Publications

A. Boudou, O. Caumont, S. Viguier-Pla (2004)
Principal Components Analysis in the Frequency Domain.
COMPSTAT 2004 proceedings
ISSN:

Classification

Keywords

principal components analysis, time series, stationarity, random measures, spectral analysis, applications

Abstract : David Brillinger (2001) proposes a method in order to summarize a p-dimensional times series (Xn), n in Z, by a q-dimensional times series (X'n), n in Z, q < p, where X'n=sum for m of C'mX(n-m). This method is performed with a principal components analysis (P.C.A.) of each spectral component, so it combines harmonic analysis and P.C.A.. However, when the spectrum is continuous, it cannot be put into practice because it would need the diagonalization of an infinity of matrices, and so we cannot compute the coefficients ..., C'(-1),C'(0),C'(1),.... We get round this difficulty by using a discretization of the spectrum and we substitute for this analysis an analysis which requires the diagonalization of a finite set of matrices. We show that, under certain assumptions, when the mesh spacing of the discretization tends to 0, the quality of the summary, that is to say of the ``approximate solution'', tends to the quality of the summary (X'n), n in Z. An application on meteorological data is given.