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ANOVA分解提供了ANOVA泛函（AFs），其中包含输入变量对输出变量贡献的信息。通过将AFs嵌入到关于其分布的适当再生核希尔伯特空间中，我们提出了一种高效的输入变量与输出变量之间独立性的统计检验方法。由此产生的检验统计量导出了输入与输出之间新的依赖关联度量，该度量具有以下特性：i) 可处理包括柯西分布在内的任意AF分布，ii) 能够考虑AF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 Uncertainty Quantification 中，现有多种度量方法均为该框架的特例。我们进一步提供了统一且通用的全局灵敏度指标及其相合估计量（包括渐近分布）。对于高斯分布的AFs，我们利用二次核得到了Sobol指标和依赖广义灵敏度指标。