On one dimensional weighted Poincare inequalities for Global Sensitivity Analysis
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929618169102336 |
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| author | Heredia, David Joulin, Aldéric Roustant, Olivier |
| author_facet | Heredia, David Joulin, Aldéric Roustant, Olivier |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04918 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On one dimensional weighted Poincare inequalities for Global Sensitivity Analysis Heredia, David Joulin, Aldéric Roustant, Olivier Probability Statistics Theory 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. |
| title | On one dimensional weighted Poincare inequalities for Global Sensitivity Analysis |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2412.04918 |