼1  f ð jÞ  1 Y;X

ð26Þ

i

where jqNj is the largest integer not greater than qN. However, for RIMCDF, there needs a numerical integration process after fY ðyÞ and fY;Xi ðy; xi Þ are estimated. It can be formulated as Z F1 ðqÞ Z Z y Xi  1  ðfY ðuÞfXi ðxi Þ RIMCDFXi ðqÞ ¼  jEðYÞj 1  1  ð27Þ  fY;Xi ðu; xi ÞÞdudy dxi where EðYÞ can be estimated using the samples in Step 1. By the procedure described above, the computational burden of regional moment-independent importance is partly eased by a single-loop simulation process. It should be noted that estimation of RIMCDF may become difficult in some cases as numerical integration is involved, and the double-loop method can be used in such cases. Estimation of distribution-based sensitivity indices have long been a tricky issue, and many methods have been studied to address this issue.25–27 Nevertheless, the emphasis of this paper is to introduce a novel idea of regional momentindependent importance analysis, and more comprehensive discussions concerning the computational strategies will be made in the upcoming research work. 4. Applications In this section, the regional moment-independent importance analysis is tested by two engineering examples. The results will

Fig. 2

be discussed in a detailed manner, to demonstrate the significance of the newly proposed concepts of RIMPDF and RIMCDF. 4.1. Example 1: A riveting process In the aircraft industry, sheet metal parts are widely used, and riveting is the most common method to assemble them.28,29 Many factors have to be considered during the riveting process, which may directly affect the riveting quality. One main factor is the squeeze force. If the squeeze force is too low, rivets may be too loose to efficiently join different parts, while a too high squeeze force may induce excessive residual stresses which will result in stress concentrations (around holes and rivets) and initial cracks. Both cases are dangerous for aircraft. Therefore, it is quite necessary to determine the squeeze force properly for the sake of flight safety. The real riveting process is very complex. In this work, we take a headless rivet as an example, and generalize the riveting process into two stages as shown in Fig. 2. In Stage 1, the rivet is punched from State A (the initial state of the rivet before impact, without any deformation) to State B (an intermediate state of the rivet when there is no clearance between the rivet and the hole), and then in Stage 2, the rivet is further punched from State B to State C (the final state of the rivet after impact, with rivet heads formed). Throughout the riveting process, we assume that the hole diameter keeps unchanged. To establish the relationship between the geometric dimensions of a rivet and the squeeze force, the following ideal conditions should be assumed: (1) The hole is not enlarged in the riveting process. (2) The change of the rivet volume in the process can be neglected. (3) After impact, the rivet driven head has a cylindrical shape. (4) The material of the rivet is isotropic. The initial volume V0 of the rivet before the impact (in State A) is obtained as V0 ¼

p 2 Dh 4

ð28Þ

where D and h are the rivet diameter and length in State A, respectively. At the end of Stage 1, in State B, the volume of the rivet, V1, can be obtained as

A simplified riveting process.

Regional moment-independent sensitivity analysis p V1 ¼ D20 h1 4

1037 ð29Þ

where D0 and h1 are the rivet diameter and length in State B, respectively. After Stage 2, we assume the top and bottom heads of the formed rivet is in State C with the same dimension, and then the volume of the rivet in State C, V2, can be obtained as p p ð30Þ V2 ¼ D20 ts þ 2  D21 H 4 4 where ts is the whole thickness of the two sheets, and D1 and H are the diameter and height of a rivet head in State C, respectively. The maximum squeeze force needed in the riveting process can be expressed as p Fmax ¼ D21 rmax ð31Þ 4 where rmax is the maximum squeeze stress of a rivet head in its formation. According to power hardening theory, the true maximum squeeze stress in the y-direction can be obtained as rmax ¼ Kðey ÞnSHE

ð32Þ

where K is the strength coefficient, nSHE is the strain hardening exponent of the rivet material, and ey is the true strain in the ydirection of a rivet head in its formation. In this model, the true strain ey is composed of two parts: the strain ey1 caused in Stage 1 and the strain ey2 caused in Stage 2, and then the true strain ey can be expressed as ey ¼ ey1 þ ey2

ð33Þ

where ey1 ¼ lnðh=h1 Þ and ey2 ¼ ln½ðh1  ts Þ=2H. Combining Eqs. (28)–(33), under the ideal assumed conditions, one can obtain the maximum squeeze force needed for a certain riveting process as  n p D2 h  D20 ts D2 h  D20 ts SHE K ln ð34Þ Fmax ¼  4 2H 2HD2

