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.

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. 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. 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. 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. 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.