Functional Partial Least-Squares: Adaptive Estimation and Inference

We study the linear regression model with a scalar response and a functional predictor, a canonical example of an ill-posed inverse problem. We show that the functional partial least-squares (PLS) estimator achieves convergence rates that are nearly minimax-optimal over a class of ellipsoids and propose an adaptive early-stopping procedure for selecting the number of PLS components. In addition, we develop a new test that detects parametric local alternatives. The test can be inverted to construct confidence sets for the functional slope parameter. Simulation results show that the estimator performs favorably relative to several existing methods, and that the proposed test has good power. We apply our methodology to evaluate the nonlinear effects of temperature on corn and soybean yields. We provide a Python software library, fpls, implementing our method. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

本研究聚焦于标量响应（scalar response）-函数型预测器（functional predictor）线性回归模型，这是不适定逆问题（ill-posed inverse problem）的经典范例。研究表明，函数型偏最小二乘（PLS）估计器在一类椭球（ellipsoids）假设下的收敛速率近乎达到极小极大最优（minimax-optimal），并提出了用于选取PLS分量个数的自适应早停程序（adaptive early-stopping procedure）。此外，本文构建了一种可检测参数型局部备择假设（parametric local alternatives）的全新检验方法，通过反转该检验可构建函数型斜率参数（functional slope parameter）的置信集（confidence sets）。模拟实验结果显示，所提估计器相较于多种现有方法表现更优，且本文提出的检验方法具备优良的检验功效。本文将所提研究方法应用于评估气温对玉米及大豆产量的非线性影响。同时，本文提供了实现所提方法的Python软件库fpls。本文的补充材料可在线获取，其中包含了用于复现本研究工作的标准化材料说明。