The processes which govern the function of biological organisms are inherently dynamic and studying their behaviour over time is critical for gaining insight into their underlying mechanisms. They are also incredibly complex with tens of thousands of interacting variables comprising their state. In recent years, the development of high-throughput assaying technologies such as microarrays and nuclear magnetic resonance spectroscopy have revolutionised the fields of genomics and metabolomics respectively with their ability to quickly and easily interrogate these states at a single moment in time. When these assaying technologies are used to collect measurements repeatedly on the same biological unit, such as a human patient, laboratory rat or cell line, then the temporal behaviour of the system can begin to emerge. Furthermore, when several of these units are studied simultaneously then the experiment is said to be biologically replicated and such data sets permit the inference of systemic behaviour in the population as a whole. The time series data sets arising from these replicated `omics experiments possess unique characteristics that make for challenging statistical analysis. They are very short (3-10 time points is typical), heterogeneous, noisy, frequently irregularly sampled and often have missing observations, in addition to being very highly dimensional. To overcome some of these difficulties, researchers in the field of genomics have turned to functional data analysis, which has proven to be successful in modelling unreplicated data sets. Replicated data sets, however, have received far less attention, due to the complexity introduced by the extremely small sample sizes and multiple levels of variation - the between-variable and the between-replicate. Furthermore, despite the remarkable similarities between genomics and metabolomics time series data sets, these methods have been far less successful at establishing themselves in the latter field. In this thesis we present a general statistical framework for the analysis of replicated, high-dimensional biological time series data sets. Supported by three case studies, we develop novel models and algorithms for tackling the unique challenges that each data set presents. We show how these fitted models can be used in dimensionality reduction, summarising the thousands of observed time series into a small number of representative temporal profiles that are eminently biologically interpretable. We introduce a novel moderated functional t -statistic that can be used for detecting variables that differ significantly between two biological groups, leveraging the high dimensionality of the data in order to increase power. In all instances detailed simulation studies are used to demonstrate that the methods outperform existing state-of-the-art approaches. With practical data analysis in mind, careful consideration is given to the implementation of the methods in software that is computationally efficient, with parallel programming exploited wherever possible. In most instances, the methods have resulted in novel biological findings when applied to real data, and represent, as far as we are aware, the first application of such functional data analysis models to metabolomics time series experiments.

Cette thèse est consacrée à l'analyse théorique de problèmes non supervisés impliquant des séries temporelles à forte dépendance. Plus particulièrement, nous abordons les deux problèmes fondamentaux que sont le problème d'estimation des points de rupture et le partitionnement de séries temporelles. Ces problèmes sont abordés dans un cadre extrêmement général où les données sont générées par des processus stochastiques ergodiques stationnaires. Il s'agit de l'une des hypothèses les plus faibles en statistiques, comprenant non seulement, les hypothèses de modèles et les hypothèses paramétriques habituelles dans la littérature scientifique, mais aussi des hypothèses classiques d’indépendance, de contraintes sur l'espace mémoire ou encore des hypothèses de mélange.En particulier, aucune restriction n'est faite sur la forme ou la nature des dépendances, de telles sortes que les échantillons peuvent être arbitrairement dépendants. Pour chaque problème abordé, nous proposons de nouvelles méthodes non paramétriques et nous prouvons de plus qu'elles sont, dans ce cadre, asymptotiquement consistantes. Pour l'estimation de points de rupture, la consistance asymptotique se rapporte à la capacité de l'algorithme à produire des estimations des points de rupture qui sont asymptotiquement arbitrairement proches des vrais points de rupture. D'autre part, un algorithme de partitionnement est asymptotiquement consistant si le partitionnement qu'il produit, restreint à chaque lot de séquences, coïncides, à partir d'un certain temps et de manière consistante, avec le partitionnement cible. Nous montrons que les algorithmes proposés sont implémentables efficacement, et nous accompagnons nos résultats théoriques par des évaluations expérimentales.L'analyse statistique dans le cadre stationnaire ergodique est extrêmement difficile. De manière générale, il est prouvé que les vitesses de convergence sont impossibles à obtenir. Dès lors, pour deux échantillons générés indépendamment par des processus ergodiques stationnaires, il est prouvé qu'il est impossible de distinguer le cas où les échantillons sont générés par le même processus de celui où ils sont générés par des processus différents. Ceci implique que des problèmes tels le partitionnement de séries temporelles sans la connaissance du nombre de partitions ou du nombre de points de rupture ne peut admettre de solutions consistantes. En conséquence, une tâche difficile est de découvrir les formulations du problème qui en permettent une résolution dans ce cadre général. La principale contribution de cette thèse est de démontrer (par construction) que malgré ces résultats d'impossibilités théoriques, des formulations naturelles des problèmes considérés existent et admettent des solutions consistantes dans ce cadre général. Ceci inclut la démonstration du fait que le nombre de points de rupture corrects peut être trouvé, sans recourir à des hypothèses plus fortes sur les processus stochastiques. Il en résulte que, dans cette formulation, le problème des points de rupture peut être réduit à du partitionnement de séries temporelles.Les résultats présentés dans ce travail formulent les fondations théoriques pour l'analyse des données séquentielles dans un espace d'applications bien plus large. / This thesis is devoted to the theoretical analysis of unsupervised learning problems involving highly dependent time-series. Specifically, two fundamental problems are considered, namely, the problem of change point estimation as well as time-series clustering. The problems are considered in an extremely general framework, where the data are assumed to be generated by arbitrary, unknown stationary ergodic process distributions. This is one of the weakest assumptions in statistics: not only is it more general than the parametric and model-based settings, but it also subsumes most of the non-parametric frameworks considered for this class of problems, which typically include the assumption that each time-series consists of independent and identically distributed observations or that it satisfies certain mixing conditions. For each of the considered problems, novel nonparametric methods are proposed, and are further shown to be asymptotically consistent in this general framework. For change point estimation, asymptotic consistency refers to the algorithm's ability to produce change point estimates that are asymptotically arbitrarily close to the true change points. On the other hand, a clustering algorithm is asymptotically consistent, if the output clustering, restricted to each fixed batch of sequences, consistently coincides with the target clustering from some time on. The proposed algorithms are shown to be efficiently implementable, and the theoretical results are complemented with experimental evaluations. Statistical analysis in the stationary ergodic framework is extremely challenging. In general for this class of processes, rates of convergence (even of frequencies to respective probabilities) are provably impossible to obtain. As a result, given a pair of samples generated independently by stationary ergodic process distributions, it is provably impossible to distinguish between the case where they are generated by the same process or by two different ones. This in turn, implies that such problems as time-series clustering with unknown number of clusters, or change point detection, cannot possibly admit consistent solutions. Thus, a challenging task is to discover the problem formulations which admit consistent solutions in this general framework. The main contribution of this thesis is to constructively demonstrate that despite these theoretical impossibility results, natural formulations of the considered problems exist to admit consistent solutions in this general framework. Specifically, natural formulations of change-point estimation and time-series clustering are proposed, and efficient algorithms are provided, which are shown to be asymptotically consistent under the assumption that the process distributions are stationary ergodic. This includes the demonstration of the fact that the correct number of change points can be found, without the need to impose stronger assumptions on the process distributions. The results presented in this work lay down the theoretical foundations for the analysis of sequential data in a much broader range of real-world applications.

