Estimating Mean Dimensionality of Analysis of Variance Decompositions
Analysis of variance (ANOVA) is now often applied to functions defined on the unit cube, where it serves as a tool for the exploratory analysis of functions. The mean dimension of a function, defined as a natural weighted combination of its ANOVA mean squares, provides one measure of how hard or easy it is to integrate the function by quasi-Monte Carlo sampling. This article presents some new identities relating the mean dimension, and some analogously defined higher moments, to the variance importance measures of I.M. Sobol. As a result, we are able to measure the mean dimension of certain functions arising in computational finance. We produce an unbiased and nonnegative estimate of the variance contribution of the highest-order interaction that avoids the cancellation problems of previous estimates. In an application to extreme value theory, we find that, among other things, the minimum of d independent U[0, 1] random variables has a mean dimension of 2(d + 1)/(d + 3).