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Victor Panaretos, EPFL Print
Friday, 19 April 2013, 14:30 - 15:30

Victor Panaretos, EPFL

Doubly Spectral Analysis of Stationary Functional Time Series

Abstract: The spectral representation of random functions afforded by the celebrated Karhunen-Loève (KL) expansion has evolved into the canonical means of statistical analysis of independent functionaldata: allowing technology transfer from multivariate statistics, appearing as the natural means of regularization in inferential problems, and providing optimal finite dimensional reductions. With the aim of obtaining a similarly canonical representation of dependent functional data, we develop a doubly spectral analysis of a stationary functional time series, decomposing it into an integral of uncorrelated functional frequency components (Cramér representation), each of which is in turn is 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.

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