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

New sensitivity indices focused on quantiles can be obtained using the square of the quantile deviation *l*. The basic concept of this quadratic form of quantile-oriented sensitivity analysis was introduced in [62] (p. 16). Unlike contrast *Q* indices, the sensitivity measure is expressed in the same unit as the variance. The decomposition of *l* 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].

The first-order *K<sup>i</sup>* index can be written as

$$K\_i = \frac{l^2 - E\left(l^2 | \mathbf{X}\_i\right)}{l^2}.\tag{13}$$

The second-order index *Kij* is computed similarly with fixing of pairs *X<sup>i</sup>* , *X<sup>j</sup>*

$$K\_{ij} = \frac{l^2 - E\left(l^2 \left|X\_{i\nu} X\_j\right.\right)}{l^2} - K\_i - K\_j. \tag{14}$$

The third-order sensitivity index *Kijk* is computed analogously

$$K\_{ijk} = \frac{l^2 - E\left(l^2 \middle| \mathcal{X}\_{i\prime} \mathcal{X}\_{j\prime} \mathcal{X}\_k\right)}{l^2} - K\_i - K\_j - K\_k - K\_{ij} - K\_{ik} - K\_{jk}.\tag{15}$$

The other higher-order indices are obtained similarly. Statistically independent input random variables are considered. The sum of all indices must be equal to one

$$\sum\_{i} \mathcal{K}\_{i} + \sum\_{i} \sum\_{j>i} \mathcal{K}\_{ij} + \sum\_{i} \sum\_{j>i} \sum\_{k>j} \mathcal{K}\_{ijk} + \dots + \mathcal{K}\_{123\dots M} = 1. \tag{16}$$

The total index *KTi* can be written as

$$K\_{\rm Ti} = 1 - \frac{l^2 - E\left(l^2 |X\_{\sim i}\right)}{l^2},\tag{17}$$

where the second term in the numerator contains the conditional parameter *l* 2 evaluated for input random variable *X<sup>i</sup>* and fixed variables (*X*1, *X*2, . . . , *Xi–*1, *Xi+*1, . . . , *XM*). Equations (13)–(17) can be used for all quantiles, i.e., they are not limited to small and large quantiles.

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