  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.
