# From Probabilistic to Quantile-Oriented Sensitivity Analysis: New Indices of Design Quantiles

## Abstract

**:**

_{f}of a structure. Although the reliability is usually expressed via P

_{f}, Eurocode building design standards assess the reliability using design quantiles of resistance and load. The presented case study showed that quantile-oriented SA can provide the same sensitivity ranking as P

_{f}-oriented SA or local SA based on P

_{f}derivatives. The first two SAs are global, so the input variables are ranked based on total sensitivity indices subordinated to contrasts. The presented studies were performed for P

_{f}ranging from 9.35 × 10

^{−8}to 1–1.51 × 10

^{−8}. The use of quantile-oriented global SA can be significant in engineering tasks, especially for very small P

_{f}. The proposed concept provided an opportunity to go much further. Left-right symmetry of contrast functions and sensitivity indices were observed. The article presents a new view of contrasts associated with quantiles as the distance between the average value of the population before and after the quantile. This distance has symmetric hyperbola asymptotes for small and large quantiles of any probability distribution. Following this idea, new quantile-oriented sensitivity indices based on measuring the distance between a quantile and the average value of the model output are formulated in this article.

## 1. Introduction

_{f}, which is estimated using stochastic models [2]. Failure occurs when the load action is greater than the resistance. In this respect, the key issue is the identification of the significance of input random variables with regard to P

_{f}.

_{f}. It is argued that sensitivity analysis (SA) should be used “in tandem” with uncertainty analysis and the latter should precede the former in practical applications [3]. This can encumber the entire computational process, especially in cases of very small P

_{f}.

_{f}can be replaced by a reliability assessment based on design quantiles, can the SA of P

_{f}be replaced by the SA of design quantiles? For this purpose, new types of sensitivity indices oriented to both design quantiles and P

_{f}can be investigated in engineering applications.

_{f}and conditional P

_{f}defined in [22]. Indices can be derived in different variants, depending on whether the square of the importance measure [20] or the absolute value of the importance measure [23,24] is considered, but only the variant [20] after Sobol is based on decomposition, with the sum of all indices equal to one.

_{f}(referred to as contrast P

_{f}indices in this article). Furthermore, it can be shown that the classical Sobol indices [6,7] are Fort contrast indices [25] associated with variance. In general, the type of Fort contrast index [25] varies, according to the type of contrast used. Contrast functions permit the estimation of various parameters associated with a probability distribution. By changing the contrast, SA can change its key quantity of interest. The contrast may or may not be reliability-oriented.

_{f}, α-quantile, variance, etc.) with regard to the variability of the inputs over their entire distribution ranges and they provide the interaction effect between different input variables. On the other hand, contrast functions account for the variability of the inputs regionally, according to the type of key quantity of interest, e.g., changes around the mean value are important for variance, changes around the quantile are important for the quantile, etc.

## 2. Probability-Based Assessment of Structural Reliability

_{1}, X

_{2}, …, X

_{M}are random variables employed for its computation. The classical theory of structural reliability [27] expresses Equation (1) as a limit state using two statistically independent random variables, the load effect (action F), and the load-carrying capacity of the structure (resistance R).

_{Z}is the mean value of Z and σ

_{Z}is its standard deviation. By modifying Equation (3), we can express μ

_{Z}−β·σ

_{Z}= 0. The failure probability P

_{f}can then be expressed as

_{U}(·) is the cumulative distribution function of the normalized Gaussian probability density function (pdf). Reliability is defined as P

_{s}= (1 − P

_{f}). For other distributions of Z, β is merely a conventional measure of reliability. Equation (3) can be modified for normally distributed Z, F, and R as

_{F}and α

_{R}are values of the first-order reliability method (FORM) sensitivity factors.

_{F}and α

_{R}: S

_{F}= ${\alpha}_{F}^{2}$ and S

_{R}= ${\alpha}_{R}^{2}$, respectively [19]. By applying α

_{F}and α

_{R}according to Equation (6), Equation (5) can be written with formally separated random variables as

_{F}, σ

_{F}, μ

_{R}, and σ

_{R}, from which β, α

_{F}, and α

_{R}are computed. The left side in Equation (7) is the design load F

_{d}(upper quantile) and the right side is the design resistance R

_{d}(lower quantile).

