# Global Sensitivity Analysis of Quantiles: New Importance Measure Based on Superquantiles and Subquantiles

## Abstract

**:**

## 1. Introduction

#### 1.1. A Brief Review of Sensitivity Analysis in Civil Engineering

#### 1.2. Reliability-Oriented Sensitivity Analysis

## 2. Quantile-Oriented and Sobol Global Sensitivity Indices

_{1}, X

_{2},... X

_{M}}, and R is a one-dimensional model output. The uncertain model inputs are considered as statistically independent random variables. This incurs no loss of generality, because mutual relations are created through the computational model on the path to the output.

#### 2.1. Linear Form of Quantile-Oriented Sensitivity Indices—Contrast Q Indices

_{1}+ l

_{2}. l

_{1}is the mean absolute deviation from θ* below θ*, l

_{2}is the mean absolute deviation from θ* above the quantile θ* (in short quantile deviation l), and f(r) is the probability density function (pdf) of the model output.

_{j}can be estimated as

_{1}is the total number of observations below the α-quantile, N

_{2}= N – N

_{1}is the total number of observations above the α-quantile, where α-quantile θ* can be estimated so that α·N observations are smaller than θ* and (1-α)·N observations are greater than θ*.

_{R}= 0 and standard deviation σ

_{R}= 1. Figure 2a depicts the Uniform probability density function (pdf). Figure 2b and Figure 3a,b depict a four-parameter Hermite pdf, where the third and fourth parameters are skewness and kurtosis.

_{i}index defined in [63] has a form that can be rewritten using the quantile deviation l as Equation (7)

_{i}. The new form of the contrast index Q

_{i}is

_{i}can change all the statistical characteristics of output R. Only the changes in l caused by changes in Xi are important for the value of index Q

_{i}; see Equation (8). What statistical characteristics does l depend on? The quantile deviation l is not dependent on μ

_{R}. Changes in l would hypothetically depend only on changes in σ

_{R}provided that the shape of the pdf does not change (e.g., still Gaussian output in additive model with Gaussian inputs). However, this cannot be generally assumed.

_{R}= 1 and σ

_{R}= 1 is considered. Analogously, changing the shape of the pdf can change σ

_{R}, but not l. Figure 3 shows an example where changing the pdf shape does not cause a change in l when μ

_{R}= 1 and σ

_{R}= 1 is considered. Therefore, changing the shape of the pdf may or may not affect l. The skewness and kurtosis may or may not affect l. In general, l does not depend on the change of μ

_{R}itself, but depends on the pdf shape where the influence of moments acts in combinations, which can have a greater or lesser influence on l depending on the specific model type. These questions are examined in more detail in the case study presented in Chapter 5.

_{ij}is derived similarly by fixing of pairs X

_{i}, X

_{j}

_{i}and X

_{j}. The third-order sensitivity index Q

_{ijk}is computed analogously

_{Ti}can be written as

_{i}and fixed variables (X

_{1}, X

_{2},…, X

_{i–}

_{1}, X

_{i+}

_{1},…, X

_{M}).

#### 2.2. Quadratic Form of Quantile-Oriented Sensitivity Indices—K Indices

^{2}can be performed in a similar manner to the decomposition of the variance in Sobol sensitivity indices [62]. The asymptotic form of these indices has been denoted as QE indices [62].

_{i}index can be written as

_{ij}is computed similarly with fixing of pairs X

_{i}, X

_{j}

_{ijk}is computed analogously

_{Ti}can be written as

^{2}evaluated for input random variable X

_{i}and fixed variables (X

_{1}, X

_{2}, …, X

_{i–}

_{1}, X

_{i+}

_{1},…, X

_{M}). Equations (13)–(17) can be used for all quantiles, i.e., they are not limited to small and large quantiles.

#### 2.3. Sobol Sensitivity Indices—Sobol Indices

_{i}index can be written as

_{i}. The total effect index S

_{Ti}, which measures first and higher-order effects (interactions) of variable X

_{i}, is another popular variance-based measure [1]

_{i}and fixed variables (X

_{1}, X

_{2}, …, X

_{i–}

_{1}, X

_{i+}

_{1}, …, X

_{M}).

