Functional Principal Components Analysis by Choice of Norm

The functional principal components analysis (PCA) involves new considerations on the mechanism of measuring distances (the norm). Some properties arising in functional framework (e.g., smoothing) could be taken into account through an inner product in the data space. But this proposed inner product could make, for example, interpretational or (and) computational abilities worse. The results obtained in this paper establish equivalences between the PCA with the proposed inner product and certain PCA with a given well-suited inner product. These results have been proved in the theoretical framework given by Hilbert valued random variables, in which multivariate and functional PCAs appear jointly as particular cases.