Regression models to relate a scalar Y to a functional predictor X(t) are
becoming increasingly common. Work in this area has concentrated on estimating
a coefficient function, β(t), with Y related to X(t) through
∫β(t)X(t)dt. Regions where β(t)î€ =0 correspond to places where
there is a relationship between X(t) and Y. Alternatively, points where
β(t)=0 indicate no relationship. Hence, for interpretation purposes, it
is desirable for a regression procedure to be capable of producing estimates of
β(t) that are exactly zero over regions with no apparent relationship and
have simple structures over the remaining regions. Unfortunately, most fitting
procedures result in an estimate for β(t) that is rarely exactly zero and
has unnatural wiggles making the curve hard to interpret. In this article we
introduce a new approach which uses variable selection ideas, applied to
various derivatives of β(t), to produce estimates that are both
interpretable, flexible and accurate. We call our method "Functional Linear
Regression That's Interpretable" (FLiRTI) and demonstrate it on simulated and
real-world data sets. In addition, non-asymptotic theoretical bounds on the
estimation error are presented. The bounds provide strong theoretical
motivation for our approach.Comment: Published in at http://dx.doi.org/10.1214/08-AOS641 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org