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变量、纵向、空间或时间序列设计记录了多个功能观察。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性的典型函数数据分析不同，它解释了函数观测值之间的复杂依赖关系，并且需要更高级的方法。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，由于需要对复杂依赖性进行建模而产生的关键方法复杂性。对于所呈现的四种类型的第二代函数数据（多变量函数数据、纵向函数数据、函数时间序列和空间函数数据）中的每一种，我们讨论了如何将广泛流行的函数主成分分析扩展到这些设置，以定义、识别变化的主要方向并描述函数之间的依赖关系。除了建模之外，我们还讨论了预测、统计推断和在聚类中的应用。最后，我们将讨论这一领域的未来发展方向。