*3.1. Local ROSA*

The partial derivative δ*P<sup>f</sup>* /δµ*xi* with respect to the mean value µ of the input variable *X<sup>i</sup>* presents a classical measure of change in *P<sup>f</sup>* (see, e.g., [28–32]). 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.

The first disadvantage is that the derivative measures only change at the point (local SA) where it is numerically realized. If the algorithms on the computer are of the "black-box" type, then only a numerical evaluation of the derivative is possible. The second disadvantage is that a large absolute value of the derivative does not necessarily mean a large influence of the input on the output if the distribution range of the input variable is small compared to other variables.

A better proportional degree of sensitivity is obtained when the derivative is multiplied by the standard deviation σ*Xi* of the input variable.

$$D\_i = \frac{\partial P\_f}{\partial \mu\_{X\_i}} \sigma\_{X\_i} \tag{9}$$

The advantage of using Equation (9) is the inclusion of σ*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.

Regarding quantiles, the use of partial derivatives as an indicator of sensitivity analogously to Equation (9) is not offered. For example, for the additive model *X*<sup>1</sup> + *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.
