Doubly Spectral Analysis of Dependent Functional Data
Séminaire Probabilités & Statistique
27/03/2014 - 14:00 Victor M. Panaretos (Swiss Federal Institute of Technology (EPFL) / Institute of Mathematics) Salle 1 - Tour IRMA
Functional data analysis deals with statistical problems where each datum in a sample is an entire realization of a random function, the aim being to infer characteristics of the law of the random function on the basis of the data. Especially the second-order structure of functional data, as encoded by the covariance operator and its spectrum, is fundamental in the development of corresponding statistical methodology, through the celebrated Karhunen-Loève (KL) expansion: an 'optimal' random Fourier series representation of the function, that, on the one hand, serves as a basis for motivating methodology by analogy to multivariate statistics, and on the other hand, appears as the natural means of regularization in problems such as regression, testing and prediction, which are ill-posed in the functional case. With the aim of obtaining a similarly canonical representation of dependent functional data, we develop a doubly spectral analysis of a stationary functional stochastic process, decomposing it into an integral of uncorrelated functional frequency components (Cramér representation), each of which is in turn expanded into a KL series. This Cramér-Karhunen-Loève representation separates temporal from intrinsic curve variation, and it is seen to yield a harmonic principal component analysis when truncated: a finite dimensional proxy of the time series that optimally captures both within and between curve variation. The construction is based on the spectral density operator, the functional analogue of the spectral density matrix, whose eigenvalues and eigenfunctions at different frequencies provide the building blocks of the representation. Empirical versions are introduced, and a rigorous analysis of their large-sample behaviour is provided. (Based on joint work with S. Tavakoli, EPFL).