Publication
This work is about time series of functional data (functional time series), and consists of three main parts. In the first part (Chapter 2), we develop a doubly spectral decomposition for functional time series that generalizes the Karhunen–Loève expansion. In the second part (Chapter 3), we develop the theory of estimation for the spectral density operators, which are the main tool involved in the doubly spectral decomposition. The third part (Chapter 4) is concerned with the problem of understanding and comparing the dynamics of DNA. It proposes a methodology for comparing the dynamics of DNA minicircles that are vibrating in solution, using tools developed in this thesis. In the first part, we develop a doubly spectral representation of a stationary functional time series that generalizes the Karhunen–Loève expansion to the functional time series setting. The representation decomposes the time series into an integral of uncorrelated frequency components (Cramér representation), each of which is in turn expanded in a Karhunen-Loève series, thus yielding a Cramér–Karhunen–Loève decomposition of the series. The construction is based on the spectral density operators—whose Fourier coefficients are the lag-t autocovariance operators—which characterise the second-order dynamics of the process. The spectral density operators are the functional analogues of the spectral density matrices, whose eigenvalues and eigenfunctions at different frequencies provide the building blocks of the representation. By truncating the representation at a finite level, we obtain a harmonic principal component analysis of the time series, an optimal finite dimensional reduction of the time series that captures both the temporal dynamics of the process, and the within-curve dynamics, and dominates functional PCA. The proofs rely on the construction of a stochastic integral of operator-valued functions, whose construction is similar to that of the Itô integral. In practice, the spectral density operators are unknown. In the second part, we therefore develop the basic theory of a frequency domain framework for drawing statistical inferences on the spectral density operators of a stationary functional time series. Our main tool is the functional Discrete Fourier Transform(fDFT).We derive an asymptotic Gaussian representation of the fDFT, thus allowing the transformation of the original collection of dependent random functions into a collection of approximately independent complex-valued Gaussian random functions. Our results are then employed in order to construct estimators of the spectral density operators based on smoothed versions of the periodogram kernel, the functional generalisation of the periodogram matrix. The consistency and asymptotic law of these estimators are studied in detail. As immediate consequences, we obtain central limit theorems for the mean and the long-run covariance operator of a stationary functional time series. Our results do not depend on struct