*6.2. New Quantile-Oriented Sensitivity Indices for Small and Large Quantiles: QE Indices*

In Equation (33) to (35), replacing the absolute values with squares (*Q*(*Y*) <sup>−</sup> *<sup>E</sup>*(*Y*))<sup>2</sup> , (*Q*(*Y*|*X<sup>i</sup>* ) − *E*(*Y*|*X<sup>i</sup>* ))2 , etc., leads to new sensitivity indices, which we denote as QE indices. The new first-order quantile-oriented index is defined as *Symmetry* **2020**, *12*, x FOR PEER REVIEW 15 of 20

$$K\_i = \frac{(Q(Y) - E(Y))^2 - E\left((Q(Y|X\_i) - E(Y|X\_i))^2\right)}{\left(Q(Y) - E(Y)\right)^2}.\tag{36}$$

The new second-order QE index is defined as quantile-oriented index is defined as

The new second-order QE index is defined as 
$$\mathbf{K}\_{ij} = \frac{(Q(Y) - E(Y))^2 - E\left(\left(Q\{\mathbf{Y}|\mathbf{X}\_i, \mathbf{X}\_j\} - E\{\mathbf{Y}|\mathbf{X}\_i, \mathbf{X}\_j\}\right)^2\right)}{\left(Q(Y) - E(Y)\right)^2} - \mathbf{K}\_i - \mathbf{K}\_j. \tag{37}$$
 The new third-order QE index is defined as 
$$\mathbf{K}\_{ij} = \frac{\sum\_{i,j} \mathbf{K}\_{ij}^2 \mathbf{K}\_{ij}^2 - \sum\_{i,j} \mathbf{K}\_{ij}^2 \mathbf{K}\_{ij}^2}{\sigma\_{ij}}$$

*ji ji*

, ,

2 2

The new third-order QE index is defined as *XXYEXXYQEYEYQ* 2 2

$$\text{The new third-order QE index is defined as}$$

$$K\_{ijk} = 1 - \frac{\mathbb{E}\left(\left(Q\{\mathbf{Y}|\mathbf{X}\_{i}, \mathbf{X}\_{j}, \mathbf{X}\_{k}\right) - \mathbb{E}\left(\mathbf{Y}|\mathbf{X}\_{i}, \mathbf{X}\_{j}, \mathbf{X}\_{k}\right)\right)^{2}}{\left(Q(\mathbf{Y}) - \mathbb{E}(\mathbf{Y})\right)^{2}} - K\_{i} - K\_{j} - K\_{k} - K\_{ij} - K\_{ik} - K\_{jk}.\tag{38}$$

Sensitivity indices *K<sup>i</sup>* , *Kij*, and *Kijk* were formulated via analogies to Equations (33)–(35) and were tested by numerical experiments using linear and non-linear *Y* functions and LHS simulations. Only low and high quantiles can be studied. The sum of the indices of all orders was equal to one in all cases. The total index *KTi* can be formulated analogously to Equation (21). *kji jkikij kji kji ijk KKKKKK YEYQ XXXYEXXXYQE K* <sup>2</sup> ,, ,, 1 . (38) Sensitivity indices *Ki*, *Kij*, and *Kijk* were formulated via analogies to Equations (33)–(35) and were

The new sensitivity indices can be explained using an analogy to Sobol sensitivity indices. The classical Sobol's first-order sensitivity index has the form tested by numerical experiments using linear and non-linear *Y* functions and LHS simulations. Only low and high quantiles can be studied. The sum of the indices of all orders was equal to one in all cases. The total index *KTi* can be formulated analogously to Equation (21).

$$\mathbf{S}\_{\bar{i}} = \frac{V(\mathbf{Y}) - E(V(\mathbf{Y}|\mathbf{X}\_{\bar{i}}))}{V(\mathbf{Y})}.\tag{39}$$

 

Equation (36) can be interpreted using Equation (39). The key idea is to introduce *l* <sup>2</sup> as a variance. Equation (36) can be rewritten analogously to Equation (39) in the form *YV S i i* . (39) Equation (36) can be interpreted using Equation (39). The key idea is to introduce *l* <sup>2</sup> as a variance.

