Functional principal components analysis by choice of norm

The functional principal components analysis ( PCA) involves new considerat ions on the mechanism of measuring distances (the norm). Some properties ar ising in functional framework (e.g., smoothing) could be taken into account through an inner product in the data space. But this proposed inner produc t could make, for example, interpretational or (and) computational abilitie s worse. The results obtained in this paper establish equivalences between the PCA with the proposed inner product and certain PCA with a given well-s uited inner product. These results have been proved in the theoretical fram ework given by Hilbert valued random variables, in which multivariate and f unctional PCAs appear jointly as particular cases. (C) 1999 Academic Press AMS classification numbers: 60G12, 46C05, 47B40, 46A35.