Statistical Learning for Functional Data

Functional data analysis (FDA) is a rapidly growing field in modern statistics that provides powerful tools for analyzing data observed as curves, surfaces, or more general functions. Unlike traditional multivariate methods, FDA explicitly accounts for the smooth and continuous nature of functional data, enabling more accurate modeling and interpretation. Traditional FDA methods, such as functional principal component analysis, functional regression, and functional classification, rely on linear assumptions and basis function expansions, which can limit their effectiveness when applied to nonlinear, high-dimensional, or irregularly sampled data. Recent advances in neural networks provide promising alternatives to these traditional approaches. Deep learning methods offer several key advantages: They naturally capture nonlinear relationships, scale to high-dimensional data without explicit dimension reduction, learn task-specific representations directly from raw observations, and handle sparse or irregular sampling without requiring imputation. This article reviews recent methodological developments in FDA, with a focus on the integration of deep learning techniques. Through this comparative review, we highlight the strengths and limitations of classical and modern approaches, providing practical guidance and future directions.