高阶Sobol指数
Higher order Sobol' indices
AI总结：
本文提出高阶Sobol指数，用于衡量高维函数对变量子集的依赖性，通过高阶矩和谱分解方法扩展传统L²分解，适用于p>2的L^p测度，量化不同变量重要性。
摘要：
Sobol指数衡量高维函数对单位立方体[0,1]^d上变量组的依赖性，基于ANOVA分解，是一种L²分解。本文讨论了Sobol指数的推广，以获得L^p测度。我们关注p>2的情况，因为此时变量重要性更关注函数f的极值。我们引入了两种方法：一种基于ANOVA项的高阶矩，另一种基于函数谱分解的高阶范数，包括傅里叶和Haar变体。两种推广都可以表示为[0,1]^{kd}上的积分，允许直接使用蒙特卡洛或拟蒙特卡洛估计。我们发现它们对f的不同方面敏感，从而量化不同的变量重要性概念。
英文摘要：
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.
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