Functional principal components analysis via penalized rank one approximation

Abstract

Two existing approaches to functional principal components analysis (FPCA) are due to Rice and Silverman (1991) and Silverman (1996), both based on maximizing variance but introducing penalization in different ways. In this article we propose an alternative approach to FPCA using penalized rank one approximation to the data matrix. Our contributions are four-fold: (1) by considering invariance under scale transformation of the measurements, the new formulation sheds light on how regularization should be performed for FPCA and suggests an efficient power algorithm for computation; (2) it naturally incorporates spline smoothing of discretized functional data; (3) the connection with smoothing splines also facilitates construction of cross-validation or generalized cross-validation criteria for smoothing parameter selection that allows efficient computation; (4) different smoothing parameters are permitted for different FPCs. The methodology is illustrated with a real data example and a simulation.

Published in at http://dx.doi.org/10.1214/08-EJS218 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Functional linear regression analysis for longitudinal data

Cited by in corpus (17)

• Smooth Principal Component Analysis over two-dimensional manifolds with an application to Neuroimaging
• Robust Regression For Image Binarization Under Heavy Noises and Nonuniform Background
• Sparse and Functional Principal Components Analysis
• Robust regularized singular value decomposition with application to mortality data
• Sampled forms of functional PCA in reproducing kernel Hilbert spaces
• Regularized Principal Component Analysis for Spatial Data
• Regularized Tensor Factorizations and Higher-Order Principal Components Analysis
• A two-way regularization method for MEG source reconstruction
• A Generalized Least Squares Matrix Decomposition
• Dynamic Multivariate Functional Data Modeling via Sparse Subspace Learning
• Multi-Rank Sparse and Functional PCA: Manifold Optimization and Iterative Deflation Techniques
• Functional Principal Component Analysis for Extrapolating Multi-stream Longitudinal Data
• Weighted least squares methods for prediction in the functional data linear model
• Representation and reconstruction of covariance operators in linear inverse problems
• Principal Component Analysis of Two-dimensional Functional Data with Serial Correlation
• Localized Functional Principal Component Analysis
• Structured penalties for functional linear models---partially empirical eigenvectors for regression