Abstract

Recent technological developments have enabled us to collect complex and high-dimensional data in many scientific fields, such as population health, meteorology, econometrics, geology, and psychology. It is common to encounter such datasets collected repeatedly over a continuum. Functional data, whose sample elements are functions in the graphical forms of curves, images, and shapes, characterize these data types. Functional data analysis techniques reduce the complex structure of these data and focus on the dependences within and (possibly) between the curves. A common research question is to investigate the relationships in regression models that involve at least one functional variable. However, the performance of functional regression models depends on several factors, such as the smoothing technique, the number of basis functions, and the estimation method. This article provides a selective comparison for function-on-function regression models where both the response and predictor(s) are functions, to determine the optimal choice of basis function from a set of model evaluation criteria. We also propose a bootstrap method to construct a confidence interval for the response function. The numerical comparisons are implemented through Monte Carlo simulations and two real data examples.

摘要

最近的技术发展使我们能够收集许多科学领域的复杂和高维数据，例如人口健康、气象学、计量经济学、地质学和心理学。经常会遇到连续不断地重复收集的此类数据集。函数数据（其样本元素是曲线、图像和形状的图形形式的函数）表征这些数据类型。功能数据分析技术减少了这些数据的复杂结构，并专注于曲线内部和（可能）曲线之间的依赖关系。一个常见的研究问题是调查涉及至少一个函数变量的回归模型中的关系。然而，函数回归模型的性能取决于几个因素，例如平滑技术、基函数的数量、和估计方法。本文提供了函数对函数回归模型的选择性比较，其中响应变量和预测变量都是函数，以从一组模型评估标准中确定基函数的最佳选择。我们还提出了一种引导方法来构建响应函数的置信区间。数值比较是通过蒙特卡罗模拟和两个真实数据示例来实现的。