Furthermore, as we have declared in Section 3.4, the singleloop method is obviously more efficient in terms of the number of model evaluations. Now, let us study the contributions of the inputs to the output uncertainty. It is noticed, in this example, that the importance rankings of the inputs based on di and ðCDFÞ Si are the same. This phenomenon seems rather reasonable. If one input has an important effect on the PDF of the model output, then there is no obvious improperness to anticipate that this input may as well greatly affect the output CDF, since PDF and CDF are closely related with each other. However, we should point out that this phenomenon is just possible rather than universal, as no theoretical evidence has been found to justify it. The importance measure results in Table 2 reflect that h is the most important, followed by K, D, ts, and D0, no matter whether the PDF or CDF of the output is considered. Thus, we can say that h is the most influential input on the whole output distribution while D0 is the least influential. In fact, this should be all the information we can obtain by the traditional importance analysis. However, by conducting the regional moment-independent importance analysis, more in-depth information can be extracted. In Fig. 3, the CSV plots as well as the RIMPDF and RIMCDF plots for the inputs D and h are illustrated. It can be seen that both the RIMPDF and RIMCDF plots exhibit differences from the CSV plot, which means that the former two can provide information from a different aspect from the latter to consider the contributions of inputs. For the input D, the CSV plot nearly coincides with the diagonal, and this phenomenon indicates that the contribution of this input to the output variance is almost constant through the whole distribution range. For the input h, its contribution to the output variance turns out to be uneven through its distribution range. The CSV plot is above the diagonal in the left tail of h, which means that its contribution to the output variance is higher than the average, and similarly, its contribution is lower than the average in the right tail as the CSV plot goes below the

In this work, the material of the rivet is 2017–T4, and we already know nSHE ¼ 0:15, H ¼ 2:2 mm, while the other inputs are all assumed as random variables, of which the distribution parameters are listed in Table 1. As the riveting quality is of significance to the safety of an aircraft, it is necessary and meaningful to find out the contributions of the inputs to the output uncertainty from different aspects. Firstly, we obtain the moment-independent importance measures by the single-loop and double-loop methods described in Section 3.4, and list the results in Table 2. Table 2 indicates that the results of the single-loop method match well with those of the traditional double-loop method.

Table 2 Input

D h D0 ts K

Table 1 Distribution information of inputs for riveting example. Input

Mean

Standard deviation

Distribution

D h D0 ts K

5 mm 20 mm 5.1 mm 5 mm 547.2 MPa

0.01 mm 0.3 mm 0.0102 mm 0.05 mm 5.472 MPa

Normal Normal Normal Normal Normal

Importance measures of inputs for riveting example. ðCDFÞ

di

Si

Double-loop

Single-loop

Double-loop

Single-loop

0.069 (3) 0.48 (1) 0.019 (5) 0.048 (4) 0.13 (2)

0.068 0.40 0.019 0.048 0.13

0.0047 (3) 0.021 (1) 0.0012 (5) 0.0031 (4) 0.0081 (2)

0.0044 0.021 0.0013 0.0031 0.0082

Note: The numbers in the brackets denote the rankings of the inputs.

1038

C. ZHOU et al.

Fig. 3

CSV, RIMPDF, and RIMCDF plots of inputs D and h in riveting example.

diagonal in this region. Now let us compare the CSV plot with the RIMPDF and RIMCDF plots. The RIMPDF plots of different inputs show a similar trend with each other, so do the RIMCDF plots. Besides, the RIMPDF and RIMCDF plots indicate that both inputs have uneven contributions to the output PDF or CDF at different regions, while only the input h shows this property for the CSV plot. This again demonstrates that different conclusions may be drawn depending on whether the output uncertainty is measured by variance or PDF/CDF. Similar discussions can be applied for the other inputs. According to Property III shown in Section 3.3, we define two ratio functions for RIMPDF and RIMCDF as Hi ðq1 ; q2 Þ ¼

½q1 ;q2 

di

di

¼

RIMPDFXi ðq2 Þ  RIMPDFXi ðq1 Þ q2  q1

ð35Þ

and Ki ðq1 ; q2 Þ ¼

ðCDFÞ½q1 ;q2 

Si

ðCDFÞ

Si

¼

RIMCDFXi ðq2 Þ  RIMCDFXi ðq1 Þ q2  q1 ð36Þ

The ratio functions, Hi ðq1 ; q2 Þ and Ki ðq1 ; q2 Þ, can reflect the reductions of the two moment-independent importance measures by reducing the range of Xi from ð1; þ1Þ to 1 ½xi;1 ; xi;2  (noting that F1 Xi ðq1 Þ ¼ xi;1 and FXi ðq2 Þ ¼ xi;2 ). Take the most important input, h, for example. The ratio functions of RIMPDF and RIMCDF of h are shown in Fig. 4. In the 3D plots, the lower the values of Hh ðq1 ; q2 Þ and Kh ðq1 ; q2 Þ are, the more reductions of the contribution of h to the output distribution will be obtained by reducing its range. Similar discussions can be performed on the other inputs, and they are not listed here to save space. Despite of the difference between the absolute values in Figs. 4(a) and (b), the two plots exhibit a similar trend, which is possibly due to the fact that PDF is highly related to CDF. In Ref.,20 Tarantola et al. derived several interesting observations for the 3D plots of variance ratio functions. In a similar manner, the ratio functions of RIMPDF and RIMCDF can be interpreted as follows: (1) The contribution to the PDF or CDF of the output can be estimated for any region ½q1 ; q2  of an input. (2) The counter-diagonal line of the 3D plot of H i ðq1 ; q2 Þ or K i ðq1 ; q2 Þ represents the ratio function by the symmetric

