Credibility distribution function based global and regional sensitivity analysis under fuzzy uncertainty

Abstract

Global sensitivity analysis (GSA) is useful to recognize important inputs for assigning priority and unimportant inputs for simplifying models by exploring whole distribution ranges. Meanwhile, regional sensitivity analysis (RSA) is also studied for finding the contribution of the critical region of an input, which can be viewed as the complementary of GSA. However, there is a lack of GSA and RSA research in presence of fuzzy uncertainty. Thus, a new global sensitivity index (GSI) under the fuzzy uncertainty is devoted on the credibility distribution function (CrDF), a comprehensive distribution description under the fuzzy uncertainty. The CrDF-based GSI is defined by the fuzzy expectation of the difference between the CrDF and the conditional CrDF of the output on fixing the fuzzy input over its whole distribution range, which can quantify the contribution of the fuzzy input to the output CrDF. Then, a new fuzzy RSA technique, the contribution to this CrDF based index (shortened by CCI) plot, is also proposed, and it can assess the effect of given regions of important inputs on output CrDF. Besides, mathematical properties of the CrDF based GSI and the CCI plot are discussed, and their solution are established by use of the fuzzy simulation with the same set of samples. After the accuracy of established fuzzy simulation solution for the CrDF based GSI and the CCI plot are verified by an analytical example, other examples are used to demonstrate the reasonability and applicability of proposed CrDF based GSI and CCI plot under fuzzy uncertainty.

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References

  1. 1.

    Saltelli A, Ratto M, Andres T, Campolongo F (2008) Global sensitivity analysis: the primer. Wiley, Chichester

    Google Scholar 

  2. 2.

    Cheng K, Lu ZZ, Ling CY, Zhou ST (2020) Surrogated-assisted global sensitivity analysis: an overview. Struct Multidiscip Optim 61(3):1187–1213

    MathSciNet  Article  Google Scholar 

  3. 3.

    Sobol’ IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55:271–280

    MathSciNet  Article  Google Scholar 

  4. 4.

    Saltelli A (2002) Sensitivity analysis for importance assessment. Risk Anal 3(22):579–590

    Article  Google Scholar 

  5. 5.

    Kucherenko S, Song SF, Wang L (2019) Quantile based global sensitivity measures. Reliab Eng Syst Saf 185:35–48

    Article  Google Scholar 

  6. 6.

    Borgonovo E (2007) A new uncertainty importance measure. Reliab Eng Syst Saf 92(6):771–784

    Article  Google Scholar 

  7. 7.

    Liu Q, Homma T (2010) A new importance measure for sensitivity analysis. J Nucl Sci Technol 47(1):53–61

    Article  Google Scholar 

  8. 8.

    Cui LJ, Lu ZZ, Zhao XP (2010) Moment-independent importance measure of basic random variable and its probability density evolution solution. Sci China Technol Sci 53(4):1138–1145

    Article  Google Scholar 

  9. 9.

    Li LY, Lu ZZ, Feng J, Wang B (2012) Moment-independent importance measure of basic variable and its state dependent parameter solution. Struct Saf 38:40–47

    Article  Google Scholar 

  10. 10.

    Yun WY, Lu ZZ, Zhang Y, Jiang X (2018) An efficient global reliability sensitivity analysis algorithm based on classification of model output and subset simulation. Struct Saf 74:49–57

    Article  Google Scholar 

  11. 11.

    Kucherenko S, Rodriguez-Fernandez M, Pantelides C, Shah N (2009) Monte Carlo evaluation of derivative based global sensitivity measures. Reliab Eng Syst Saf 94(7):1135–1148

    Article  Google Scholar 

  12. 12.

    Song SF, Zhou T, Wang L, Kucherenko S, Lu ZZ (2019) Derivative-based new upper bound of Sobol’ sensitivity measure. Reliab Eng Syst Saf 187:142–148

    Article  Google Scholar 

  13. 13.

    Wei PF, Lu ZZ, Wu DQ, Zhou CC (2013) Moment-independent regional sensitivity analysis: application to an environmental model. Environ Model Softw 47:55–63

    Article  Google Scholar 

  14. 14.

