Inverse regression for ridge recovery: A data-driven approach for parameter reduction in computer experiments
This work provides a theoretical foundation and warning for computational scientists using SDR for parameter reduction in deterministic computer models.
The paper addresses the problem of using sufficient dimension reduction (SDR) methods like SIR and SAVE for parameter reduction in computer experiments, where deterministic simulation data violate SDR assumptions. The authors show that these methods can be interpreted as estimating ridge function directions, but caution that they may miss important directions for non-ridge functions.
Parameter reduction can enable otherwise infeasible design and uncertainty studies with modern computational science models that contain several input parameters. In statistical regression, techniques for sufficient dimension reduction (SDR) use data to reduce the predictor dimension of a regression problem. A computational scientist hoping to use SDR for parameter reduction encounters a problem: a computer prediction is best represented by a deterministic function of the inputs, so data comprised of computer simulation queries fail to satisfy the SDR assumptions. To address this problem, we interpret SDR methods sliced inverse regression (SIR) and sliced average variance estimation (SAVE) as estimating the directions of a ridge function, which is a composition of a low-dimensional linear transformation with a nonlinear function. Within this interpretation, SIR and SAVE estimate matrices of integrals whose column spaces are contained in the ridge directions' span; we analyze and numerically verify convergence of these column spaces as the number of computer model queries increases. Moreover, we show example functions that are not ridge functions but whose inverse conditional moment matrices are low-rank. Consequently, the computational scientist should beware when using SIR and SAVE for parameter reduction, since SIR and SAVE may mistakenly suggest that truly important directions are unimportant.