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Functional Regression

机译：功能回归

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Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay & Silverman's (1997) textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article focuses on functional regression, the area of FDA that has received the most attention in applications and methodological development. First, there is an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: (a) functional predictor regression (scalar-on-function), (b) functional response regression (function-on-scalar), and (c) function-on-function regression. For each, the role of replication and regularization is discussed andthe methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. The review concludes with a brief discussion describing potential areas of future development in this field.
机译：功能数据分析（FDA）涉及对数据的分析，这些数据的理想观察单位是在某个连续域上定义的功能，并且观察到的数据包括从某个总体中获取的功能样本，并在离散网格上进行采样。 Ramsay＆Silverman（1997）的教科书激发了该领域的发展，在过去的十年中，该领域的发展加速，成为统计领域增长最快的领域之一，这是由于产生此类数据的应用程序数量不断增加所推动的。 FDA的一个独特特征是需要在功能内和功能内合并信息，Ramsay和Silverman分别将其称为复制和正则化。本文关注功能回归，这是在应用和方法学开发中受到最多关注的FDA领域。首先，介绍基本函数，功能回归方法中用于正则化的关键构建块，然后概述功能回归方法，分为三类：（a）功能预测变量回归（标量函数），（b ）功能响应回归（标量函数）和（c）功能对功能回归。对于每种方法，都讨论了复制和正则化的作用，并大致按时间顺序描述了方法的发展，有时会偏离历史时间表，将相似的方法组合在一起。主要侧重于建模和方法论，重点介绍已开发的建模结构以及所采用的各种正则化方法。回顾以简短的讨论结束，描述了该领域未来发展的潜在领域。

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