_{d}, according to the equation β ≥ β

_{d}, which transforms Equation (7) into the design condition of reliability:

_{F}and α

_{R}may be considered as 0.7 and 0.8, respectively [4].

## 3. Sensitivity Analysis

_{f}and the design quantiles F

_{d}and R

_{d}. In order to analyse the reliability, ROSA must be focused on the same key quantity of interest: P

_{f}, F

_{d}, and R

_{d}. Local and global types of ROSA are applied in this article.

#### 3.1. Local ROSA

_{f}/δμ

_{xi}with respect to the mean value μ of the input variable X

_{i}presents a classical measure of change in P

_{f}(see, e.g., [28,29,30,31,32]). The derivative-based approach has the advantage of being very efficient in terms of the computation time. There are two main disadvantages of using the derivative as an indicator of sensitivity.

_{Xi}of the input variable.

_{Xi}and the possibility of introducing a correlation between the input random variables. A limitation of the derivative-based approach occurs when the analysed variable is of an unknown linearity.

_{1}+ X

_{2}, the derivative of the quantile with respect to the mean value is always equal to one. Conversely, in non-additive models, the derivative of the quantile with respect to the mean value may give very high or low values, and thus, the derivative of the quantile does not appear to be a useful measure of sensitivity.

#### 3.2. Global ROSA

_{f}[33,34], the design quantiles F

_{d}and R

_{d}[35], or the median [36].

_{f}indices) are based on quadratic-type contrast functions [25]. However, contrast P

_{f}indices can be defined more easily based on the probability of failure and the conditional probabilities of failure [19]. A formula that does not require the evaluation of contrast functions can be used for practical computation. For practical use, the first-order probability contrast index C

_{i}can be rewritten in the form of [19]

_{i}measures, on average, the effect of fixing X

_{i}on P

_{f}, where P

_{f}= P(Z < 0) is the failure probability and P

_{f}|X

_{i}= P((Z|X

_{i}) < 0) is the conditional failure probability. The mean value E[·] is taken over X

_{i}. In Equation (10), the term P

_{f}(1 − P

_{f}) is derived for probability estimator θ* = Argmin ψ(θ) = P

_{f}from the minimum of contrast $\underset{\theta}{\mathrm{min}}\mathsf{\psi}\left(\theta \right)$:

_{Z<0}) is the variance in the case where there are only two outcomes of 0 and 1, with one having a probability of P

_{f}. The largest variance occurs if P

_{f}= 0.5, with each outcome given an equal chance. The contrast function ψ(θ) = E(1

_{Z<0}− θ)

^{2}vs. θ is convex and symmetrical in the interval across the vertical axis θ*. The plot of P

_{f}(1 − P

_{f}) vs. P

_{f}is a concave function with left-right symmetry. The contrast for conditional probability is expressed in a similar manner as (P

_{f}|X

_{i})(1 − P

_{f}|X

_{i}).

_{ij}is computed similarly:

_{f}|X

_{i},X

_{j}= P((Z|X

_{i},X

_{j}) < 0) is the conditional failure probability for fixed X

_{i}and X

_{j}. E[·] is taken over X

_{i}and X

_{j}. The index C

_{ij}measures the joint effect of X

_{i}and X

_{j}on P

_{f}minus the first-order effects of the same factors. The third-order sensitivity index C

_{ijk}is computed similarly:

_{f}|X

_{i},X

_{j},X

_{k}= P((Z|X

_{i},X

_{j},X

_{k}) < 0) is the conditional failure probability for fixed triples X

_{i}, X

_{j}, and X

_{k}. The other indices are computed analogously. All input random variables are considered statistically independent. The sum of all indices must be equal to one:

_{f}indices can also be derived by rewriting Sobol indices in the context of ROSA [21]. Estimating all sensitivity indices in Equation (14) can be highly computationally challenging and difficult to evaluate. For a large number of input variables, it may be better to analyse the effects of input variables using the total effect index (in short, the total index) C

_{Ti}.

