Modern studies from a variety of fields record multiple functional observations according to either multivariate, longitudinal, spatial, or time series designs. We refer to such data as second-generation functional data because their analysis—unlike typical functional data analysis, which assumes independence of the functions—accounts for the complex dependence between the functional observations and requires more advanced methods. In this article, we provide an overview of the techniques for analyzing second-generation functional data with a focus on highlighting the key methodological intricacies that stem from the need for modeling complex dependence, compared with independent functional data. For each of the four types of second-generation functional data presented—multivariate functional data, longitudinal functional data, functional time series and spatially functional data—we discuss how the widely popular functional principal component analysis can be extended to these settings to define, identify main directions of variation, and describe dependence among the functions. In addition to modeling, we also discuss prediction, statistical inference, and application to clustering. We close by discussing future directions in this area.

中文翻译：

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第二代功能数据

来自各个领域的现代研究根据多元、纵向、空间或时间序列设计记录了多种功能观察。我们将此类数据称为第二代函数数据，因为它们的分析与假设函数独立的典型函数数据分析不同，解释了函数观察之间的复杂依赖性，并且需要更先进的方法。在本文中，我们概述了分析第二代功能数据的技术，重点是强调与独立功能数据相比，由于对复杂依赖性进行建模的需要而产生的关键方法的复杂性。对于所提出的四种类型的第二代函数数据（多元函数数据、纵向函数数据、函数时间序列和空间函数数据）中的每一种，我们讨论如何将广泛流行的函数主成分分析扩展到这些设置来定义、识别主要变化方向，并描述功能之间的依赖性。除了建模之外，我们还讨论预测、统计推断和聚类应用。我们最后讨论该领域的未来方向。