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

Sensitivity indices subordinated to contrasts associated with quantiles [63] (in short, Contrast *Q* indices) are based on linear contrast functions. The contrast function ψ associated with the α-quantile of output *R* can be expressed using parameter *θ* as

$$\psi(\theta) = E(\psi(R,\theta)) = E((R-\theta)(\alpha - \mathbf{1}\_{R < \theta})),\tag{1}$$

where *R* is a scalar. Equation (16) attains its minimum if the argument *θ* has a value of *α*-quantile of *R*, see Equation (2)

$$\theta^\* = \underset{\theta}{\text{Argmin}} \,\psi(\theta) = \underset{\theta}{\text{Argmin}} \, E((R-\theta)(a-1\_{R<\theta})) \,\tag{2}$$

where *θ*\* is the *α*-quantile of *R*. The minimum of Equation (1) can be expressed using *θ*\* as

$$\min\_{\theta} \psi(\theta) = \psi(\theta^\*) = E((R - \theta^\*)(\mathfrak{a} - \mathbf{1}\_{R < \theta^\*})) = l \cdot \mathfrak{a} \cdot (1 - \mathfrak{a}),\tag{3}$$

where *l* is the absolute difference (distance) between the mean value of the population below the *α*-quantile *θ*\* and mean value of the population above the *α*-quantile *θ*\*. Let *l* be the quantile deviation. The quantile deviation *l* is the difference between superquantile *E*(*R*|*R* ≥ *θ*\*) and subquantile *E*(*R*|*R* < *θ*\*)

consequential.

$$l = \underbrace{\frac{1}{1 - a} \int\_{\theta^\*}^{\infty} r \cdot f(r) dr}\_{\text{Superquantile}} - \underbrace{\frac{1}{a} \int\_{-\infty}^{\theta^\*} r \cdot f(r) dr}\_{\text{Subquantile}} = \underbrace{\frac{1}{1 - a} \int\_{\theta^\*}^{\infty} |r - \theta^\*| \cdot f(r) dr}\_{l\_2} - \underbrace{\frac{1}{a} \int\_{-\infty}^{\theta^\*} |r - \theta^\*| \cdot f(r) dr}\_{l\_1} \tag{4}$$

where *l* = *l*<sup>1</sup> + *l*2. *l*<sup>1</sup> is the mean absolute deviation from *θ*\* below *θ*\*, *l*<sup>2</sup> 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. \* \* \* \* \* 1 \* 1 1 1 1 1 *rfrl dr rfr dr rfr dr rfr dr* , (4)

The introduction of superquantile and subquantile in Equation (4) introduces quantile deviation *l* as a new quantile sensitivity measure. 1 2 *l Superquantile eSubquantil l* where *l* = *l*<sup>1</sup> + *l*2. *l*1 is the mean absolute deviation from *θ*\* below *θ*\*, *l*2 is the mean absolute

It can be noted that superquantiles are fundamental building blocks for estimates of risk in finance [64] and engineering [65]. In finance, the superquantile has various names, such as expected tail loss [66], conditional value-at-risk (CVaR) [67–70] or tail value-atrisk [71], average value at risk [72], expected shortfall [73,74]. Subquantile is not such a widespread concept. 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. The introduction of superquantile and subquantile in Equation (4) introduces quantile deviation *l* as a new quantile sensitivity measure. It can be noted that superquantiles are fundamental building blocks for estimates of

In the context of SA, *l* was first introduced as a new sensitivity measure in [62]. A property of the quantile deviation *l* is that it is expressed in the same unit as the data. The quantile deviation *l* is a robust statistic, which, compared with the standard deviation, is more resilient to outliers in a dataset. This is due to the fact that in the case of standard deviation, i.e., the square root of variance, the distances from the mean are squared, so that large deviations are weighted more and can, therefore, be strongly influenced by outliers. Regarding quantile deviation *l*, the deviations of a small number of outliers are inconsequential. risk in finance [64] and engineering [65]. In finance, the superquantile has various names, such as expected tail loss [66], conditional value-at-risk (CVaR) [67–70] or tail value-atrisk [71], average value at risk [72], expected shortfall [73,74]. Subquantile is not such a widespread concept. In the context of SA, *l* was first introduced as a new sensitivity measure in [62]. A property of the quantile deviation *l* is that it is expressed in the same unit as the data. The quantile deviation *l* is a robust statistic, which, compared with the standard deviation, is more resilient to outliers in a dataset. This is due to the fact that in the case of standard deviation, i.e., the square root of variance, the distances from the mean are squared, so

