Functional principal components analysis via penalized rank one approximation

TL;DR

This paper introduces a novel penalized rank one approximation method for functional principal components analysis (FPCA), which improves regularization, incorporates spline smoothing, and allows flexible smoothing parameters, demonstrated through real data and simulations.

Contribution

It proposes an alternative FPCA approach using penalized rank one approximation, enhancing regularization, smoothing, and computational efficiency.

Findings

Efficient power algorithm for FPCA computation.

Incorporates spline smoothing naturally.

Flexible smoothing parameters for different FPCs.

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.