函数交叉偏导数和代用数的最优估计值

Pub Date : 2024-07-05 DOI:10.3390/stats7030042
M. Lamboni
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引用次数: 0

摘要

使用较少的模型运行计算交叉偏导数与建模有关,例如随机逼近、基于导数的方差分析、探索复杂模型和活动子空间。本文通过在 N 个随机点评估函数并使用一组 L 约束,引入了所有函数交叉部分导数的代用值。随机点依赖于独立、中心和对称变量。基于 NL 模型运行的相关估计值达到了最佳收敛率(即 O(N-1)),而且我们的近似值的偏差不会受到多种函数的维度诅咒。这些结果可用于:(i) 计算灵敏度指数的主界和上限;(ii) 利用基于导数的方差分析推导出模拟器的仿真器或函数的代用器。模拟结果表明了我们的模拟器和敏感度指数估算器的准确性。使用一个样本的 U 统计量对指数进行的插件估计在数值上非常稳定。
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Optimal Estimators of Cross-Partial Derivatives and Surrogates of Functions
Computing cross-partial derivatives using fewer model runs is relevant in modeling, such as stochastic approximation, derivative-based ANOVA, exploring complex models, and active subspaces. This paper introduces surrogates of all the cross-partial derivatives of functions by evaluating such functions at N randomized points and using a set of L constraints. Randomized points rely on independent, central, and symmetric variables. The associated estimators, based on NL model runs, reach the optimal rates of convergence (i.e., O(N−1)), and the biases of our approximations do not suffer from the curse of dimensionality for a wide class of functions. Such results are used for (i) computing the main and upper bounds of sensitivity indices, and (ii) deriving emulators of simulators or surrogates of functions thanks to the derivative-based ANOVA. Simulations are presented to show the accuracy of our emulators and estimators of sensitivity indices. The plug-in estimates of indices using the U-statistics of one sample are numerically much stable.
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