Fig. 4

Ratio functions of RIMPDF and RIMCDF for input h.

reduction of the range from both sides of the range, i.e., from the quantile interval ½0; 1 to ½q; 1  q with 0 6 q 6 0:5. The one-dimensional ratio functions of RIMPDF and RIMCDF for all the inputs are shown in Fig. 5 (3) The diagonal lines of the 3D plots represent the situation q1 ¼ q2 ¼ q or the derivatives of RIMPDF and RIMCDF, i.e., H i ðq; qÞ ¼ dðRIMPDFX i ðqÞÞ=dq and K i ðq; qÞ ¼ dðRIMCDFX i ðqÞÞ=dq. It measures the effects on the moment-independent importance measures by fixing an input at quantile q. The functions H i ðq; qÞ and K i ðq; qÞ for all the inputs are illustrated in Fig. 6.

Regional moment-independent sensitivity analysis

Fig. 5 Ratio functions of RIMPDF and RIMCDF by symmetrically reducing full quantile range to [q, 1  q] in riveting example.

In Fig. 5, conclusions can be drawn that the contributions of the inputs to the output PDF and CDF decrease with the symmetric reduction of the input ranges. Quantitative information is also available. In Fig. 5(a), it can be found that Hh ð0:2; 1  0:2Þ ¼ 0:8, which indicates that by reducing the distribution range of h from the quantile interval ½0; 1 to ½0:2; 0:8, the contribution of h to the output PDF can be reduced by 20%. Meanwhile, in Fig. 5(b), we see Kh ð0:2; 1  0:2Þ ¼ 0:65, which indicates that the contribution of h to the output CDF can be reduced by 35% by reducing the range of h from the quantile interval ½0; 1 to ½0:2; 0:8. The contribution can be further reduced if continuing to reduce the range of h. Similar studies can be conducted on the other inputs. Furthermore, Fig. 5 both show that for each input, the tail area is more concerned with the output uncertainty than the center area, since the one-dimensional plots change more steeply in the tail area. Fig. 6 can offer information about how much the contribution of an input to the output uncertainty can be changed by fixing the input at different values of q. As declared above, the one-dimensional plots in Fig. 6 are derivatives of the corresponding RIMPDF and RIMCDF. Taking the input h for example, in Fig. 6(a), it shows that Hh ð0:5; 0:5Þ ¼ 0:75, which indicates that its contribution to the output PDF can be reduced by as much as 25% if we fix h at its median value. An interesting phenomenon is noticed here, i.e., its contribution to the output uncertainty tends to increase when fixing an input near the tail area. These conclusions are rather informative for analysts when uncertainty of the output needs to be controlled.

1039

Fig. 6 Ratio functions of RIMPDF and RIMCDF by fixing input at quantile q in riveting example.

4.2. Example 2: A ten-bar structure Consider a ten-bar structure with 15 input variables, which is depicted in Fig. 7. The length and sectional area of horizontal and vertical bars are denoted as L and Ai ði ¼ 1; 2; . . . ; 6Þ, while pffiffiffi the length and sectional area of diagonal bars are 2L and Ai ði ¼ 7; 8; 9; 10Þ. The elastic modulus of all bars is denoted as E. P1 ,P2 ; and P3 are the external loads. In this problem, the 15 input variables are denoted as X ¼ ½Ai ði ¼ 1; 2; . . . ; 10Þ; L; P1 ; P2 ; P3 ; E, with the distribution information listed in Table 3. Taking the perpendicular displacement of Node 2, d, not exceeding 0.004 m as the constraint condition, the model can be established as Y ¼ gðXÞ ¼ 0:004  jdj

ð37Þ

where d is an implicit function of the inputs. The model in this example needs to be evaluated by calling ANSYS codes, and thus the computation process will be time-

Fig. 7

Diagram of a ten-bar structure.

