Functional Regression
Functional Regression
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DOI：
10.1146/annurev-statistics-010814-020413
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发表时间：
2015-01-01
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通讯作者：
Morris, Jeffrey S.
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文献类型：
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作者：
Morris, Jeffrey S.
功能数据分析（FDA）涉及对数据的分析，这些数据的理想观测单位是在某个连续域上定义的函数，观察到的数据由从某个总体中抽取的函数样本组成，在一个离散网格上采样。Ramsay和Silverman（1997）的教科书激发了这一领域的发展，在过去10年里，由于产生这类数据的应用程序越来越多，这一领域已经加速成为统计学中发展最快的领域之一。FDA的一个独特特点是需要将功能之间和功能内部的信息结合起来，Ramsay和Silverman分别称之为复制和正则化。本文重点介绍了功能回归，这是FDA在应用和方法发展方面最受关注的领域。首先，介绍了基函数，函数回归方法中正则化的关键构建块，然后概述了函数回归方法，分为三种类型：(a)函数预测回归（函数上的标量），(b)函数响应回归（函数上的标量）和(c)函数上的函数回归。对于每一种方法，都讨论了复制和正则化的作用，并以大致按时间顺序描述了方法的发展，有时会偏离历史时间轴，将类似的方法组合在一起。主要的焦点是建模和方法论，强调已经开发的建模结构和所采用的各种正则化方法。本文最后简要讨论了该领域未来发展的潜在领域。
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 and the 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.