## 3. Resistance of Steel Member under Compression

_{d}. Standard [59] enables the determination of design value R

_{d}as 0.1 percentile [77,78,79,80,81].

_{0}/(1 − F/F

_{cr}), where F

_{cr}is Euler’s critical load. Increasing the external load action F increases the compressive stress σ

_{x}until the yield strength f

_{y}is attained in the middle of the span in the lower (extremely compressed) part of the cross-section; see Figure 4a. Hooke’s law with Young’s modulus E is considered. The dependence of σ

_{x}on F is non-linear if e

_{0}> 0, where F < F

_{cr}. The elastic resistance R (unit Newton) is the maximum load action F; a higher value of force F would cause overstressing and structural failure. The resistance R can be computed using the response function [82]

_{0}is the amplitude of initial axis curvature, L is the member length, h is the cross-sectional height, b is the cross-sectional width, t

_{1}is the web thickness and t

_{2}is the flange thickness. These variables are used to further compute the following variables: A is cross-sectional area and I

_{z}is second moment of area around axis z.

_{0}is the amplitude of pure geometrical imperfection with an idealized shape according to the elastic critical buckling mode [83]. Amplitude e

_{0}is not an equivalent geometrical imperfection [84,85,86], which would replace the influence of other imperfections, such as the residual stress. In Equation (20), the influence of residual stress is neglected.

_{1}and h have a minimal influence on R. Therefore, these variables can be considered as deterministic with values t

_{1}= 6 mm and h = 171 mm. The input random variables are listed in Table 1. All random variables are statistically independent of each other.

_{y}> 0, E > 0, t

_{2}> 0 and b > 0. However, negative realizations of random variables f

_{y}, E, t

_{2}and b practically never occur if the LHS method [92,93] is used with no more than tens of millions of runs. Theoretically, if f

_{y}→ 0 then R → 0 (due to no stress), if E → 0 then R → 0 (due to zero stiffness), if e

_{0}→ 0 then R → F

_{cr}or R → f

_{y}·A (pure buckling for high L or simple compression for low L), if L → 0 then R → f

_{y}·A (simple compression).

## 4. Results of Sensitivity Analysis

_{0}/10, where L

_{0}= 4.244 m is the length of the member with non-dimensional slenderness [94] $\overline{\lambda}$= 1.0. The common non-dimensional slenderness of a strut in an efficient structural system is around one, but struts usually do not have non-dimensional slenderness higher than two [95]. The slenderness is directly proportional to the length. It is possible, for the presented case study, to write the transformation L =$\overline{\lambda}$· L

_{0}, which makes it easier to understand the lengths.

^{2}or variance), which are estimated using an inner loop algorithm. The inner loop is repeated 4 million times (4 million LHS runs) to compute statistics (l, l

^{2}or variance) with some random realizations fixed by the outer loop.

_{y}is dominant for low values of L (low slenderness), imperfection e

_{0}is dominant for intermediate lengths L (intermediate slenderness), Young’s modulus and flange thickness gain dominance in the case of long members (high slenderness).

_{T}indices shown in Figure 14 provide very similar (but not the same) information as the first-order K

_{i}indices depicted in Figure 12.

_{i}.

_{1}being 37% smaller than S

_{1}, and K

_{1}being 3% smaller than S

_{1}. Imperfection e

_{0}has the greatest influence for L ≈ 3.8 m, with Q

_{3}being 46% smaller than S

_{3}, and K

_{3}being 16% smaller than S

_{3}. K

_{i}indices are closer to S

_{i}indices (compared to Q

_{i}indices).

_{T}

_{1}being 22% greater than S

_{T}

_{1}, and K

_{T}

_{1}being 7% greater than S

_{T}

_{1}. Imperfection e

_{0}has the greatest influence for L ≈ 3.8 m, with Q

_{T}

_{3}being 13% greater than S

_{T}

_{3}and K

_{T}

_{3}being 3% smaller than S

_{T}

_{3}. K

_{Ti}indices are closer to S

_{Ti}indices (compared to Q

_{Ti}indices).

