Total interaction index: A variance-based sensitivity index for interaction screening

Jana Fruth (1) , Olivier Roustant (2, 3) , Sonja Kuhnt (1)

1 TU - Technische Universität Dortmund [Dortmund]
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
3 DEMO-ENSMSE - Département Décision en Entreprise : Modélisation, Optimisation

We consider the problem of investigating the interaction structure of a multivariate function, possibly containing high order interactions. By analogy with the total index, used to detect the most influential variables, interaction screening can be done with the so-called \textit{total interaction index} (TII), defined as the superset importance (\citet{Liu_Owen_2006}) of a pair of variables. One motivation is that block-additive structures can be recovered from the graph of active total interaction indices. Our aim is to investigate the total interaction index. At the theoretical level, it is connected to (usual) total indices and closed indices, and we also show that the TII is obtained by averaging the second order interaction of a 2-dimensional function obtained by fixing the original one. Then estimation is considered, and we prove the asymptotical efficiency of Liu and Owen's estimate. Its superiority is confirmed empirically, compared to other recent methods using FAST and RBD-FAST techniques. Finally, an application is given to recover a block-additive structure of a function, without knowledge about the interaction orders nor about the blocks.

Mots clés: Sensitivity analysis; FANOVA decomposition; Sobol indices; High-order interactions; Superset importance; Additive structure; Computer experiments

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Calcul [stat.CO]

Jana Fruth, Olivier Roustant, Sonja Kuhnt. Total interaction index: A variance-based sensitivity index for interaction screening. 2012. ⟨hal-00631066v2⟩