No abstract

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. Academic PressAMS classification numbers: 60G12, 46C05, 47B40, 46A35.

Functional data analysis, Hilbert spaces, Principal components, Covariance estimation, Orthogonal projection,

The following result in the theory of numerical ranges in Banach algebras is well known (see [3, Theorem 12.2]). The numerical range of an element F in the bidual of a unital Banach algebra A is the closure of the set of values at F of the w*-continuous states of . As a consequence of the results in this paper the following

In this paper, a linear model for forecasting a continuous‐time stochastic process in a future interval in terms of its evolution in a past interval is developed. This model is based on linear regression of the principal components in the future against the principal components in the past. In order to approximate the principal factors from discrete observations of a set of regular sample paths, cubic spline interpolation is used. An application for forecasting tourism evolution in Granada is also included. © 1997 by John Wiley & Sons, Ltd.