^{2}, which behaves similarly to variance ${\sigma}_{R}^{2}$.

## 5. Static Dependencies between l and σ_{R} and Other Connections

^{2}while Sobol sensitivity indices are based on variance ${\sigma}_{R}^{2}$. The subject of interest of both quantile-oriented SA is the 0.001-quantile of R.

_{R}is performed when X

_{i}is fixed. The aim is to identify similarities and differences between l and σ

_{R}, rather than to accurately quantify sensitivity using Equations (8), (13) and (18). Samples σ

_{R}|X

_{i}and l|X

_{i}are plotted for 400 LHS runs of X

_{i}, otherwise the solution is the same as in the previous chapter. Skewness a

_{R}|X

_{i}and kurtosis k

_{R}|X

_{i}are added for selected samples, see Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21.

_{0}with a mean value of zero. Only the absolute value of this variable is applied in Equation (20). The output R is not monotonically dependent on e

_{0}. The practical consequence is that in the case of an even number of LHS runs, it is sufficient to compute the nested loop in Equations (8), (13) and (18) only once for the positive value of random realization e

_{0}, because the solution is the same for a negative value. This reduces the computational cost of estimating indices Q

_{3}, K

_{3}and S

_{3}by half.

_{R}|X

_{i}, the more fixing of X

_{i}reduces the uncertainty of the output in terms of variance, which measures change around μ

_{R}. The smaller the estimated l|X

_{i}, the more the fixing of X

_{i}reduces the uncertainty of the output in terms of parameter l, which measures change around θ. Imperfection e

_{0}(X

_{3}) has the greatest influence in both cases, see low values on the vertical axes in Figure 19.

_{i}vs. σ

_{R}|X

_{i}is approximately linear with the exception of the concave course on the right in Figure 16. Pearson correlation coefficient between 400 samples l|X

_{i}vs. σ

_{R}|X

_{i}is approximately 0.66. The concave course and lower correlation (compared to other variables) is due to the conflicting influences of σ

_{R}, a

_{R}and k

_{R}. By approximating R using the Hermite distribution R~H(μ

_{R}, σ

_{R}, a

_{R}, k

_{R}), the effects of μ

_{R}, σ

_{R}, a

_{R}and k

_{R}on l can be observed separately as follows: change in μ

_{R}has no influence on l, increasing σ

_{R}increases l, increasing a

_{R}decreases l, increasing k

_{R}increases l, assuming small values of changes.

_{1}is interesting. Figure 16, on the left, shows that with increasing X

_{1}, σ

_{R}|X

_{1}has an approximately decreasing plot, with the exception of the beginning on the left. Figure 17 shows that a

_{R}|X

_{1}has an approximately decreasing course, k

_{R}|X

_{1}has an increasing course. At the beginning on the left, increasing X

_{1}causes an increase in σ

_{R}|X

_{1}, k

_{R}|X

_{1}and a

_{R}|X

_{1}, which, taken together, increases l|X

_{1}due to the dominance of the joint effect of σ

_{R}|X

_{1}, k

_{R}|X

_{1}and a

_{R}|X

_{1}. The region where σ

_{R}|X

_{1}starts decreasing but l still increases is interesting. Although the standard deviation is an important output characteristic, a change in the input variable can have a stronger influence on the quantile through skewness and kurtosis. At the opposite end (right), increasing X

_{1}causes a decrease in σ

_{R}|X

_{1}, a decrease in a

_{R}|X

_{1}and an increase in k

_{R}|X

_{1}, which together reduces l|X

_{1}, because the decreasing sole effect of σ

_{R}|X

_{1}is dominant. The whole course of l|X

_{1}vs. X

_{1}is shown in Figure 16 in the middle. The example shows the combined effect of standard deviation, skewness and kurtosis on the quantile deviation l, which is the core of the computation of quantile-oriented sensitivity indices Q

_{i}and K

_{i}.

_{2}, X

_{3}, X

_{4}and X

_{5}the range of σ

_{R}|X

_{i}is significantly larger than that of σ

_{R}|X

_{1}(99.1 − 90.3 = 8.8 MPa) and σ

_{R}|X

_{i}has a crucial influence on l|X

_{i}. Hence, the dependences l|X

_{i}vs. σ

_{R}|X

_{i}, i = 2, 3, 4, 5 are approximately linear.

