Functional linear regression that’s interpretable

TLDR

Regression models linking a scalar outcome to a functional predictor increasingly use a coefficient function β(t) whose nonzero values identify regions of association and zeros indicate no relationship. The goal is to develop a regression procedure that yields β(t) estimates that are exactly zero where no relationship exists and have simple, interpretable structures elsewhere. FLiRTI applies variable‑selection to derivatives of β(t) to produce interpretable, flexible, and accurate estimates. The method achieves strong theoretical guarantees and performs well on both simulated and real datasets.

Abstract

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 $\intβ(t)X(t) dt$. Regions where $β(t)\ne0$ 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.