  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 $\beta(t)\ne0$ 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.
