Optimal estimators of cross-partial derivatives and surrogates of functions

• URL: http://arxiv.org/abs/2407.11035v1
• Date: Fri, 5 Jul 2024 03:39:06 GMT
• Title: Optimal estimators of cross-partial derivatives and surrogates of functions
• Authors: Matieyendou Lamboni,
• Abstract summary: This paper introduces surrogates of all the cross-partial derivatives of functions by evaluating such functions at $N$ randomized points. The associated estimators, based on $NL$ model runs, reach the optimal rates of convergence. 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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., $\mathcal{O}(N^{-1})$), 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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