*XYVEYV*

$$\mathbf{K}\_{\bar{i}} = \frac{l^2 - E\left(\left(l\|\mathbf{X}\_i\right)^2\right)}{l^2},\tag{40}$$

where *l* is the standard deviation of the "artificial" two-point probability mass function (pmf) having left-right symmetry around quantile *Q*(*Y*) (see Figure 15). Half of the population is mirrored behind the quantile *Q*(*Y*) and replaced by a dot on each side of *Q*(*Y*). In SA, only low and high quantiles of *Y* can be analysed, indicating high *l* and low σ*<sup>Y</sup>* in unconditional and conditional pdfs. *l* where *l* is the standard deviation of the "artificial" two-point probability mass function (pmf) having left-right symmetry around quantile *Q*(*Y*) (see Figure 15). Half of the population is mirrored behind the quantile *Q*(*Y*) and replaced by a dot on each side of *Q*(*Y*). In SA, only low and high quantiles of

*i*

*Y* can be analysed, indicating high *l* and low *σ<sup>Y</sup>* in unconditional and conditional pdfs.

2

**Figure 15.** Introduction of *l* as the standard deviation of the two-point probability mass function. **Figure 15.** Introduction of *l* as the standard deviation of the two-point probability mass function.

*QT*2 = 0.50 and *QT*3 = 0.74, *QT*4 = 0.34, and *QT5* = 0.02, the order of importance of input variables can be determined as *F*1 and *F*2 and *fy, t*, and *b*. The sensitivity ranking based on all five *QTi* is *fy*, *F*1, *F*2, *t*, and

Let *μP* = −79.592 kN (*Pf* = 7.2 × 10−5). In the case study, QE indices were obtained on the load action

Let <sup>µ</sup>*<sup>P</sup>* <sup>=</sup> <sup>−</sup>79.592 kN (*P<sup>f</sup>* <sup>=</sup> 7.2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> ). In the case study, QE indices were obtained on the load action side as *K*<sup>1</sup> = 0.50, *K*<sup>2</sup> = 0.49, and *K*<sup>12</sup> = 0.01 and on the resistance side as *K*<sup>3</sup> = 0.65, *K*<sup>4</sup> = 0.25, *K*<sup>5</sup> = 0.01, *K*<sup>34</sup> = 0.08, *K*<sup>35</sup> = 0.00, *K*<sup>35</sup> = 0.00, and *K*<sup>345</sup> = 0.01 (see Figure 16). By computing the total indices *QT*<sup>1</sup> = 0.51, *QT*<sup>2</sup> = 0.50 and *QT*<sup>3</sup> = 0.74, *QT*<sup>4</sup> = 0.34, and *QT*<sup>5</sup> = 0.02, the order of importance of input variables can be determined as *F*<sup>1</sup> and *F*<sup>2</sup> and *fy, t*, and *b*. The sensitivity ranking based on all five *QTi* is *fy*, *F*1, *F*2, *t*, and *b*. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 16 of 20

**Figure 16.** Contrast *QE* indices: (**a**) 0.9963-quantile of *F* ad (**b**) 0.0036-quantile of *R*. **Figure 16.** Contrast *QE* indices: (**a**) 0.9963-quantile of *F* ad (**b**) 0.0036-quantile of *R*.

#### **7. Discussion 7. Discussion**

In the case study, input variables were listed in decreasing order of sensitivity as *fy*, *F*1, *F*2, *t,* and *b*. Although the values of sensitivity indices of the different ROSA types vary, each ROSA gives the same sensitivity ranking: In the case study, input variables were listed in decreasing order of sensitivity as *fy*, *F*1, *F*2, *t,* and *b*. Although the values of sensitivity indices of the different ROSA types vary, each ROSA gives the same sensitivity ranking:


the proposed concept cannot be used.

These results were obtained for *Pf* = 7.2 × 10−5 and the corresponding design quantiles (see previous sections). Contrast *Q* and *Pf* indices of higher-orders have a significant share in both types of ROSA; therefore, key information is provided by total indices. Regarding the sensitivity ranking, the total indices of design quantiles are a good proxy of the total indices of *Pf*. However, the result cannot be generalized beyond the Gaussian (or approximately Gaussian) design reliability These results were obtained for *<sup>P</sup><sup>f</sup>* <sup>=</sup> 7.2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> and the corresponding design quantiles (see previous sections). Contrast *Q* and *P<sup>f</sup>* indices of higher-orders have a significant share in both types of ROSA; therefore, key information is provided by total indices. Regarding the sensitivity ranking, the total indices of design quantiles are a good proxy of the total indices of *P<sup>f</sup>* . However, the result cannot be generalized beyond the Gaussian (or approximately Gaussian) design reliability conditions.

