  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.