With regard to random sampling, the quantile deviation *l* of a finite observation of size *N* with values *r<sup>j</sup>* can be estimated as that large deviations are weighted more and can, therefore, be strongly influenced by outliers. Regarding quantile deviation *l*, the deviations of a small number of outliers are in-

$$l \approx \underbrace{\frac{1}{\mathcal{N}\_2} \sum\_{j: r\_j \ge \theta^\*} r\_j}\_{\text{Superquantile}} - \underbrace{\frac{1}{\mathcal{N}\_1} \sum\_{j: r\_j < \theta^\*} r\_j}\_{\text{Subquantile}} = \underbrace{\frac{1}{\mathcal{N}\_2} \sum\_{j: r\_j \ge \theta^\*} |r\_j - \theta^\*|}\_{l\_2} + \underbrace{\frac{1}{\mathcal{N}\_1} \sum\_{j: r\_j < \theta^\*} |r\_j - \theta^\*|}\_{l\_1} \tag{5}$$

where *N*<sup>1</sup> is the total number of observations below the α-quantile, *N*<sup>2</sup> = *N* – *N*<sup>1</sup> 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\**. 1 2 *l l eSubquantil Superquantile* where *N*1 is the total number of observations below the α-quantile, *N*<sup>2</sup> = *N* – *N*1 is the total number of observations above the α-quantile, where α-quantile *θ*\* can be estimated so

*j j j j*

Figures 2 and 3 depict examples of symmetric and asymmetric probability density functions (pdfs), where the value of *l* is expressed as the distance between the centres of gravity of the green and yellow areas. All probability density functions (pdfs) have mean value *µ<sup>R</sup>* = 0 and standard deviation *σ<sup>R</sup>* = 1. Figure 2a depicts the Uniform probability density function (pdf). Figures 2b and 3a,b depict a four-parameter Hermite pdf, where the third and fourth parameters are skewness and kurtosis. that α·*N* observations are smaller than *θ\** and (1-α)·*N* observations are greater than *θ\**. Figures 2 and 3 depict examples of symmetric and asymmetric probability density functions (pdfs), where the value of *l* is expressed as the distance between the centres of gravity of the green and yellow areas. All probability density functions (pdfs) have mean value *μR* = 0 and standard deviation *σR* = 1. Figure 2a depicts the Uniform probability density function (pdf). Figures 2b and 3a,b depict a four-parameter Hermite pdf, where the third and fourth parameters are skewness and kurtosis.

**Figure 2.** Quantile deviation *l* of 0.4-quantile of: (**a**) Uniform symmetric pdf; (**b**) Hermite asymmetric pdf. **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 3.** Quantile deviation *l* of 0.4-quantile of: (**a**) Hermite asymmetric pdf; (**b**) Hermite symmetric pdf.

*Xi*, *X<sup>j</sup>*

puted analogously

By modifying Equation (3), quantile deviation *l* can be computed using the probability density function according to Equation (6) By modifying Equation (3), quantile deviation *l* can be computed using the probability density function according to Equation (6)

dflying Equation (3), quantile deviation  $l$  can be computed using the probability action according to Equation (6)

$$l = \frac{1}{\mathfrak{a} \cdot (1 - \mathfrak{a})} \psi(\theta^\*) = \frac{1}{\mathfrak{a} \cdot (1 - \mathfrak{a})} E((\mathcal{R} - \theta^\*)(\mathfrak{a} - 1\_{R < \theta^\*})),\tag{6}$$

where the value of α·(1 − *α*) is constant. The first-order contrast *Qi* index defined in [63] has a form that can be rewritten using the quantile deviation *l* as Equation (7) where the value of α·(1 − *α*) is constant. The first-order contrast *Q<sup>i</sup>* index defined in [63] has a form that can be rewritten using the quantile deviation *l* as Equation (7)

$$\begin{aligned} \text{where the value of } \boldsymbol{a} \cdot (1 - \boldsymbol{a}) \text{ is constant. The first-order contrast } Q\_i \text{ index defined in [63]}\\ \text{has a form that can be rewritten using the quantile deviation } l \text{ as Equation (7)}\\ \boldsymbol{Q}\_i = \frac{\min\_{\boldsymbol{\theta}} \psi(\boldsymbol{\theta}) - E \left( \min\_{\boldsymbol{\theta}} E(\psi(\mathbf{R}, \boldsymbol{\theta}) | \mathbf{X}\_i) \right)}{\min\_{\boldsymbol{\theta}} \psi(\boldsymbol{\theta})} = \frac{l \cdot \boldsymbol{a} \cdot (1 - \boldsymbol{a}) - E((l|\boldsymbol{X}\_i) \cdot \boldsymbol{a} \cdot (1 - \boldsymbol{a}))}{l \cdot \boldsymbol{a} \cdot (1 - \boldsymbol{a})} = \frac{l - E(l|\boldsymbol{X}\_i)}{l}, \end{aligned} \tag{7}$$

