  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.
