Abstract Weighted Poincare-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 Poincare-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 → R n with n ∈ N ∗ , we show that Var μ f = ∫ Ω × Ω ∇ f x ∇ f x ′ T F min ( x , x ′ ) − F ( x ) F ( x ′ ) ρ ( x ) ρ ( x ′ ) d μ ( x ) d μ ( x ′ ) , and Var μ f ⪯ 1 2 ∫ Ω ∇ f x ∇ f x T F ( x ) 1 − F ( x ) ρ ( x ) 2 d μ ( x ) , with Var μ f = ∫ Ω f f T d μ − ∫ Ω f d μ ∫ Ω f T d μ , F and ρ the distribution and the density functions, respectively. Such results are generalized to cope with any function f : R d → R n 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 Poincare inequalities.

中文翻译：

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基于导数的积分等式和不等式：敏感性分析的代理度量

Abstract 加权庞加莱类型和相关的不等式提供函数方差的上限。它们在灵敏度分析中的应用允许快速识别活动输入。尽管对输入进行优先排序的效率取决于这些上限，但后者可以采用更高的值，因此在实践中毫无用处。在本文中，针对多种概率度量研究了最优加权 Poincare 型不等式和基于梯度的方差表达（积分等式）。对于函数 f : R → R n 且 n ∈ N ∗ ，我们证明 Var μ f = ∫ Ω × Ω ∇ fx ∇ fx ′ TF min ( x , x ′ ) − F ( x ) F ( x ′ ) ρ ( x ) ρ ( x ′ ) d μ ( x ) d μ ( x ′ ) 和 Var μ f ⪯ 1 2 ∫ Ω ∇ fx ∇ fx TF ( x ) 1 − F ( x ) ρ ( x ) 2 d μ ( x ) ，其中 Var μ f = ∫ Ω ff T d μ − ∫ Ω fd μ ∫ Ω f T d μ ，F 和 ρ 分别是分布函数和密度函数。这样的结果被推广到使用交叉偏导数来处理任何函数 f : R d → R n 。新结果允许为敏感性分析提出新的代理措施。最后，分析测试和数值模拟显示了我们的代理测量通过改进 Poincare 不等式的上限来识别重要输入的相关性。