Review of Functional Data Analysis

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.

47 pages

References in corpus (13)

• Prediction in functional linear regression
• Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data
• Defining probability density for a distribution of random functions
• Local polynomial regression on unknown manifolds
• Distance-based clustering of sparsely observed stochastic processes, with applications to online auctions
• Optimal estimation of the mean function based on discretely sampled functional data: Phase transition
• Functional single index models for longitudinal data
• Robust functional principal components: A projection-pursuit approach
• Estimation and model selection in generalized additive partial linear models for correlated data with diverging number of covariates
• Nonlinear manifold representations for functional data
• Functional linear regression via canonical analysis
• Factor models and variable selection in high-dimensional regression analysis
• Functional response additive model estimation with online virtual stock markets

Cited by in corpus (16)

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• On the use of reproducing kernel Hilbert spaces in functional classification
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• Learning the smoothness of noisy curves with application to online curve estimation
• Simultaneous inference for misaligned multivariate functional data
• Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres
• Universal kernel-type estimation of random fields
• A frequency domain analysis of the error distribution from noisy high-frequency data
• Search for Evergreens in Science: A Functional Data Analysis
• Functional Inference on Rotational Curves and Identification of Human Gait at the Knee Joint
• Sparse Wavelet Estimation in Quantile Regression with Multiple Functional Predictors
• Robust Functional Principal Component Analysis via Functional Pairwise Spatial Signs
• Factor Models for High-Dimensional Functional Time Series
• Estimating Historical Functional Linear Models with a Nested Group Bridge Approach
• Hypothesis Testing in Nonlinear Function on Scalar Regression with Application to Child Growth Study
• Parameter clustering in Bayesian functional PCA of fMRI data