1040 Table 3 example.

C. ZHOU et al. Distribution information of inputs for ten-bar

Variable

Mean

Standard deviation

Distribution

Ai L P1 P2 P3 E

1  103 m2 1m 80 kN 10 kN 10 kN 100 GPa

5  105 m2 0.05 m 4 kN 0.5 kN 0.5 kN 5 GPa

Normal Normal Normal Normal Normal Normal

consuming if the double-loop method is used. The results of the two moment-independent importance measures, di and ðCDFÞ Si , are obtained by the single-loop method and presented in Fig. 8. The results in Fig. 8 can reflect the contributions of the inputs to the output uncertainty from the viewpoint of probabilistic distribution. The inputs can be ranked according to the importance measures in a descending order as L, E, P1, P2, A1, A3, A7, A8, P3, A9, A5, A10, A4, A6, and A2. In fact, the contributions by A2, A4, A5, A6, and A10 are so tiny that they can almost be neglected compared to the rest. The importance analysis can thus provide helpful advice for the improvement of the model, and dig out some underlying information associated with the inputs. We now take a further step to perform regional importance analysis on the three most important inputs, L, E, and P1. As in this example, the importance ranking based on di is the same ðCDFÞ , we will only study the RIMPDF of the as that based on Si three inputs. The RIMPDF plots for the inputs L, E, and P1 are presented in Fig. 9, where the plots show a similar trend. Taking the input L for example, its contribution to the output PDF turns out to be uneven through its distribution range. The RIMPDF plot is above the diagonal in the left tail of L, which means that its contribution to the output PDF is higher than the average, and its contribution is lower than the average in the right tail as the plot goes below the diagonal in this region. Similar conclusions can be drawn for the inputs E and P1. As pointed out in the last example, the ratio functions, Hi ðq1 ; q2 Þ and Ki ðq1 ; q2 Þ, can represent the ratios of the moment-independent importance measures before and after the input range is reduced from the quantile interval ½0; 1 to ½q1 ; q2 . Here, Hi ðq1 ; q2 Þ is plotted for the top three important inputs, L, E, and P1, in the formulation of the 3D plots by Fig. 10. It is noticed that the 3D plots are similar to each other, which indicates that the information obtained by performing

Fig. 8

Importance measure results for ten-bar example.

the regional importance analysis for one input possibly holds true for another. Again, in the 3D plots, the lower the value of Hi ðq1 ; q2 Þ is, the more reduction of input contribution is obtained by reducing the input range. The counter-diagonal lines corresponding to the 3D plots in Fig. 10 are plotted in Fig. 11. These lines reflect the reductions

Fig. 9

RIMPDF plots of inputs L, P1, and E in ten-bar example.

Fig. 10

Ratio functions of RIMPDF for inputs L, P1, E.

Regional moment-independent sensitivity analysis

1041 the output uncertainty, as PDF and CDF are more sufficient to describe the uncertainty than solely depending on the mean or variance. Meanwhile, RIMPDF and RIMCDF can be obtained with the same samples used to estimate di and ðCDFÞ Si , and thus they can be viewed as byproducts of the standard moment-independent importance analysis, without a need of extra model evaluations. The idea of RIMPDF and RIMCDF provides not a substitution but a viable supplement to sensitivity analysis. Discussions on applying RIMPDF and RIMCDF in two engineering cases demonstrate that the regional moment-independent importance analysis can add more information concerning the contributions of model inputs.

Fig. 11 Ratio function of RIMPDF by symmetrically reducing full quantile range to [q, 1  q] in ten-bar example.

Acknowledgements This study was supported by the National Natural Science Foundation of China (No. NSFC51608446) and the Fundamental Research Fund for Central Universities of China (No. 3102016ZY015). References

Fig. 12 Ratio function of RIMPDF by fixing an input at quantile q in ten-bar example.

of the contributions of the inputs L, E, and P1 to the output PDF by symmetrically reducing the ranges from both sides of the ranges. It shows that the counter-diagonal line of the input L almost coincides with that of E. Taking the input P1 for example, we see HP1 ð0:25; 1  0:25Þ ¼ 0:5, which indicates that by reducing the distribution range of P1 from the quantile interval ½0; 1 to ½0:25; 0:75, the contribution of P1 to the output PDF can be reduced by as much as 50%. Meanwhile, the diagonal lines corresponding to the 3D plots in Fig. 10 are plotted in Fig. 12, which can reflect how much the contribution of an input to the output uncertainty can be changed by fixing the input at different values of q. As declared in last example, the diagonal line can be mathematically interpreted as the derivative of the RIMPDF. 5. Conclusions Enlightened by CSM and CSV, the idea of regional analysis is extended to the moment-independent importance analysis in this work. Two new definitions, RIMPDF and RIMCDF, aiming to evaluate the contributions of specific regions of an input to the output PDF and CDF, are introduced. The properties are discussed, as well as the corresponding computational strategies. By performing the regional moment-independent importance analysis, information concerning how the regions inside the inputs affect the whole output distribution can be obtained. Such information is helpful for analysts to control

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