Review of Functional Data Analysis

TL;DR

This paper provides a comprehensive overview of Functional Data Analysis (FDA), covering core techniques like FPCA, functional regression, clustering, and nonlinear methods, highlighting recent advances and future directions.

Contribution

It offers a broad summary of FDA methodologies, including recent developments in nonlinear approaches and applications, serving as a foundational reference.

Findings

FPCA is effective for dimension reduction and data imputation.

Functional linear regression is a key technique for modeling relationships.

Nonlinear methods like time warping and manifold learning show promise.

Abstract

With the advance of modern technology, more and more data are being recorded continuously during a time interval or intermittently at several discrete time points. They are both examples of "functional data", which have become a prevailing type of data. Functional Data Analysis (FDA) encompasses the statistical methodology for such data. Broadly interpreted, FDA deals with the analysis and theory of data that are in the form of functions. This paper provides an overview of FDA, starting with simple statistical notions such as mean and covariance functions, then covering some core techniques, the most popular of which is Functional Principal Component Analysis (FPCA). FPCA is an important dimension reduction tool and in sparse data situations can be used to impute functional data that are sparsely observed. Other dimension reduction approaches are also discussed. In addition, we review another core technique, functional linear regression, as well as clustering and classification of functional data. Beyond linear and single or multiple index methods we touch upon a few nonlinear approaches that are promising for certain applications. They include additive and other nonlinear functional regression models, such as time warping, manifold learning, and dynamic modeling with empirical differential equations. The paper concludes with a brief discussion of future directions.