arXiv subjects

David Heredia

Publications and source records attributed to David Heredia.

3 recordsLinked to original sources

arXiv2026

On weighted Poincar{\'e} inequalities for multivariate Liouville distributions -- Application to Global Sensitivity Analysis

In this work we establish weighted Poincar{\'e} inequalities for multivariate Liouville distributions, which are a generalization of the Dirichlet distribution. We also consider continuous elliptically contoured distributions, whose density levels are unions of hyperellipsoids. Our approach is based on a transport argument which allows weighted Poincar{\'e} inequalities to be transferred between probability measures. We apply our results to global sensitivity analysis and illustrate their practical use in a flood model case study, where the structure of dependence of the input variables is encoded by classical copulas.

math.FA↗

arXiv2025

Gradient-enhanced global sensitivity analysis with Poincar{\'e} chaos expansions

Spectral methods, also known as chaos expansions, are widely used in global sensitivity analysis (GSA), as they leverage orthogonal bases of L2 spaces to efficiently compute Sobol' indices, particularly in data-scarce settings. When derivatives of the model are available, a desirable property, both for modeling and GSA purposes, is for the derivatives of the basis functions to also form an orthogonal basis. We demonstrate that the only basis satisfying this property is the one associated with weighted Poincar{\'e} inequalities and Sturm--Liouville eigenvalue problems, which we call Poincar{\'e} basis. We also show that under certain conditions the Poincar{\'e} basis achieves the same convergence rate as the best polynomial approximation for classes of smooth functions. We then introduce a comprehensive framework for gradient-enhanced GSA that integrates recent advances both in the construction of the expansion - with gradient-enhanced regression - and in the construction of weights for derivative-based sensitivity analysis. Furthermore, the proposed methodology is applicable to a broad class of probability measures and various choices of weights. We illustrate its efficiency on a challenging flood modeling case study, where Sobol' indices are accurately estimated using limited data.

math.ST↗

arXiv2024

On one dimensional weighted Poincare inequalities for Global Sensitivity Analysis

One-dimensional Poincare inequalities are used in Global Sensitivity Analysis (GSA) to provide derivative-based upper bounds and approximations of Sobol indices. We add new perspectives by investigating weighted Poincare inequalities. Our contributions are twofold. In a first part, we provide new theoretical results for weighted Poincare inequalities, guided by GSA needs. We revisit the construction of weights from monotonic functions, providing a new proof from a spectral point of view. In this approach, given a monotonic function g, the weight is built such that g is the first non-trivial eigenfunction of a convenient diffusion operator. This allows us to reconsider the linear standard, i.e. the weight associated to a linear g. In particular, we construct weights that guarantee the existence of an orthonormal basis of eigenfunctions, leading to approximation of Sobol indices with Parseval formulas. In a second part, we develop specific methods for GSA. We study the equality case of the upper bound of a total Sobol index, and link the sharpness of the inequality to the proximity of the main effect to the eigenfunction. This leads us to theoretically investigate the construction of data-driven weights from estimators of the main effects when they are monotonic, another extension of the linear standard. Finally, we illustrate the benefits of using weights on a GSA study of two toy models and a real flooding application, involving the Poincare constant and/or the whole eigenbasis.

math.PR↗