_{f}|X~

_{i}= P((Z|X~

_{i}) < 0) is the conditional failure probability evaluated for a input random variable X

_{i}and fixed variables (X

_{1}, X

_{2},…, X

_{i–}

_{1}, X

_{i+}

_{1},…, X

_{M}). The total index C

_{Ti}measures the contribution of input variable X

_{i}, including all of the effects caused by its interactions, of any order, with any other input variable. The total index C

_{Ti}can also be computed if all sensitivity indices in Equation (14) are computed. For example, C

_{T}

_{1}for M = 3 can be written as C

_{T}

_{1}= C

_{1}+ C

_{12}+ C

_{13}+ C

_{123}.

_{d}or R

_{d}). The plot of contrast function ψ(θ) vs. θ is convex and, with some exceptions, asymmetric.

_{f}, because the distance (Y − θ) is considered linear. The first-order contrast Q index is defined, on the basis of Equation (16), as

_{i}is fixed. E[·] is taken over X

_{i}.

_{ij}is computed analogously, but with the fixing of pairs X

_{i}and X

_{j}:

_{ijk}is computed similarly:

_{Ti}can be written analogously to Equation (15) as:

_{i}and fixed variables (X

_{1}, X

_{2}, …, X

_{i–}

_{1}, X

_{i+}

_{1}, …, X

_{M}). Equation (21) is analogous to Equation (15), but for the quantile.

#### 3.3. Specific Properties of Contrasts Associated with Quantiles

_{Y}

_{<−0.253})) = 0.386, where the weight 0.6 favors the minority population over the 0.4-quantile and the weight 0.4 puts the majority population after the 0.4-quantile at a disadvantage. In this specific example, it can be observed that the function ψ(θ*) vs. θ* has an N(0, 1) course and therefore, ψ(−0.253) = ϕ(−0.253, 0, 1) = 0.386. In the case of the general Gaussian pdf Y ~ N(μ, σ

^{2}), function ψ(θ*) can be written in a specific form:

_{i}, etc., makes the use of Equation (23) problematic in black box tasks, where skewness and kurtosis can have non-Gaussian values.

^{2}− (θ*)

^{2}≈ 1.6

^{2}with asymptotes l = ±θ* (see Figure 4). In a more general case of Y ~ N(μ, σ

^{2}), the dependence between l and θ* is a hyperbola l

^{2}− (θ* − μ)

^{2}≈ σ

^{2}·1.6

^{2}with asymptotes l = ±(θ* − μ). The intersection of two asymptotes is at the center of symmetry of the hyperbola, which is the mean value μ = E(Y). The skewness and kurtosis (departure from the Gaussian pdf) lead to asymmetric and symmetric deviations from this hyperbola, but asymptotes of such a curve remain l = ±(θ* − μ). Figure 4 illustrates an example with the so-called Hermite pdf with a mean value of 0, standard deviation of 1, skewness of 0.9, and kurtosis of 2.9. Although deviations from the hyperbola are significant around the mean value, the dependence l vs. θ* approaches the asymptotes l = ±(θ* − μ) in the regions of design quantiles (see Figure 4b). The observation can be generalized to any pdf or histogram of Y.

_{1}+ X

_{2}that corr(Q(Y|X

_{i}), E(Y|X

_{i})) ≈ 1, where Q(Y|X

_{i}) is the conditional α-quantile and E(Y|X

_{i}) is the conditional mean value. Changing X

_{i}causes synchronous changes in the α-quantile Q(Y|X

_{i}) and mean value E(Y|X

_{i}).

## 4. Case Study of the Ultimate Limit State

_{1}and F

_{2}, both of which have a Gaussian pdf (see Figure 5b and Table 1). Parameter μ

_{P}changes the mean value of the axial load of the bar, while the standard deviation of F is constant. The resulting force F = F

_{1}+ F

_{2}has a Gaussian pdf with a mean value of ${\mu}_{F}$ = ${\mu}_{{F}_{1}}$ + ${\mu}_{{F}_{2}}$ = 309.56 kN + μ

_{P}and standard deviation ${\sigma}_{F}$ = (${\sigma}_{{F}_{1}}^{2}$ + ${\sigma}_{{F}_{2}}^{2}$)

^{0.5}= 33.94 kN.

_{y}; plate thickness t; and plate width b [40]:

_{y}, t, and b, whose random variabilities are considered according to the results of experimental research [41,42]. Random variables f

_{y}, t, and b are statistically independent and are introduced with Gaussian pdfs (see Table 2).

