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 Walsh variants. Both of our generalizations have representations as integrals over [0, 1](kd) for some 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. In a numerical example, we study a model for the cycle time of a piston in terms of seven variables. The surface area of the piston is most important for driving it to high values, while the initial gas volume is most important for driving the cycle time down.