Functional principal components analysis via penalized rank one approximation

Two existing approaches to functional principal components analysis(FPCA) are due to Rice and Silverman(1991) andSilverman(1996), both based on maximizing variance but introducing penalization in differ- ent 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 suggestsan efficient power algorithmfor 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) differ- ent smoothing parameters are permitted for different FPCs. The method- ology is illustrated with a real data example and a simulation. AMS 2000 subject classifications: Primary 62G08, 62H25; secondary 65F30.