Integral equalities and inequalities: a proxy-measure for multivariate sensitivity analysis

TL;DR

This paper introduces an optimal integral equality and inequality for variance estimation in multivariate sensitivity analysis, providing a new proxy-measure that improves input importance ranking.

Contribution

It develops a novel gradient-based integral equality and inequality for variance, extending to multivariate functions, and proposes a new proxy-measure for sensitivity analysis that enhances input prioritization.

Findings

The new proxy-measure effectively identifies important inputs.

It improves the upper bounds of variance estimates compared to traditional Poincaré inequalities.

Numerical tests confirm the relevance of the proxy-measure in practical sensitivity analysis.

Abstract

Weighted Poincar\'e-type and related inequalities provide upper bounds of the variance of functions. Their application in sensitivity analysis allows for quickly identifying the active inputs. Although the efficiency in prioritizing inputs depends on the upper bounds, the latter can be big, and therefore useless in practice. In this paper, an optimal weighted Poincar\'e-type inequality and gradient-based expression of the variance (integral equality) are studied for a wide class of probability measures. For a function $f : \mathbb{R} \to \mathbb{R}^n$, we show that $${\rm Var}_\mu(f) = \int_{\Omega \times \Omega} \nabla f (x) \nabla f (x')^T \frac{F(\min(x,\, x') - F(x)F(x')}{\rho(x) \rho(x')} \, d\mu(x) d\mu(x')\, ,$$ and $${\rm Var}_\mu(f) \preceq \frac{1}{2}\int_\Omega \nabla f(x) \nabla f(x)^T \frac{F(x) (1-F(x))}{[\rho(x)]^2}\, d\mu(x) \, ,$$ with ${\rm Var}_\mu(f) = \int_\Omega ff^T\, d\mu -\int_\Omega f \, d\mu \int_\Omega f^T\, d\mu$, $F$ and $\rho$ the distribution and the density functions, respectively. These results are generalized to cope with multivariate functions by making use of cross-partial derivatives, and they allow for proposing a new proxy-measure for multivariate sensitivity analysis, including Sobol' indices. Finally, the analytical and numerical tests show the relevance of our proxy-measure for identifying important inputs by improving the upper bounds from Poincar\'e inequalities.