Functional Data Analysis

Abstract

This chapter introduces functional data analysis (FDA) and selective topics in FDA, including functional principal component analysis (FPCA) and functional linear regression (FLR), with real data applications using a software package, which is publicly available. The methods in this chapter are based on local polynomial regression, a basic and important smoothing technique in nonparametric and semiparametric statistics. The approaches included in this chapter are not limited to the analysis of dense functional data but can also be used for the analysis of sparse functional/longitudinal data. In Sect. 4.1, we introduce FDA with some interesting examples of functional data and briefly describe FPCA and FLR. Section 4.2 details FPCA, one of the most important topics and tools in FDA. Topics such as the estimation of mean and covariance functions using nonparametric smoothing, choosing the number of principal components (PC) using subjective and objective methods, and prediction of trajectories are included and illustrated using a publicly available bike-sharing data set. Section 4.3 presents FLR based on FPCA described in Sect. 4.2. FLR is a generalization of traditional linear regression to the case of functional data. It is a powerful tool to model the relationship between functional/scalar response and functional predictors. This section is also illustrated using the same bike-sharing data set. We focus on the case when both response and predictor are functions in this section, but we mentioned other types of FLR topics in Sect. 4.4. Section 4.4 presents a short overview of other selected topics and software packages in FDA. These topics are either about functional data with more complex features than the simple and basic ones included in the previous two sections or about other statistical estimation and inference not covered before. The statistical software packages used in this chapter are written in Matlab and may be appropriate for the analysis of some basic types of functional data but not for others. Section 4.4 described other software packages written in different languages, such as R, and those packages have the flexibility to analyze various problems in functional data and different types of functional data.