_{R}, standard deviation σ

_{R}, and standard skewness a

_{R}of resistance R can be expressed using equations (see [40]), based on arithmetic means μ

_{fy}, μ

_{t}, and μ

_{b}and standard deviations σ

_{fy}, σ

_{t}, and σ

_{b}presented in Table 2.

_{R}= 412.68 kN, σ

_{R}= 34.057 kN, and a

_{R}= 0.111.

^{−19}are estimated relatively accurately using the approximation of probability density R by a three-parameter lognormal pdf with parameters μ

_{R}, σ

_{R}, and a

_{R}. This approximation is also suitable when one variable in Equation (26) is fixed. Fixing two variables leads to R with a Gaussian pdf with parameters μ

_{R}and σ

_{R}.

_{F}(y) is the pdf of load action, Φ

_{R}(y) is the distribution function of resistance, and y denotes a general point of the force (the observed variable) with the unit of Newton. The integration in Equation (30) is performed in the case study numerically using Simpson’s rule, with more than ten thousand integration steps over the interval [μ

_{Z}− 10σ

_{Z}, μ

_{Z}+ 10σ

_{Z}].

## 5. Computation of Sensitivity Indices

_{1}, F

_{2}, f

_{y}, t, and b on the failure probability P

_{f}or design quantiles F

_{d}and R

_{d}.

_{P}, which changes with the step Δμ

_{P}= 10 kN. Although μ

_{P}is the computation parameter, sensitivity indices are preferably plotted, depending on P

_{f}, because P

_{f}has a clear relevance to reliability. The transformation of μ

_{P}to P

_{f}is expressed using Equation (30) (see Figure 6a).

_{P}is selected, the sensitivity indices and P

_{f}are computed, and the indices vs. P

_{f}are then plotted. If the design quantiles are the key quantities of interest, then the dependency between P

_{f}and the probabilities of the design quantiles can be considered, according to Figure 6b.

_{d}and R

_{d}is considered under the condition F

_{d}= R

_{d}in Equation (7) and σ

_{F}= σ

_{R}. Perfect biaxial symmetry of the curves in Figure 6b is only observed for perfect σ

_{F}= σ

_{R}; otherwise, the curve of the variable with the smaller standard deviation has a steeper slope. In the case study, for β = 3.8 (P

_{f}= 7.2 × 10

^{−5}), P(F < F

_{d}) = 0.9963, and P(R < R

_{d}) = 0.0036, where F

_{d}= R

_{d}= 321.01 kN (μ

_{F}= 229.97 kN, μ

_{R}= 412.68 kN, and σ

_{F}= 33.94 kN ≈ σ

_{R}= 34.057 kN).

#### 5.1. Local ROSA—Sensitivity Indices Based on Derivatives

_{f}with respect to the mean values μ

_{xi}. Although the partial derivative of P

_{f}with respect to μ

_{t}has the greatest value, t is not the most influential input variable in terms of the absolute change of P

_{f}due to the uncertainty (variance) of the input variable t. A better measure of sensitivity is obtained by multiplying the partial derivatives by the standard deviations of the respective input variables (see Figure 7b). Ranking according to D

_{i}gives the sensitivity ranking of input variables as f

_{y}, F

_{1}, F

_{2}, t, and b.

#### 5.2. Global ROSA—Contrast P_{f} Indices

_{f}indices are depicted in Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. All contrast P

_{f}indices were computed numerically using Equation (30) for the interval P

_{f}∈ [9.35 × 10

^{−8}, 1–1.51 × 10

^{−8}].

_{f}∈[0.1, 0.9], the plot of C

_{i}is a concave function with approximately left-right symmetry. The sum of indices C

_{1}+ C

_{2}is the same as what would have been obtained had we introduced only one random variable for F with a Gaussian pdf with a mean value of μ

_{F}= 309.56 kN +μ

_{P}and standard deviation of σ

_{F}= 33.94 kN: C

_{2}+ C

_{1}= C

_{F}. The sum of indices C

_{3}+ C

_{4}+ C

_{5}is the same as what would have been obtained had we introduced only one random variable for R with a three-parameter lognormal pdf with parameters μ

_{R}= 412.68 kN, σ

_{R}= 34.057 kN, and a

_{R}= 0.111: C

_{3}+ C

_{4}+ C

_{5}= C

_{R}.

