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.

本文提出了一种名为 FLiRTI 的方法，基于可变选择的思想，应用于 $eta (t)$ 的各种导数，以产生精确、灵活且可解释的估计，该方法在模拟和实际数据集上进行了演示，并提供了非渐近理论误差界限，为我们的方法提供了强有力的理论动机。

可解释的函数线性回归

Functional linear regression that's interpretable

We propose a generalized functional linear regression model for a regression
situation where the response variable is a scalar and the predictor is a random
function. A linear predictor is obtained by forming the scalar product of the
predictor function with a smooth parameter function, and the expected value of
the response is related to this linear predictor via a link function. If, in
addition, a variance function is specified, this leads to a functional
estimating equation which corresponds to maximizing a functional
quasi-likelihood. This general approach includes the special cases of the
functional linear model, as well as functional Poisson regression and
functional binomial regression. The latter leads to procedures for
classification and discrimination of stochastic processes and functional data.
We also consider the situation where the link and variance functions are
unknown and are estimated nonparametrically from the data, using a
semiparametric quasi-likelihood procedure. An essential step in our proposal is
dimension reduction by approximating the predictor processes with a truncated
Karhunen-Loeve expansion. We develop asymptotic inference for the proposed
class of generalized regression models. In the proposed asymptotic approach,
the truncation parameter increases with sample size, and a martingale central
limit theorem is applied to establish the resulting increasing dimension
asymptotics. We establish asymptotic normality for a properly scaled distance
between estimated and true functions that corresponds to a suitable L^2 metric
and is defined through a generalized covariance operator.

本文提出了一种广义的函数线性回归模型，包括函数线性模型、泊松回归和二项回归，并使用截断 Karhunen-Loeve 展开逼近预测过程进行降维，开发了一种适用于广义回归模型的渐近推断方法，其中截断参数随样本大小而增加，并应用控制变量的中心极限定理来建立维数的渐近增加，建立合适的 L^2 度量下的估计和真实函数之间的适当缩放距离的渐近正态性。