Penalized Nonparametric Scalar-on-Function Regression via Principal Coordinates

A number of classical approaches to nonparametric regression have recently been extended to the case of functional predictors. This article introduces a new method of this type, which extends intermediate-rank penalized smoothing to scalar-on-function regression. In the proposed method, which we call <i>principal coordinate ridge regression</i>, one regresses the response on leading principal coordinates defined by a relevant distance among the functional predictors, while applying a ridge penalty. Our publicly available implementation, based on generalized additive modeling software, allows for fast optimal tuning parameter selection and for extensions to multiple functional predictors, exponential family-valued responses, and mixed-effects models. In an application to signature verification data, principal coordinate ridge regression, with dynamic time warping distance used to define the principal coordinates, is shown to outperform a functional generalized linear model. Supplementary materials for this article are available online.

近年来，诸多经典非参数回归（nonparametric regression）方法已被推广至函数型预测变量（functional predictors）场景。本文提出一种此类新方法，将中间秩惩罚平滑推广至标量对函数回归（scalar-on-function regression）场景。所提方法被命名为主坐标岭回归（principal coordinate ridge regression）：该方法基于函数型预测变量间的相关距离定义主坐标，对前导主坐标进行响应变量回归，并施加岭惩罚。本文基于广义可加模型（generalized additive modeling）软件开发了公开可用的实现工具，可快速完成最优调参选择，且支持拓展至多函数型预测变量、指数族分布响应变量以及混合效应模型（mixed-effects models）。在签名验证数据集的应用中，采用动态时间规整（dynamic time warping）距离定义主坐标的主坐标岭回归，其性能优于函数型广义线性模型（functional generalized linear model）。本文补充材料可在线获取。