*Q*

*i*

contrast index *Q<sup>i</sup>* is where the mean value *E*(·) is considered over all likely values of *X<sup>i</sup>* . The new form of the contrast index *Q<sup>i</sup>* is

$$Q\_{\bar{l}} = \frac{l - E(l \mid \mathbf{X}\_{\bar{l}})}{l}. \tag{8}$$

In a general model, fixing *Xi* 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 *Qi*; see Equation (8). What statistical characteristics does *l* depend on? The quantile deviation *l* is not dependent on *μR*. Changes in *l* would hypothetically depend only on changes in *σ<sup>R</sup>* 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. In a general model, fixing *X<sup>i</sup>* 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<sup>i</sup>* ; see Equation (8). What statistical characteristics does *l* depend on? The quantile deviation *l* is not dependent on *µR*. Changes in *l* would hypothetically depend only on changes in *σ<sup>R</sup>* 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.

Figure 2 shows an example where changing the shape of the pdf causes a change in *l* from *l* = 1.73 to *l* = 1.47 when *μR* = 1 and *σR* = 1 is considered. Analogously, changing the shape of the pdf can change *σR*, but not *l*. Figure 3 shows an example where changing the pdf shape does not cause a change in *l* when *μR* = 1 and *σR* = 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 *μR* 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. Figure 2 shows an example where changing the shape of the pdf causes a change in *l* from *l* = 1.73 to *l* = 1.47 when *µ<sup>R</sup>* = 1 and *σ<sup>R</sup>* = 1 is considered. Analogously, changing the shape of the pdf can change *σR*, but not *l*. Figure 3 shows an example where changing the pdf shape does not cause a change in *l* when *µ<sup>R</sup>* = 1 and *σ<sup>R</sup>* = 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<sup>R</sup>* 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.

The second-order α-quantile contrast index *Qij* is derived similarly by fixing of pairs The second-order α-quantile contrast index *Qij* is derived similarly by fixing of pairs *Xi* , *X<sup>j</sup>*

$$Q\_{ij} = \frac{l - E(l \mid \mathbf{X}\_{i\prime} \mathbf{X}\_j)}{l} - Q\_i - Q\_{j\prime} \tag{9}$$

*l* where *E*(·) is considered across all *Xi* and *Xj*. The third-order sensitivity index *Qijk* is comwhere *E*(·) is considered across all *X<sup>i</sup>* and *X<sup>j</sup>* . The third-order sensitivity index *Qijk* is computed analogously

(1) is considered across all  $\mathbf{X}\_{i}$  and  $\mathbf{X}\_{j}$ . The third-order sensitivity index  $\mathbb{Q}\_{ijk}$  is (and analogously)

$$\mathbf{Q}\_{ijk} = \frac{l - \mathbb{E}\left(l\,|\mathbf{X}\_{i}, \mathbf{X}\_{j}, \mathbf{X}\_{k}\right)}{l} - \mathbf{Q}\_{i} - \mathbf{Q}\_{j} - \mathbf{Q}\_{k} - \mathbf{Q}\_{ij} - \mathbf{Q}\_{ik} - \mathbf{Q}\_{jk} - \mathbf{Q}\_{jk} \tag{10}$$

Statistically independent input random variables are considered. The sum of all indices must be equal to one

$$\sum\_{i} Q\_{i} + \sum\_{i} \sum\_{j>i} Q\_{ij} + \sum\_{i} \sum\_{j>i} \sum\_{k>j} Q\_{ijk} + \dots + Q\_{123\dots M} = 1. \tag{11}$$

The total index *QTi* can be written as

$$Q\_{\overline{l}\overline{l}} = 1 - \frac{l - E(l|X\_{\sim \overline{l}})}{l},\tag{12}$$

where the second term in the numerator contains the conditional quantile deviation *l* evaluated for input random variable *X<sup>i</sup>* and fixed variables (*X*1, *X*2, . . . , *Xi–*1, *Xi+*1, . . . , *XM*).

Contrast *Q* indices expressed using the quantile deviation *l* are the same as the indices based on contrasts defined in [63], but are obtained in a different way. Contrast *Q* indices can also be written in an asymptotic form [62] (p. 15), which is based on measuring the distance between an α-quantile *θ*\* and the mean value *µ* of the model output *l* ≈ ±(*θ*\* − *µ*), but limited to only large and small quantiles.

The contrast *Q* indices described in this chapter use quantile deviation *l* in linear form.
