ANOVA decomposition of function with random input variables provides ANOVA functionals (AFs), which contain information about the contributions of the input variables on the output variable(s). By embedding AFs into an appropriate reproducing kernel Hilbert space regarding their distributions, we propose an efficient statistical test of independence between the input variables and output variable(s). The resulting test statistic leads to new dependent measures of association between inputs and outputs that allow for i) dealing with any distribution of AFs, including the Cauchy distribution, ii) accounting for the necessary or desirable moments of AFs and the interactions among the input variables. In uncertainty quantification for mathematical models, a number of existing measures are special cases of this framework. We then provide unified and general global sensitivity indices and their consistent estimators, including asymptotic distributions. For Gaussian-distributed AFs, we obtain Sobol' indices and dependent generalized sensitivity indices using quadratic kernels.

翻译：具有随机输入变量的函数方差分解可提供方差分析泛函，这些泛函包含输入变量对输出变量贡献的信息。通过将方差分析泛函依据其分布嵌入合适的再生核希尔伯特空间，我们提出一种有效的输入变量与输出变量间独立性的统计检验方法。由此生成的检验统计量引出了输入与输出间新的依赖关联度量，其优势在于：i) 可处理包括柯西分布在内的任意方差分析泛函分布；ii) 能考虑方差分析泛函的必要或期望矩以及输入变量间的交互作用。在数学模型的不确定性量化中，现有多种度量方法均是该框架的特例。我们进一步提出统一通用的全局灵敏度指标及其一致估计量，包含渐近分布。对于服从高斯分布的方差分析泛函，我们采用二次核获得了Sobol'指标与依赖广义灵敏度指标。