conditions. The proposed SA concept is applicable in tasks where the reliability can be assessed by comparing two *α*-quantiles of two statistically independent variables analogous to *R* and *F* (see Equation (2)). The pdfs of *R* and *F* should be close to Gaussian (see Equation (8)), with condition *σ<sup>F</sup>* ≈ *σR*. Then, ROSA can be effectively evaluated using the SA of design quantiles *Rd* and *Fd*, without having to analyse either *P<sup>f</sup>* or the interactions between *R* and *F*. This is advantageous because estimates of contrast *Q* indices are usually numerically easier than estimates of contrast *Pf* indices, The proposed SA concept is applicable in tasks where the reliability can be assessed by comparing two α-quantiles of two statistically independent variables analogous to *R* and *F* (see Equation (2)). The pdfs of *R* and *F* should be close to Gaussian (see Equation (8)), with condition σ*<sup>F</sup>* ≈ σ*R*. Then, ROSA can be effectively evaluated using the SA of design quantiles *R<sup>d</sup>* and *Fd*, without having to analyse either *P<sup>f</sup>* or the interactions between *R* and *F*. This is advantageous because estimates of contrast *Q* indices are usually numerically easier than estimates of contrast *P<sup>f</sup>* indices, especially for small values of *P<sup>f</sup>* .

especially for small values of *Pf*. For inequalities *σ<sup>F</sup>* ≠ *σR*, the total indices of design quantiles should be corrected using weights based on the sensitivity factors *αF* and *αR* from Equation (6). For example, if *σ<sup>F</sup>* → 0, then *α<sup>F</sup>* → 0 and *α<sup>R</sup>* → 1. When the influence of input variables on the load action side approaches zero, the reliability is only influenced by the variables on the resistance side. In the presented case study, the corrections of *QTi* indices are as follows: *αF*·*QT*1, *αF*·*QT*2, *αR*·*QT*3, *αR*·*QT*4, and *αR*·*QT*5. The correction of indices *KTi* can be performed similarly. If *σ<sup>F</sup>* = *σR*, corrections are not necessary because *α<sup>F</sup>* = *αF* = 0.7071. Initial studies have shown the rationality of this approach; however, further analysis is necessary. Corrections of indices *CTi* are not performed. If *σ<sup>F</sup>* → 0, then *CTi* of the variables on the load action side approaches zero naturally. If an extreme value distribution is used, such as a Gumbel or Weibull pdf [45,46], then For inequalities σ*<sup>F</sup>* , σ*R*, the total indices of design quantiles should be corrected using weights based on the sensitivity factors α*<sup>F</sup>* and α*<sup>R</sup>* from Equation (6). For example, if σ*<sup>F</sup>* → 0, then α*<sup>F</sup>* → 0 and α*<sup>R</sup>* → 1. When the influence of input variables on the load action side approaches zero, the reliability is only influenced by the variables on the resistance side. In the presented case study, the corrections of *QTi* indices are as follows: α*F*·*QT*1, α*F*·*QT*2, α*R*·*QT*3, α*R*·*QT*4, and α*R*·*QT*5. The correction of indices *KTi* can be performed similarly. If σ*<sup>F</sup>* = σ*R*, corrections are not necessary because α*<sup>F</sup>* = α*<sup>F</sup>* = 0.7071. Initial studies have shown the rationality of this approach; however, further analysis is necessary. Corrections of indices *CTi* are not performed. If σ*<sup>F</sup>* → 0, then *CTi* of the variables on the load action side approaches zero naturally. If an extreme value distribution is used, such as a Gumbel or Weibull pdf [45,46], then the proposed concept cannot be used.

functions (see Figure 4). Although contrast *Q* indices do not have an analogy to the variance decomposition offered by Sobol's indices through the Hoeffding theorem, studies of contrasts in applications [35,36] show some similarities between contrast *Q* indices and Sobol's indices. The new QE indices and Sobol's indices have formulas based on the squares of the distances from the average

value and therefore, their comparison may be interesting in further work.

Contrast *Q* indices are based on measuring the fluctuations around the quantile, which is the

Contrast *Q* indices are based on measuring the fluctuations around the quantile, which is the distance *l* between the average value of the population before and after the quantile (see Figure 3). For low and high quantiles, contrast *Q* indices can be rewritten using asymptotes *l* = ±θ\* of hyperbolic functions (see Figure 4). Although contrast *Q* indices do not have an analogy to the variance decomposition offered by Sobol's indices through the Hoeffding theorem, studies of contrasts in applications [35,36] show some similarities between contrast *Q* indices and Sobol's indices. The new QE indices and Sobol's indices have formulas based on the squares of the distances from the average value and therefore, their comparison may be interesting in further work.