    Tarantola S, Kopustinskas V, Boladolavin R, Kaliatka A, Uspuras E, Vaisnoras M (2012) Sensitivity analysis using contribution to sample variance plot: application to a water hammer model. Reliab Eng Syst Saf 99:62–73

    Article  Google Scholar 

  15. 15.

    Bolado-Lavin R, Castaings W, Tarantola S (2009) Contribution to the sample mean plot for graphical and numerical sensitivity analysis. Reliab Eng Syst Saf 94:1041–1049

    Article  Google Scholar 

  16. 16.

    Li LY, Lu ZZ (2013) Regional importance effect analysis of the input variables on failure probability. Comput Struct 125:74–85

    Article  Google Scholar 

  17. 17.

    Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1(1):3–28

    MathSciNet  Article  Google Scholar 

  18. 18.

    Dubois D, Prade H (1988) Possibility theory: an approach to computerized processing of uncertainty. Plenum Press, New York

    Google Scholar 

  19. 19.

    Wang L, Lu ZZ, Jia BX (2020) A decoupled method for credibility-based design optimization with fuzzy variables. Int J Fuzzy Syst 22:844–858

    MathSciNet  Article  Google Scholar 

  20. 20.

    Liu B (2002) Uncertainty theory, 2nd edn. Springer, Berlin

    Google Scholar 

  21. 21.

    Ling CY, Lu ZZ, Feng KX (2019) An efficient method combining adaptive Kriging and fuzzy simulation for estimating failure credibility. Aerosp Sci Technol 92:620–634. https://doi.org/10.1016/j.ast.2019.06.037

    Article  Google Scholar 

  22. 22.

    Gauger U, Turrin S, Hanss M (2008) A new uncertainty analysis for the transformation method. Fuzzy Sets Syst 159(11):1273–1291

    MathSciNet  Article  Google Scholar 

  23. 23.

    Song SF, Lu ZZ, Cui LJ (2012) A generalized Borgonovo’s importance measure for fuzzy input uncertainty. Fuzzy Sets Syst 189(1):53–62

    MathSciNet  Article  Google Scholar 

  24. 24.

    Shi Y, Lu ZZ, Zhou YC (2018) Global sensitivity analysis for fuzzy inputs based on the decomposition of fuzzy output entropy. Eng Optim 50(6):1078–1096

    MathSciNet  Article  Google Scholar 

  25. 25.

    Wang JQ, Lu ZZ, Shi Y (2018) Aircraft icing safety analysis method in presence of fuzzy inputs and fuzzy state. Aerosp Sci Technol 82–83:172–184. https://doi.org/10.1016/j.ast.2018.09.003

    Article  Google Scholar 

  26. 26.

    Liu B, Liu YK (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Trans Fuzzy Syst 10:445–450

    Article  Google Scholar 

  27. 27.

    Liu B (2012) Membership functions and operational law of uncertain sets. Fuzzy Optim Decis Making 11:387–410

    MathSciNet  Article  Google Scholar 

  28. 28.

    Medasani S, Kim J, Krishnapuram R (1998) An overview of membership function generation techniques for pattern recognition. Int J Approx Reason 19(3–4):391–417

    MathSciNet  Article  Google Scholar 

  29. 29.

    Wang C, Martthies HG, Qiu ZP (2017) Optimization-based inverse analysis for membership function identification in fuzzy steady-state heat transfer problem. Struct Multidiscip Optim 57(4):1495–1505

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The support by the National Natural Science Foundation of China (Project 52075442 and 11702281) and National Science and Technology Major Project (2017-IV-0009-0046) are gratefully acknowledged.

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Correspondence to Zhenzhou Lu.

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Appendices

Appendix 1: Regular membership function

A membership function \(\rho_X (x)\) is said to be regular if there exists a point \(x_0\) such that \(\rho_X (x_0 ) = 1\) and \(\rho_X (x)\) is unimodal about the mode \(x_0\) [27]. That is, \(\rho_X (x)\) is increasing on \(( - \infty , \, x_0 ]\) and decreasing on \([x_0 , + \infty )\). Commonly used regular fuzzy distributions and their characteristics are listed in Table

Table 6 Common regular fuzzy distributions and their characteristics

6.

