As part of the study of an input-output numerical simulator, performing a sensitivity analysis allows to identify the input parameters having the greatest influence on the output variability. Since the variance-based approach (also known as the ANOVA framework) is too expensive, and the kernel-based approach (leading to HSIC indices) lacks interpretability, the HSIC-ANOVA framework has recently emerged to marry the advantages of both. A major particularity of this new methodology is the need to use the unanchored Sobolev kernels. This paper investigates how sensitivity is measured according to the chosen Sobolev kernel. To achieve this, at least one explicit feature map is extracted from each Sobolev kernel and this helps identify the dependence patterns captured by HSIC-ANOVA indices. For the Sobolev kernel of order r = 1, three different proof techniques are proposed to disclose its Mercer feature map. For higher-order Sobolev kernels (r ≥ 2), it is proved that the Mercer feature map does not have a closed-form expression. In response, a slightly relaxed feature map is obtained after considering a sub-kernel decomposition. This latter feature map allows to justify why the Sobolev kernels of order r ≥ 2 should not be used to estimate HSIC-ANOVA indices.
Authors
Gabriel Sarazin, Amandine Marrel, Sebastien da Veiga, Vincent Chabridon
- Bibliographic Reference
- Gabriel Sarazin, Amandine Marrel, Sebastien da Veiga, Vincent Chabridon. New insights into the feature maps of Sobolev kernels: application in global sensitivity analysis. 2023. ⟨cea-04320711⟩
- Department
- Service d'Etudes des Systèmes Innovants
- HAL Collection
- ["CEA - Commissariat à l'énergie atomique", 'Université Toulouse 2', 'Université Paul Sabatier - Toulouse III', "Groupe des Écoles Nationales d'Économie et Statistique", 'CNRS - Centre national de la recherche scientifique', 'Institut National des Sciences Appliquées de Toulouse', 'CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions', 'Institut de Mathématiques de Toulouse', "Ensai, Ecole Nationale de la Statistique et de l'Analyse de l'Information", 'Université Toulouse 1 Capitole', 'Direction des énergies', 'CEA - Université Paris-Saclay', 'Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes', 'Université Paris-Saclay', 'EDF', 'DES Cadarache', 'CEA Cadarache', 'DES Saclay', 'Groupe INSA', 'ANR', 'Université Toulouse 3', 'Université Toulouse III / Toulouse INP']
- HAL Identifier
- 4320711
- Institution
- ["Commissariat à l'énergie atomique et aux énergies alternatives", 'Université Paris-Saclay', 'Université Toulouse Capitole', 'Institut National des Sciences Appliquées - Toulouse', 'Université Toulouse - Jean Jaurès', 'Université Toulouse III - Paul Sabatier', "Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz]", 'EDF']
- Laboratory
- ['Département de Modélisation des Systèmes et Structures', 'Département Etude des Réacteurs', 'Institut de Mathématiques de Toulouse UMR5219', 'Performance, Risque Industriel, Surveillance pour la Maintenance et l’Exploitation']
- Published in
- France
