Functional linear regression that's interpretable

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, $\beta (t)$, with $Y$ related to $X(t)$ through $\int \beta (t)X(t)dt$. Regions where $\beta (t)\ne 0$ correspond to places where there is a relationship between $X(t)$ and $Y$. Alternatively, points where $\beta (t)=0$ indicate no relationship. Hence, for interpretation purposes, it is desirable for a regression procedure to be capable of producing estimates of $\beta (t)$ 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 $\beta (t)$ 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 $\beta (t)$, 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.

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)

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• Rejoinder to "Least angle regression" by Efron et al
• Least Angle Regression
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Generalized functional linear models

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