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Publication# On the rate of convergence for the autocorrelation operator in functional autoregression

Résumé

We consider the problem of estimating the autocorrelation operator of an autoregressive Hilbertian process. By means of a Tikhonov approach, we establish a general result that yields the convergence rate of the estimated autocorrelation operator as a function of the rate of convergence of the estimated lag zero and lag one autocovariance operators. The result is general in that it can accommodate any consistent estimators of the lagged autocovariances. Consequently it can be applied to processes under any mode of observation: complete, discrete, sparse, and/or with measurement errors. An appealing feature is that the result does not require delicate spectral decay assumptions on the autocovariances but instead rests on natural source conditions. The result is illustrated by application to important special cases. (C) 2022 The Author(s). Published by Elsevier B.V.

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Time series modeling and analysis is central to most financial and econometric data modeling. With increased globalization in trade, commerce and finance, national variables like gross domestic productivity (GDP) and unemployment rate, market variables like indices and stock prices and global variables like commodity prices are more tightly coupled than ever before. This translates to the use of multivariate or vector time series models and algorithms in analyzing and understanding the relationships that these variables share with each other. Autocorrelation is one of the fundamental aspects of time series modeling. However, traditional linear models, that arise from a strong observed autocorrelation in many financial and econometric time series data, are at times unable to capture the rather nonlinear relationship that characterizes many time series data. This necessitates the study of nonlinear models in analyzing such time series. The class of bilinear models is one of the simplest nonlinear models. These models are able to capture temporary erratic fluctuations that are common in many financial returns series and thus, are of tremendous interest in financial time series analysis. Another aspect of time series analysis is homoscedasticity versus heteroscedasticity. Many time series data, even after differencing, exhibit heteroscedasticity. Thus, it becomes important to incorporate this feature in the associated models. The class of conditional heteroscedastic autoregressive (ARCH) models and its variants form the primary backbone of conditional heteroscedastic time series models. Robustness is a highly underrated feature of most time series applications and models that are presently in use in the industry. With an ever increasing amount of information available for modeling, it is not uncommon for the data to have some aberrations within itself in terms of level shifts and the occasional large fluctuations. Conventional methods like the maximum likelihood and least squares are well known to be highly sensitive to such contaminations. Hence, it becomes important to use robust methods, especially in this age with high amounts of computing power readily available, to take into account such aberrations. While robustness and time series modeling have been vastly researched individually in the past, application of robust methods to estimate time series models is still quite open. The central goal of this thesis is the study of robust parameter estimation of some simple vector and nonlinear time series models. More precisely, we will briefly study some prominent linear and nonlinear models in the time series literature and apply the robust S-estimator in estimating parameters of some simple models like the vector autoregressive (VAR) model, the (0, 0, 1, 1) bilinear model and a simple conditional heteroscedastic bilinear model. In each case, we will look at the important aspect of stationarity of the model and analyze the asymptotic behavior of the S-estimator.

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 structural modeling assumptions, but only functional versions of classical cumulant mixing conditions. The effect of discrete noisy observations on the consistency of the estimators is studied in a framework general enough to apply to a wide range of smoothing techniques for converting discrete noisy observations into functional data. We also perform a simulation study to assess the finite sample performance of our estimators, and give a discussion of the technical assumptions of our results, and at what cost our weak dependence assumptions could be changed or weakened, and provide examples of processes satisfying the technical assumptions of our asymptotic results. As an application, we consider in the third part the problem of comparing the dynamics of the trajectories of two DNA minicircles that are vibrating in solution, which are obtained via Molecular Dynamics simulations. The approach we take is to view and compare the dynamics through their spectral density operators, which contain the entire second-order structure of the trajectories. As a first step, we compare the spectral density operators of the two DNA minicircles using a new test we develop, which allows us to compare the spectral density operators at a fixed frequencies. Using multiple testing procedures, we are able to localize in frequencies the differences in spectral density operators of the two DNA minicircles, while controlling a type-I error, and conduct numerical simulations to assess the performance of our method. We further investigate the differences between the two minicircles by comparing their spectral density operators within frequencies. This allows us to localize their differences both in frequencies and on the minicircles, while controlling the averaged false discovery rate over the selected frequencies. Our methodology is general enough to be applied to the comparison of the dynamics of any pair of stationary functional time series.

Functional time series is a temporally ordered sequence of not necessarily independent random curves. While the statistical analysis of such data has been traditionally carried out under the assumption of completely observed functional data, it may well happen that the statistician only has access to a relatively low number of sparse measurements for each random curve. These discrete measurements may be moreover irregularly scattered in each curve's domain, missing altogether for some curves, and be contaminated by measurement noise. This sparse sampling protocol escapes from the reach of established estimators in functional time series analysis and therefore requires development of a novel methodology.
The core objective of this thesis is development of a non-parametric statistical toolbox for analysis of sparsely observed functional time series data. Assuming smoothness of the latent curves, we construct a local-polynomial-smoother based estimator of the spectral density operator producing a consistent estimator of the complete second order structure of the data. Moreover, the spectral domain recovery approach allows for prediction of latent curve data at a given time by borrowing strength from the estimated dynamic correlations in the entire time series across time. Further to predicting the latent curves from their noisy point samples, the method fills in gaps in the sequence (curves nowhere sampled), denoises the data, and serves as a basis for forecasting.
A classical non-parametric apparatus for encoding the dependence between a pair of or among a multiple functional time series, whether sparsely or fully observed, is the functional lagged regression model. This consists of a linear filter between the regressors time series and the response. We show how to tailor the smoother based estimators for the estimation of the cross-spectral density operators and the cross-covariance operators and, by means of spectral truncation and Tikhonov regularisation techniques, how to estimate the lagged regression filter and predict the response process.
The simulation studies revealed the following findings: (i) if one has freedom to design a sampling scheme with a fixed number of measurements, it is advantageous to sparsely distribute these measurements in a longer time horizon rather than concentrating over a shorter time horizon to achieve dense measurements in order to diminish the spectral density estimation error, (ii) the developed functional recovery predictor surpasses the static predictor not exploiting the temporal dependence, (iii) neither of the two considered regularisation techniques can, in general, dominate the other for the estimation in functional lagged regression models. The new methodologies are illustrated by applications to real data: the meteorological data revolving around the fair-weather atmospheric electricity measured in Tashkent, Uzbekistan, and at Wank mountain, Germany; and a case study analysing the dependence of the US Treasury yield curve on macroeconomic variables.
As a secondary contribution, we present a novel simulation method for general stationary functional time series defined through their spectral properties. A simulation study shows universality of such approach and superiority of the spectral domain simulation over the temporal domain in some situations.