Block-diagonal covariance estimation and application to the Shapley effects in sensitivity analysis

Baptiste Broto (1) , François Bachoc (2) , Laura Clouvel , Jean-Marc Martinez (3)

1 LI3A (CEA, LIST) - Intelligence Artificielle et Apprentissage Automatique (CEA, LIST)
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 DM2S - Département de Modélisation des Systèmes et Structures

This paper deals with the estimation of sensitivity indices called “Shapley effects” when the model is linear and when the input vector is high dimensional with a Gaussian distribution. The computation cost of the Shapley effects makes it necessary to focus on the case where the input vector has a block-diagonal covariance matrix. First, we estimate a block-diagonal covariance matrix from Gaussian variables in high dimension. We prove that, under some mild assumptions, we find the block-diagonal structure of the matrix with probability that goes to one. We deduce an estimator of the covariance matrix that is as accurate as if the block-diagonal structure was known, with numerical applications. We also prove the asymptotic efficiency of a similar estimator in fixed dimension. Then, we apply this estimator for the estimation of the Shapley effects, in the Gaussian linear framework. We derive an estimator of the Shapley effects in high dimension with a relative error that converges to 0 at the parametric rate, up to a logarithmic factor. Finally, we apply the Shapley effects estimator on nuclear data.

Domaines: Statistiques [math.ST]

Baptiste Broto, François Bachoc, Laura Clouvel, Jean-Marc Martinez. Block-diagonal covariance estimation and application to the Shapley effects in sensitivity analysis. SIAM/ASA Journal on Uncertainty Quantification, 2022, 10, pp.379-403. ⟨10.1137/20M1358839⟩. ⟨hal-02196583v2⟩