Nonparametric estimation of the first order Sobol indices with bootstrap bandwidth

TL;DR

This paper introduces a bootstrap-based kernel method for estimating first order Sobol indices, addressing bias issues in bandwidth selection and improving sensitivity analysis accuracy for unknown link functions.

Contribution

It proposes a novel bootstrap procedure to correct bias in kernel-based Sobol index estimation, enhancing robustness when the link function is unknown.

Findings

Bootstrap method reduces bias in Sobol index estimates.

Simulation results demonstrate improved accuracy over traditional methods.

Method effectively handles complex functions in sensitivity analysis.

Abstract

Suppose that $Y = \psi(X_1, \ldots, X_p)$, where $(X_1,\ldots, X_p)^\top$ are random inputs, $Y$ is the output, and $\psi(\cdot)$ is an unknown link function. The Sobol indices gauge the sensitivity of each $X$ against $Y$ by estimating the regression curve's variability between them. In this paper, we estimate these curves with a kernel-based method. The method allows to estimate the first order indices when the link between the independent and dependent variables is unknown. The kernel-based methods need a bandwidth to average the observations. For finite samples, the cross-validation method is famous to decide this bandwidth. However, it produces a structural bias. To remedy this, we propose a bootstrap procedure which reconstruct the model residuals and re-estimate the non-parametric regression curve. With the new set of curves, the procedure corrects the bias in the Sobol index. To test the developed method, we implemented simulated numerical examples with complex functions.