Second-Generation Functional Data

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.