Principal Component Analysis for Functional Data

In functional principal component analysis (PCA), we treat the data that consist of functions not of vectors (Ramsay and Silverman, 1997). It is an attractive methodology, because we often meet the cases where we wish to apply PCA to such data. But, to make this method widely useful, it is desirable to study advantages and disadvantages in actual applications. As alternatives to functional PCA, we may consider multivariate PCA applied to 1) original observation data, 2) sampled functional data with appropriate intervals, and 3) coefficients of basis function expansion. Theoretical and numerical comparison is made among ordinary functional PCA, penalized functional PCA and the above three multivariate PCA.