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

Abstract

Sensitivity analysis aims at exploring which of a number of variables have an impact on a certain response. Not only are the individual variables of interest but also whether they interact or not. By analogy with the total sensitivity index, used to detect the most influential variables, a screening of interactions can be done efficiently with the so-called total interaction index (TII), defined as the superset importance of a pair of variables. Our aim is to investigate the TII, with a focus on statistical inference. At the theoretical level, we derive its connection to total and closed sensitivity indices. We present several estimation methods and prove the asymptotical efficiency of the Liu and Owen estimator. We also address the question of estimating the full set of TIIs, with a given budget of function evaluations. We observe that with the pick-and-freeze method the full set of TIIs can be estimated at a linear cost with respect to the problem dimension. The different estimators are then compared empirically. Finally, an application is given aiming at discovering a block-additive structure of a function, where no prior knowledge either about the interaction structure or about the blocks is available.

Variance Components and Generalized Sobol' Indices

“…An unbiased and nonnegative estimate is especially valuable. When the true GSI is zero, an unbiased nonnegative estimate will always return exactly zero as Fruth, Roustant, and Kuhnt (2012) remark. The Sobol' index τ 2 u is of square form, but τ 2 u is not.…”

“…Theorem 2.2 leads to a sum of squares for u |u|σ 2 u . Liu and Owen (2006) express the superset importance measure as a square: Fruth, Roustant, and Kuhnt (2012) find that a sample version of (4.1) is the best among four estimators of Υ 2 u . In classical crossed mixed effects models (Montgomery (1998)) every ANOVA expected mean square has a contribution from the highest order variance component, typically containing measurement error.…”

Mathematical Methods of Budget Modeling

“…In studies of different economic processes the concept of sensitivity [22,23] of interrelation of various technical and economic indicators is used. It can be absolute (rate of change in the indicator in response to the change in the factor) and relative (elasticity showing percentage of change in the indicator in response to a one-percent change in the factor).…”

A Parallel Optimization Algorithm based on FANOVA Decomposition

“…In the following, a short introduction of the most important concepts concerning the FANOVA decomposition is given. A more extensive overview of the theory is beyond the scope of this paper-we refer to Muehlenstaedt et al, 17 Sobol 27 and Fruth et al 28 for a comprehensive summary.…”

“…Apart from the metamodel-based computational costs, it can be shown that choosing an appropriate TII estimation method guarantees a linear increase of complexity with increasing problem dimensions. See Fruth et al 28 for a review of the estimation procedure.…”