FUNCTIONAL LINEAR REGRESSION THAT'S INTERPRETABLE

摘要

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, _(t), with Y related to X(t) through ∫β(t)X (t) dt. Regions where _(t) ≠ 0 correspond to places where there is a relationship between X(t) and Y. Alternatively, points where _(t) = 0 in_dicate no relationship. Hence, for interpretation purposes, it is desirable for a regression procedure to be capable of producing estimates of _(t) that are exactly zero over regions with no apparent relationship and have simple struc_tures over the remaining regions. Unfortunately, most fitting procedures re_sult in an estimate for β(t) that is rarely exactly zero and has unnatural wig_gles 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 accu_rate. We call our method “Functional Linear Regression That's Interpretable” (FLiRTI) and demonstrate it on simulated and real-world data sets. In addi_tion, non-asymptotic theoretical bounds on the estimation error are presented. The bounds provide strong theoretical motivation for our approach.