_{i}. For example, for P

_{f}= 0.3, indices C

_{1}, C

_{2}, and C

_{12}(load action) have slightly smaller values and indices C

_{3}, C

_{4}, C

_{5}, C

_{34}, C

_{35}, C

_{45}, and C

_{345}(resistance) have slightly higher values, compared to the perfect symmetry. For the other indices, there is a mix of both influences.

_{f}∈[0.1, 0.9], the first-, fourth-, and fifth-order indices generally have higher values than the second- and third-order indices.

_{f}for reliability classes RC1, RC2, and RC3 taken from [4] are 8.5 × 10

^{−6}, 7.2 × 10

^{−5}, and 4.8 × 10

^{−4}(also see [19]). Figure 11b shows the contribution of all 31 indices for target value P

_{f}= 7.2 × 10

^{−5}. First-order indices are represented minimally, where ∑S

_{i}= 0.017. On the contrary, the representation of higher-order indices is significant, especially those related to f

_{y}, F

_{1}, and F

_{2}(see Figure 11b).

_{y}occurs in all significant parts of the graph, but the same is true for F

_{1}or F

_{2}. Determining the order of importance of input variables using 31 indices can be difficult. The use of total indices C

_{Ti}is more practical. Input variables are ranked based on C

_{Ti}as f

_{y}, F

_{1}, F

_{2}, t, and b (see Figure 12). This is the same ranking as was found using index D

_{i}(Figure 7b).

_{f}, which are relevant for the design of building structures. Figure 13 shows the local extremes of some sensitivity indices in the interval of small P

_{f}. Interestingly, the sensitivity indices of small P

_{f}have plots that are not obvious (cannot be extrapolated) from the plots in the interval P

_{f}$\in $ [0.1, 0.9]. Similar local extremes as in Figure 13 were not observed for D

_{i}in Figure 7.

#### 5.3. Global ROSA—Contrast Q Indices

_{i}were estimated from Equation (17) using double-nested-loop computation. In the outer loop, E[·] was computed using one thousand runs of the LHS method. In the nested loop, conditional contrast values were computed using four million runs of the LHS method. The unconditional contrast value in the denominator was computed using four million runs of the LHS method. Higher-order indices were estimated similarly.

_{f}= 7.2 × 10

^{−5}is considered according to [4]. In Equation (7), the design value of resistance R

_{d}is considered as the 0.0036-quantile and the design load value F

_{d}is considered as the 0.9963-quantile (see Figure 6). Sensitivity analysis is performed for R with a three-parameter lognormal pdf when no or one variable in Equation (26) is fixed; otherwise, a Gaussian pdf is used in the stochastic model.

_{d}= R

_{d}= 321.01 kN computed using Equation (7) consider F and R with a Gaussian pdf. However, the design resistance value computed using a three-parameter lognormal pdf (stochastic model) is 325.00 kN. The small difference is because the skewness a

_{R}= 0.111 was neglected in Equation (7).

_{P}for P

_{f}= 7.2 × 10

^{−5}is μ

_{P}= −79.592 kN. Input random variables for R are considered according to Table 2. The results of SA of the 0.0036-quantile of R are depicted in Figure 14b.

_{T}

_{1}= 0.71, Q

_{T}

_{2}= 0.70 and Q

_{T}

_{3}= 0.86, Q

_{T}

_{4}= 0.59, and Q

_{T}

_{5}= 0.13, the order of importance of input variables can be determined as F

_{1}and F

_{2}and f

_{y}, t, and b. Variables F and R have the same weight in Equation (2) and therefore, the order of importance of all five input variables can be determined as f

_{y}, F

_{1}, F

_{2}, t, and b, based on the estimates of all Q

_{Ti}.

_{f}; it is “only” based on the SA of design quantiles R

_{d}and F

_{d}.

_{P}(generally a change in μ

_{F}) is not reflected in the results of contrast Q indices.

