Higher order Sobol’ indices

Sobol’ indices measure the dependence of a high dimensional function on groups of variables defined on the unit cube [0, 1]. They are based on the ANOVA decomposition of functions, which is an L decomposition. In this paper we discuss generalizations of Sobol’ indices which yield L measures of the dependence of f on subsets of variables. Our interest is in values p > 2 because then variable importance becomes more about reaching the extremes of f . We introduce two methods. One based on higher order moments of the ANOVA terms and another based on higher order norms of a spectral decomposition of f , including Fourier and Haar variants. Both of our generalizations have representations as integrals over [0, 1] for k > 1, allowing direct Monte Carlo or quasi-Monte Carlo estimation. We find that they are sensitive to different aspects of f , and thus quantify different notions of variable importance.