## 6. Discussion

_{R}, because it measures the variability of the population around the quantile. Quantile deviation l has good resistance to outlier values around the quantile. When X

_{i}changes deterministically, the quantile deviation l is found to change, but not always monotonically, despite a monotonic variation in the standard deviation σ

_{R}. The correlation between l and σ

_{R}may or may not be strong even though a dependence exists; see Figure 16.

^{2}, Q indices can be rewritten as the new K indices, which are based on the decomposition of l

^{2}, similarly to the way Sobol indices are based on the decomposition of variance; see Equations (13)–(17).

_{i}on the output. In the case study, four input variables X

_{i}, i = 2, 3, 4, 5 have approximately linear dependence l|X

_{i}vs. σ

_{R}|X

_{i}, where l and σ

_{R}are computed for fixed X

_{i}while the other X

_{~i}are considered as random. However, this does not apply to input variable X

_{1}(yield strength f

_{y}), which leads to a non-linear concave dependence l|X

_{i}vs. σ

_{R}|X

_{i}. The example shows the strong influence of the shape of the distribution (skewness and kurtosis) on the quantile deviation l as one of the causes leading to differences in K indices from Sobol indices.

## 7. Conclusions

^{2}expressed in the same unit as variance, thus approaching Sobol sensitivity indices with their properties. l

^{2}has a significance similar to variance, but around quantile. The unit consistency between K indices and Sobol indices makes K indices attractive in stochastic models, where more parameters (goals) of the probability distribution of the model output need to be analysed. Overall, the new K indices can be considered effective in solving the effect of input random variables on design quantiles.

_{R}may or may not be strong. Although l correlates with σ

_{R}, l is also related to the shape of the probability distribution.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Number of publications in civil engineering Web of Science categories (Web of Science core collection database, 29 January 2021) on the topic “sensitivity analysis “and “buckling”.

**Figure 2.**Quantile deviation l of 0.4-quantile of: (

**a**) Uniform symmetric pdf; (

**b**) Hermite asymmetric pdf.

**Figure 3.**Quantile deviation l of 0.4-quantile of: (

**a**) Hermite asymmetric pdf; (

**b**) Hermite symmetric pdf.

**Figure 5.**Comparison of three types of sensitivity analysis (SA) for L = 0 m ($\overline{\lambda}$ = 0).

**Figure 10.**Q indices: (

**a**) third-order sensitivity indices; (

**b**) fourth- and fifth-order sensitivity indices.

**Figure 13.**K indices: (

**a**) third-order sensitivity indices; (

**b**) fourth- and fifth-order sensitivity indices.

**Figure 17.**Samples of skewness a

_{R}|X

_{3}and kurtosis k

_{R}|X

_{3}for L = 4.244 m ($\overline{\lambda}$ = 1.0).

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

Yield strength | 1 | f_{y} | 297.3 MPa | 16.8 MPa |

Young’s modulus | 2 | E | 210 GPa | 10 GPa |

Imperfection | 3 | e_{0} | 0 | L/1960 |

Flange thickness | 4 | t_{2} | 9.5 mm | 0.436 mm |

Flange width | 5 | b | 180 mm | 1.776 mm |

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Kala, Z.
Global Sensitivity Analysis of Quantiles: New Importance Measure Based on Superquantiles and Subquantiles. *Symmetry* **2021**, *13*, 263.
https://doi.org/10.3390/sym13020263

**AMA Style**

Kala Z.
Global Sensitivity Analysis of Quantiles: New Importance Measure Based on Superquantiles and Subquantiles. *Symmetry*. 2021; 13(2):263.
https://doi.org/10.3390/sym13020263

**Chicago/Turabian Style**

Kala, Zdeněk.
2021. "Global Sensitivity Analysis of Quantiles: New Importance Measure Based on Superquantiles and Subquantiles" *Symmetry* 13, no. 2: 263.
https://doi.org/10.3390/sym13020263