Appendix 2: Independent conditions for fuzzy variables

The independent conditions for the fuzzy variables are different from these of the random variables. And for the sake of helping readers to have a better understand of the independence of the fuzzy variables, this subsection illustrates the independent conditions for the fuzzy variables.

  1. 1.

    The fuzzy variables \({\varvec{X = }}\left\{ {X_1 ,X_2 , \ldots ,X_n } \right\}^{\varvec{T}}\) are said to be independent if \({\text{Cr}}\left\{ {\bigcap\nolimits_{i = 1}^n {\{ x_i \in B_i \} } } \right\} = \mathop {\min }\nolimits_{1 \le i \le n} {\text{Cr}}\left\{ {x_i \in B_i } \right\}\) for any sets \(B_i \ \ (i = 1,2, \ldots ,n)\) of \(R\) (\(R\) is the real number space) (Definition 3.17 in Ref. [20]).

  2. 2.

    The fuzzy variables \({\varvec{X = }}\left\{ {X_1 ,X_2 , \ldots ,X_n } \right\}^{\varvec{T}}\) are said to be independent if and only if \({\text{Cr}}\left\{ {\bigcup\nolimits_{i = 1}^n {\{ x_i \in B_i \} } } \right\} = \mathop {\max }\nolimits_{1 \le i \le n} {\text{Cr}}\left\{ {x_i \in B_i } \right\}\) for any sets \(B_i \ \ (i = 1,2, \ldots ,n)\) of \(R\) (Theorem 3.23 in Ref. [20]).

  3. 3.

    The fuzzy variables \({\varvec{X = }}\left\{ {X_1 ,X_2 , \ldots ,X_n } \right\}^{\varvec{T}}\) are said to be independent if and only if \({\text{Cr}}\left\{ {\bigcap\nolimits_{i = 1}^n {\{ x_i = x^{\prime}_i \} } } \right\} = \mathop {\min }\nolimits_{1 \le i \le n} {\text{Cr}}\left\{ {x_i = x^{\prime}_i } \right\}\) for any real numbers \(x^{\prime}_i (i = 1,2, \ldots ,n)\) with \({\text{Cr}}\left\{ {\bigcap\nolimits_{i = 1}^n {\{ x_i = x^{\prime}_i \} } } \right\} < 0.5\) (Theorem 3.24 in Ref. [20]).

  4. 4.

    The fuzzy variables \({\varvec{X = }}\left\{ {X_1 ,X_2 , \ldots ,X_n } \right\}^{\varvec{T}}\) are said to be independent if and only if \(\rho_{{\varvec{X}}} (x_1 ,x_2 , \ldots ,x_n ) = \mathop {\min }\nolimits_{1 \le i \le n} \left\{ {\rho_{X_i } (x_i )} \right\}\) for any real numbers \(x_i \ (i = 1,2, \ldots ,n)\) (Theorem 3.25 in Ref [20].).

Appendix 3: Proof of non-decreasing property of the CCI

According to the basic theorem of credibility distribution function (CrDF) shown in Ref. [20], CrDF \(\tilde{F}_X (x)\) is an increasing function with \(\mathop {\lim }\nolimits_{x \to - \infty } \tilde{F}_X (x) \le 0.5 \le \mathop {\lim }\nolimits_{x \to + \infty } \tilde{F}_X (x)\). Correspondingly, the inverse CrDF \(\tilde{F}_X^{ - 1} (q)\) is also an increasing function based on the inverse function theorem. Since \(\tilde{F}_{X_i }^{ - 1} (q)\) increases with the increase of \(q\), the realization of regional indicator function \(I_q (x_i )\) is more likely to be 1 when q increases.

