Higher order Sobol' indices

Sobol' indices measure the dependence of a high dimensional function on groups of variables defined on the unit cube $[0,1]^d$. They are based on the ANOVA decomposition of functions, which is an $L^2$ decomposition. In this paper we discuss generalizations of Sobol' indices which yield $L^p$ 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 $[0,1]^{kd}$ for $k\ge 1$, 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.