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Publication# Random Surface Covariance Estimation by Shifted Partial Tracing

Abstract

The problem of covariance estimation for replicated surface-valued processes is examined from the functional data analysis perspective. Considerations of statistical and computational efficiency often compel the use of separability of the covariance, even though the assumption may fail in practice. We consider a setting where the covariance structure may fail to be separable locally-either due to noise contamination or due to the presence of a nonseparable short-range dependent signal component. That is, the covariance is an additive perturbation of a separable component by a nonseparable but banded component. We introduce nonparametric estimators hinging on the novel concept of shifted partial tracing, enabling computationally efficient estimation of the model under dense observation. Due to the denoising properties of shifted partial tracing, our methods are shown to yield consistent estimators even under noisy discrete observation, without the need for smoothing. Further to deriving the convergence rates and limit theorems, we also show that the implementation of our estimators, including prediction, comes at no computational overhead relative to a separable model. Finally, we demonstrate empirical performance and computational feasibility of our methods in an extensive simulation study and on a real dataset. Supplementary materials for this article are available online.

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This thesis focuses on non-parametric covariance estimation for random surfaces, i.e.~functional data on a two-dimensional domain. Non-parametric covariance estimation lies at the heart of functional data analysis, andconsiderations of statistical and computational efficiency often compel the use of separability of the covariance, when working with random surfaces. We seek to provide efficient alternatives to this ambivalent assumption.In Chapter 2, we study a setting where the covariance structure may fail to be separable locally -- either due to noise contamination or due to the presence of a non-separable short-range dependent signal component. That is, the covariance is an additive perturbation of a separable component by a non-separable but banded component. We introduce non-parametric estimators hinging on shifted partial tracing -- a novel concept enjoying strong denoising properties. We illustrate the usefulness of the proposed methodology on a data set of mortality surfaces.In Chapter 3, we propose a distinctive decomposition of the covariance, which allows us to understand separability as an unconventional form of low-rankness. From this perspective, a separable covariance has rank one. Allowing for a higher rank suggests a structured class in which any covariance can be approximated up to an arbitrary precision. The key notion of the partial inner product allows us to generalize the power iteration method to general Hilbert spaces and estimate the aforementioned decomposition from data. Truncation and retention of the leading terms automatically induces a non-parametric estimator of the covariance, whose parsimony is dictated by the truncation level. Advantages of this approach, allowing for estimation beyond separability, are demonstrated on the task of classification of EEG signals.While Chapters 2 and 3 propose several generalizations of separability in the densely sampled regime, Chapter 4 deals with the sparse regime, where the latent surfaces are observed only at few irregular locations. Here, a separable covariance estimator based on local linear smoothers is proposed, which is the first non-parametric utilization of separability in the sparse regime. The assumption of separability reduces the intrinsically four-dimensional smoothing problem into several two-dimensional smoothers and allows the proposed estimator to retain the classical minimax-optimal convergence rate for two-dimensional smoothers. The proposed methodology is used for a qualitative analysis of implied volatility surfaces corresponding to call options, and for prediction of the latent surfaces based on information from the entire data set, allowing for uncertainty quantification. Our quantitative results show that the proposed methodology outperforms the common approach of pre-smoothing every implied volatility surface separately.Throughout the thesis, we put emphasis on computational aspects, since those are the main reason behind the immense popularity of separability. We show that the covariance structures of Chapters 2 and 3 come with no (asymptotic) computational overhead relative to assuming separability. In fact, the proposed covariance structures can be estimated and manipulated with the same asymptotic costs as the separable model. In particular, we develop numerical algorithms that can be used for efficient inversion, as required e.g.~for prediction. All the methods are implemented in R and available on~GitHub.

Traditional approaches to analysing functional data typically follow a two-step procedure, consisting in first smoothing and then carrying out a functional principal component analysis. The idea underlying this procedure is that functional data are well approximated by smooth functions, and that rough variations are due to noise. However, it may very well happen that localised features are rough at a global scale but still smooth at some finer scale. In this thesis we put forward a new statistical approach for functional data arising as the sum of two uncorrelated components: one smooth plus one rough. We give non-parametric conditions under which the covariance operators of the smooth and of the rough components are jointly identifiable on the basis of discretely observed data: the covariance operator corresponding to the smooth component must be of finite rank and have real analytic eigenfunctions, while the one corresponding to the rough component must have a banded covariance function. We construct consistent estimators of both covariance operators without assuming knowledge of the true rank or bandwidth. We then use them to estimate the best linear predictors of the the smooth and the rough components of each functional datum. In both the identifiability and the inference part, we do not follow the usual strategy used in functional data analysis which is to first employ smoothing and work with continuous estimate of the covariance operator. Instead, we work directly with the covariance matrix of the discretely observed data, which allows us to use results and tools from linear algebra. In fact, we show that the whole problem of uniquely recovering the covariance operator of the smooth component given the one of the raw data can be seen as a low-rank matrix completion problem, and we make great use of a classical relation between the rank and the minors of a matrix to solve this matrix completion problem. The finite-sample performance of our approach is studied by means of simulation study.

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.