Because the denominator of \(CCI_i \left( q \right)\) is a constant when the membership function of \(X_i\) is given, only the dominator \(\tilde{E}_{X_i } (I_q (X_i )s(X_i ))\) is discussed here. Since \(s(X_i ) \ge 0\) always holds,, \(\tilde{E}_{X_i } (I_q (X_i )s(X_i ))\) can be expressed as Eq. (25) based on the definition of fuzzy expectation.

$$\begin{gathered} \tilde{E}_{X_i } (I_q (X_i )s(X_i )) = \int_0^{ + \infty } {Cr\{ I_q (X_i )s(X_i ) \ge r\} {\text{d}}r} \hfill \\ \;\;\;\;\;\;\; \approx \frac{1}{2}\;\int_0^{ + \infty } {\left( {\mathop {\max }\limits_{1 \le k \le N} \left\{ {\rho_{X_i } (x_i^{(k)} )\;\left| {I_q (x_i^{(k)} )s(x_i^{(k)} )\; \ge r} \right.} \right\} - \mathop {\max }\limits_{1 \le k \le N} \left\{ {\rho_{X_i } (x_i^{(k)} )\;\left| {I_q (x_i^{(k)} )s(x_i^{(k)} )\; < r} \right.} \right\} + 1} \right){\text{d}}r} \hfill \\ \end{gathered}$$
(25)

where \(\{ x_i^{(1)} ,x_i^{(2)} ,\ldots,x_i^{(N)} \}\) are samples of \(X_i\), which are uniformly generated in its membership interval.

Assume \(q_1 < q_2\), and corresponding regional indicator functions are \(I_{q_1 } (X_i )\) and \(I_{q_2 } (X_i )\) respectively. Then, the samples satisfying \(I_{q_1 } (x_i^{(k)} )s(x_i^{(k)} )\; \ge r\) are denoted by \({{\varvec{S}}}_1^{q_1 }\) and the samples satisfying \(I_{q_1 } (x_i^{(k)} )s(x_i^{(k)} )\; < r\) are denoted by \({{\varvec{S}}}_2^{q_1 }\). Similarly, the samples satisfying \(I_{q_2 } (x_i^{(k)} )s(x_i^{(k)} )\; \ge r\) are denoted by \({{\varvec{S}}}_1^{q_2 }\) and the samples satisfying \(I_{q_2 } (x_i^{(k)} )s(x_i^{(k)} )\; < r\) are denoted by \({{\varvec{S}}}_2^{q_2 }\). For an arbitrary value of \(r\), it is not hard to find that \({{\varvec{S}}}_1^{q_1 } \subset {{\varvec{S}}}_1^{q_2 }\) and \({{\varvec{S}}}_2^{q_2 } \subset {{\varvec{S}}}_2^{q_1 }\) because \(\sum_{k = 1}^N {I_{q_1 } (x_i^{(k)} )} \le \sum_{k = 1}^N {I_{q_2 } (x_i^{(k)} )}\). So \(\mathop {\max }\nolimits_{x_i \in {{\varvec{S}}}_1^{q_{_1 } } } \left\{ {\rho_{X_i } (x_i )\;} \right\} \le \mathop {\max }\nolimits_{x_i \in {{\varvec{S}}}_1^{q_{_2 } } } \left\{ {\rho_{X_i } (x_i )\;} \right\}\) and \(\mathop {\max }\nolimits_{x_i \in {{\varvec{S}}}_2^{q_{_1 } } } \left\{ {\rho_{X_i } (x_i )\;} \right\} \ge \mathop {\max }\nolimits_{x_i \in {{\varvec{S}}}_2^{q_{_2 } } } \left\{ {\rho_{X_i } (x_i )\;} \right\}\) always hold for any r. Finally, considering the property of integral in Eq. (25), \(\tilde{E}_{X_i } (I_{q_1 } (X_i )s(X_i )) \le \tilde{E}_{X_i } (I_{q_1 } (X_i )s(X_i ))\) and \(CCI_i \left( {q_1 } \right) \le CCI_i \left( {q_2 } \right)\) subsequently, which can prove \(CCI_i \left( q \right)\) is a strict non-decreasing function of q.

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Wang, L., Zhang, X., Li, G. et al. Credibility distribution function based global and regional sensitivity analysis under fuzzy uncertainty. Engineering with Computers (2021). https://doi.org/10.1007/s00366-020-01271-w

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Keywords

  • Global sensitivity analysis
  • Regional sensitivity analysis
  • Fuzzy inputs
  • Credibility theory
  • Fuzzy simulation