## 6. New Sensitivity Indices of Small and Large Design Quantiles

#### 6.1. The Asymptotic Form of Contrast Q Indices for Small and Large Quantiles

^{2}− (θ* − μ)

^{2}= σ

^{2}·${l}_{0}^{2}$ for l, we can obtain an approximate relation for Q

_{i}:

^{2}. The non-dimensional parameter l

_{0}can be calculated from Equation (25) as l

_{0}= l

^{2}/σ

^{2}at the point θ* = μ. However, the precise value of l

_{0}is not important if |Q(Y)-E(Y)| is large and l

_{0}does not affect the asymptotes. By substituting the hyperbolic functions with their asymptotes, Equation (31) can be simplified as

_{0}. The second-order probability Q index can be rewritten analogously:

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

^{2}, (Q(Y|X

_{i}) − E(Y|X

_{i}))

^{2}, etc., leads to new sensitivity indices, which we denote as QE indices. The new first-order quantile-oriented index is defined as

_{i}, K

_{ij}, and K

_{ijk}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 K

_{Ti}can be formulated analogously to Equation (21).

^{2}as a variance. Equation (36) can be rewritten analogously to Equation (39) in the form

_{Y}in unconditional and conditional pdfs.

_{P}= −79.592 kN (P

_{f}= 7.2 × 10

^{−5}). In the case study, QE indices were obtained on the load action side as K

_{1}= 0.50, K

_{2}= 0.49, and K

_{12}= 0.01 and on the resistance side as K

_{3}= 0.65, K

_{4}= 0.25, K

_{5}= 0.01, K

_{34}= 0.08, K

_{35}= 0.00, K

_{35}= 0.00, and K

_{345}= 0.01 (see Figure 16). By computing the total indices Q

_{T}

_{1}= 0.51, Q

_{T}

_{2}= 0.50 and Q

_{T}

_{3}= 0.74, Q

_{T}

_{4}= 0.34, and Q

_{T}

_{5}= 0.02, the order of importance of input variables can be determined as F

_{1}and F

_{2}and f

_{y}, t, and b. The sensitivity ranking based on all five Q

_{Ti}is f

_{y}, F

_{1}, F

_{2}, t, and b.

## 7. Discussion

_{y}, 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:

- Q
_{T}_{3}= 0.86 > Q_{T}_{1}= 0.71 > Q_{T}_{2}= 0.70 > Q_{T}_{4}= 0.59 > Q_{T}_{5}= 0.13; - C
_{T}_{3}= 0.92 < C_{T}_{1}= 0.892 < C_{T}_{2}= 0.887 < C_{T}_{4}= 0.69 < C_{T}_{5}= 0.16; - |D
_{3}| = 1.64 × 10^{−4}> |D_{1}| = 1.52 × 10^{−4}> |D_{2}| = 1.50 × 10^{−4}> |D_{4}| = 1.02 × 10^{−4}> |D_{5}| = 0.21 × 10^{−4}; - K
_{T}_{3}= 0.74 > K_{T1}= 0.51 > K_{T}_{2}= 0.50 > K_{T}_{4}= 0.34 > K_{T}_{5}= 0.02.

_{f}= 7.2 × 10

^{−5}and the corresponding design quantiles (see previous sections). Contrast Q and P

_{f}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

_{f}. However, the result cannot be generalized beyond the Gaussian (or approximately Gaussian) design reliability conditions.

_{F}≈ σ

_{R}. Then, ROSA can be effectively evaluated using the SA of design quantiles R

_{d}and F

_{d}, without having to analyse either P

_{f}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

_{f}indices, especially for small values of P

_{f}.

_{F}≠ σ

_{R}, the total indices of design quantiles should be corrected using weights based on the sensitivity factors α

_{F}and α

_{R}from Equation (6). For example, if σ

_{F}→ 0, then α

_{F}→ 0 and α

_{R}→ 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 Q

_{Ti}indices are as follows: α

_{F}·Q

_{T}

_{1}, α

_{F}·Q

_{T}

_{2}, α

_{R}·Q

_{T}

_{3}, α

_{R}·Q

_{T}

_{4}, and α

_{R}·Q

_{T}

_{5}. The correction of indices K

_{Ti}can be performed similarly. If σ

_{F}= σ

_{R}, corrections are not necessary because α

_{F}= α

_{F}= 0.7071. Initial studies have shown the rationality of this approach; however, further analysis is necessary. Corrections of indices C

_{Ti}are not performed. If σ

_{F}→ 0, then C

_{Ti}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.

