Résumé
Weighted Poincaré-type and related inequalities provide upper bounds of the variance of functions. Their applications in sensitivity analysis allow for quickly identifying the active inputs. Although the efficiency in prioritizing inputs depends on those upper bounds, the latter can take higher values, and therefore useless in practice. In this paper, an optimal weighted Poincaré-type inequality and gradient-based expression of the variance (integral equality) are studied for a wide class of probability measures. For a function f:R→Rn with n∈N∗, we show that Varμf=∫Ω×Ω∇fx∇fx′TFmin(x,x′)−F(x)F(x′)ρ(x)ρ(x′)dμ(x)dμ(x′),and Varμf⪯12∫Ω∇fx∇fxTF(x)1−F(x)ρ(x)2dμ(x),with Varμf=∫ΩffTdμ−∫Ωfdμ∫ΩfTdμ, F and ρ the distribution and the density functions, respectively. Such results are generalized to cope with any function f:Rd→Rn using cross-partial derivatives. The new results allow for proposing a new proxy-measure for sensitivity analysis. Finally, analytical tests and numerical simulations show the relevance of our proxy-measure for identifying important inputs by improving the upper bounds from the Poincaré inequalities.
•Derivative-based expansion of any smooth function.•Derivative-based expression of the variance of any smooth function.•Optimal weighted Poincaré inequality.•Derivative-based expressions of the first-order, total and total interaction sensitivity indices.•Derivative-based proxy-measures of sensitivity indices (DGSM).