Kernel-based Measures of Association Between Inputs and Outputs Using ANOVA

Lamboni, Matieyendou

doi:10.1007/s13171-024-00354-w

Cited by 2 publications

(1 citation statement)

References 38 publications

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“…Using the fact that X ∼ F, with F(x) ̸ = ∏ d j=1 F j (x j ), we are able to model X as follows [8][9][10][11][12]14,43]:…”

Section: Stochastic Expressions Of the Gradients Of Functions With De...mentioning

confidence: 99%

Optimal and Efficient Approximations of Gradients of Functions with Nonindependent Variables

Lamboni

2024

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Gradients of smooth functions with nonindependent variables are relevant for exploring complex models and for the optimization of the functions subjected to constraints. In this paper, we investigate new and simple approximations and computations of such gradients by making use of independent, central, and symmetric variables. Such approximations are well suited for applications in which the computations of the gradients are too expansive or impossible. The derived upper bounds of the biases of our approximations do not suffer from the curse of dimensionality for any 2-smooth function, and they theoretically improve the known results. Also, our estimators of such gradients reach the optimal (mean squared error) rates of convergence (i.e., O(N−1)) for the same class of functions. Numerical comparisons based on a test case and a high-dimensional PDE model show the efficiency of our approach.

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“…Using the fact that X ∼ F, with F(x) ̸ = ∏ d j=1 F j (x j ), we are able to model X as follows [8][9][10][11][12]14,43]:…”

Section: Stochastic Expressions Of the Gradients Of Functions With De...mentioning

confidence: 99%

Optimal and Efficient Approximations of Gradients of Functions with Nonindependent Variables

Lamboni

2024

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Optimal Estimators of Cross-Partial Derivatives and Surrogates of Functions

Lamboni

2024

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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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Kernel-based Measures of Association Between Inputs and Outputs Using ANOVA

Cited by 2 publications

References 38 publications

Optimal and Efficient Approximations of Gradients of Functions with Nonindependent Variables

Optimal and Efficient Approximations of Gradients of Functions with Nonindependent Variables

Optimal Estimators of Cross-Partial Derivatives and Surrogates of Functions

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