_{i}, K

_{ij}, and K

_{ijk}give significant values of first-order indices K

_{i}(compared to Q

_{i}) 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.

_{f}with respect to a given input parameter. Although the sensitivity ranking determined on the basis of D

_{i}is the same as from C

_{Ti}, Q

_{Ti}, or K

_{Ti}, this conclusion cannot be generalized, and D

_{i}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.

_{i}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

_{i}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

_{i}are S

_{1}= 0.25, S

_{2}= 0.24, S

_{3}= 0.34, S

_{4}= 0.16, and S

_{5}= 0.01. Sobol’s higher-order sensitivity indices are negligible. Both the correlation and SSA give the same sensitivity ranking as ROSA: f

_{y}, F

_{1}, F

_{2}, t, and b. The case study shows that the normalization of the newly proposed indices K

_{Ti}leads to the classical S

_{i}, i.e., K

_{Ti}/2.11 ≈ S

_{i}. 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.

_{f}= (1 − P

_{f}) or unreliability P

_{f}leads to the same contrast P

_{f}indices, because P

_{f}(1 − P

_{f}) = P~

_{f}(1 − P~

_{f}). 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

_{f}would be perfectly symmetric in the case of a perfectly symmetric pdf of R and F, with zero skewness.

_{f}and new types of QE indices. Other types of SA of P

_{f}like [47] or SA of the quantile [48] have not been studied. Numerous other types of sensitivity measures exist, such as [49,50,51,52,53,54,55,56,57,58,59], 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.

## 8. Conclusions

## Funding

## Conflicts of Interest

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**Figure 3.**Graphical representation of the contrast in Equation (16) for the 0.4-quantile of N(0, 1).

**Figure 4.**Plot of parameter l and function ψ(θ*) vs. the α-quantile θ*: (

**a**) The Gaussian and non-Gaussian pdf; (

**b**) The same asymptotes of hyperbolic and non-hyperbolic function.

**Figure 5.**Static model: (

**a**) Bar under tension and (

**b**) probability density functions of R, F

_{1}, and F

_{2}for μ

_{p}= 0.

**Figure 6.**Probability of design α-quantiles vs. failure probability P

_{f}: (

**a**) μ

_{p}vs. P

_{f}; (

**b**) P

_{f}vs. α-quantiles.

**Figure 7.**Derivative-based local SA of failure probability P

_{f}: (

**a**) Derivatives; (

**b**) Sensitivity index D

_{i}.

**Figure 8.**Derivative-based local SA of failure probability P

_{f}: (

**a**) Derivatives; (

**b**) Sensitivity index D

_{i}.

**Figure 11.**(

**a**) Fifth-order contrast P

_{f}indices and (

**b**) all-order contrast P

_{f}indices for P

_{f}= 7.2 × 10

^{−5}.

Characteristic | Index | Symbol | Mean Value μ (kN) | Standard Deviation σ |
---|---|---|---|---|

Load Action | 1 | F_{1} | 241.4 + 0.5·μ_{P} | 24.14 kN |

Load Action | 2 | F_{2} | 68.16 + 0.5·μ_{P} | 23.86 kN |

Characteristic | Index | Symbol | Mean Value μ | Standard Deviation σ |
---|---|---|---|---|

Yield strength | 3 | f_{y} | 412.68 MPa | 27.941 MPa |

Thickness | 4 | t | 10 mm | 0.46 mm |

Width | 5 | b | 100 mm | 1 mm |

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Kala, Z.
From Probabilistic to Quantile-Oriented Sensitivity Analysis: New Indices of Design Quantiles. *Symmetry* **2020**, *12*, 1720.
https://doi.org/10.3390/sym12101720

**AMA Style**

Kala Z.
From Probabilistic to Quantile-Oriented Sensitivity Analysis: New Indices of Design Quantiles. *Symmetry*. 2020; 12(10):1720.
https://doi.org/10.3390/sym12101720

**Chicago/Turabian Style**

Kala, Zdeněk.
2020. "From Probabilistic to Quantile-Oriented Sensitivity Analysis: New Indices of Design Quantiles" *Symmetry* 12, no. 10: 1720.
https://doi.org/10.3390/sym12101720