Functional Data Analysis: Key Concepts and Applications

Functional data analysis offers a variety of approaches for effectively managing observations in the form of curves. In this chapter, we provide an overview of common techniques for functional principal component analysis, modeling, comparison, prediction, and classification. These techniques have been widely studied and applied in the literature. In the framework of functional principal component analysis, we discuss methods for estimating the eigenfunctions and eigenvalues of covariance operation. We introduce techniques for estimating the mean and covariance of functional data, along with hypothesis testing for comparing mean functions. Specifically, we present two-sample tests and analysis of variance approaches for assessing differences in mean functions. Additionally, we present confidence bands obtained by a unified approach that handles both dense and sparse functional data plus other data that are of neither type but are under a general weighing scheme. Whether the response or the predictors, or both are curves, estimation of the slope function in the functional linear regression based on principal component analysis have been discussed. Furthermore, we suggest techniques for classifying functional data that are observed discretely based on a regular or irregular design with measurements that are contaminated by noise. The numerical performance of proposed approaches has been compared.