It can be noted that QE indices *K<sup>i</sup>* , *Kij*, and *Kijk* give significant values of first-order indices *K<sup>i</sup>* (compared to *Q<sup>i</sup>* ) and relatively small values of higher-order indices, which is also a property observed in Sobol's indices in the case study [35]. QE indices are based on quadratic measures of sensitivity like Sobol, but associated with quantiles. This domain deserves much more work in order to make QE indices a useful and practical tool.

All of the presented techniques are appropriate for SA of the stochastic model type considered in this article. For a general model, an important criterion is also the ease with which the SA can be performed. The most fundamental aspect of sensitivity techniques is local SA based on partial derivatives for computing the rate of change in *P<sup>f</sup>* with respect to a given input parameter. Although the sensitivity ranking determined on the basis of *D<sup>i</sup>* is the same as from *CTi*, *QTi*, or *KTi*, this conclusion cannot be generalized, and *D<sup>i</sup>* is not suitable for application in every task. The one-at-a-time techniques are only valid for small variabilities in parameter values or linear computation models; otherwise, the partials must be recalculated for each change in the base-case scenario. In contrast, contrast-based SA does not have these limitations because computational models can generally be non-linear and sensitivity indices take into account the variability of inputs throughout their distribution range and provide interaction effects between different input variables.

The results of ROSA can be compared with traditional SA techniques, such as the correlation between input *X<sup>i</sup>* and output *Z*. Spearman's rank correlation coefficients are computed using one million LHS runs as corr(*X*1, Z) = −0.49, corr(*X*2, Z) = −0.48, corr(*X*3, Z) = 0.56, corr(*X*4, Z) = 0.38, and corr(*X*5, Z) = 0.08. The second traditional SA technique is SSA. Sobol's first-order indices *S<sup>i</sup>* are computed according to Equation (39), using double-nested-loop computation [35], whereas the inner loop has four million runs and the outer loop ten thousand runs. The model output is *Z*. The values of *S<sup>i</sup>* are *S*<sup>1</sup> = 0.25, *S*<sup>2</sup> = 0.24, *S*<sup>3</sup> = 0.34, *S*<sup>4</sup> = 0.16, and *S*<sup>5</sup> = 0.01. Sobol's higher-order sensitivity indices are negligible. Both the correlation and SSA give the same sensitivity ranking as ROSA: *fy*, *F*1, *F*2, *t*, and *b*. The case study shows that the normalization of the newly proposed indices *KTi* leads to the classical *S<sup>i</sup>* , i.e., *KTi*/2.11 ≈ *S<sup>i</sup>* . Although correlations and Sobol's indices are commonly used in SA of the limit states of structures, neither is directly reliability-oriented [19]. Further analysis of the relationship between the new QE indices and traditional Sobol indices is needed because it can provide new insights into the use of SSA in reliability tasks.

The dominance of the yield strength is an important finding for static tensile tests of steel specimens in the laboratory. In structural systems, the slender members under compression may be influenced by other initial imperfections, such as bow and out-of-plumb imperfections [9,10]. In a general steel structure, these imperfections can change the order of importance of the input random variables.

Symmetry is an important part of sensitivity indices and contrast functions (see, e.g., Equation (11) or Equation (24)). Reliability *P*~*<sup>f</sup>* = (1 − *P<sup>f</sup>* ) or unreliability *P<sup>f</sup>* leads to the same contrast *P<sup>f</sup>* indices, because *Pf* (1 − *P<sup>f</sup>* ) = *P*~*<sup>f</sup>* (1 − *P*~*<sup>f</sup>* ). In the case study, the plots of the sensitivity indices were slightly asymmetric due to the small values of skewness of *R*. The plots of sensitivity indices vs. *P<sup>f</sup>* would be perfectly symmetric in the case of a perfectly symmetric pdf of *R* and *F*, with zero skewness.

In the presented study, conclusions were made using SA subordinated to a contrast [25] and SA based on partial derivatives of *P<sup>f</sup>* and new types of QE indices. Other types of SA of *P<sup>f</sup>* like [47] or SA of the quantile [48] have not been studied. Numerous other types of sensitivity measures exist, such as [49–59], and it cannot be expected that the conclusions would be confirmed using any sensitivity index. The advantage of SA subordinated to a contrast is the use of a single platform (contrast) for the analysis of different parameters associated with a probability distribution.
