Functional linear regression that's interpretable

摘要

Regression models to relate a scalar $Y$ to a functional predictor $X(t)$ arebecoming increasingly common. Work in this area has concentrated on estimatinga coefficient function, $\beta(t)$, with $Y$ related to $X(t)$ through$\int\beta(t)X(t) dt$. Regions where $\beta(t)\ne0$ correspond to places wherethere is a relationship between $X(t)$ and $Y$. Alternatively, points where$\beta(t)=0$ indicate no relationship. Hence, for interpretation purposes, itis desirable for a regression procedure to be capable of producing estimates of$\beta(t)$ that are exactly zero over regions with no apparent relationship andhave simple structures over the remaining regions. Unfortunately, most fittingprocedures result in an estimate for $\beta(t)$ that is rarely exactly zero andhas unnatural wiggles making the curve hard to interpret. In this article weintroduce a new approach which uses variable selection ideas, applied tovarious derivatives of $\beta(t)$, to produce estimates that are bothinterpretable, flexible and accurate. We call our method "Functional LinearRegression That's Interpretable" (FLiRTI) and demonstrate it on simulated andreal-world data sets. In addition, non-asymptotic theoretical bounds on theestimation error are presented. The bounds provide strong theoreticalmotivation for our approach.