No description available.

new class of univariate time series models is developed, the Generalized Structural (GEST) time series model. The GEST model extends Gaussian structural time series models by allowing the distribution of the dependent variable to come from any parametric distribution, including highly skew and=or kurtotic distributions. Furthermore, the GEST model expands the systematic part of time series models to allow the explicit modelling of any or all of the distribution parameters as structural terms and (smoothed) functions of independent variables. The proposed GEST model primarily addresses the difficulty in modelling time-varying skewness and kurtosis (beyond location and dispersion time series models). The originality of the thesis starts from Chapter 6 and in particular Chapter 7 and Chapter 8, with applications of the GEST model in Chapter 9. Chapters 2 and 3 contain the literature review of non-Gaussian time series models, Chapter 4 is a reproduction of Chapter 17 in Pawitan (2001), which contains an alternative method for estimating the hyperparameters instead of using the Kalman filter, and Chapter 5 is an application of Chapter 4 to smoothing Gaussian structural time series models.

Nous abordons divers problèmes concernant les séries temporelles fonctionnelles. Il s'agit de processus stochastiques discrets à valeurs dans un espace fonctionnel. La principale motivation provient de l’interprétation séquentielle d'un phénomène continu. Si par exemple on observe des données météorologiques au cours du temps de manière continue, il est naturel de segmenter ce processus en une série temporelle fonctionnelle indexée par les jours. Chaque terme de la série représente la courbe journalière. Dans un premier temps, nous nous sommes intéressés à l'analyse spectrale. Plus précisément nous avons montré que sous des hypothèses très générales, la transformée de Fourier discrète d’une telle série est asymptotiquement normale et a pour variance l’opérateur de densité spectrale. Une application possible de ce résultat est de tester la présence de composantes périodiques dans une série fonctionnelle. Nous avons développé un test valable pour une fréquence arbitraire. Pour ce faire, nous avons étudié le comportement asymptotique du maximum de la norme de la transformée de Fourier. Enfin, nous avons travaillé sur la généralisation fonctionnelle du modèle GARCH. Ce modèle permet de décrire la dynamique de la volatilité, c’est-à-dire de la variance conditionnelle, dans les données financières. Nous avons proposé une méthode d’estimation des paramètres du modèle, inspirée de l’estimateur de quasi-maximum de vraisemblance. Nous avons montré que cet estimateur est convergent et asymptotiquement normal, puis nous l’avons évalué sur des simulations et appliqué à des données réelles. / In this thesis we address some issues related to functional time series, which consists in a discrete stochastic process valued in a functional space. The main motivation comes from a sequential approach of a continuous phenomenon. For example, if we observe some meteorological data continuously over time, then it is natural to segment this process into a functional series indexed by days, each term representing the daily curve. The first part is devoted to spectral analysis, more precisely we study the asymptotic behavior of the discrete Fourier transform. We show that, under very general conditions, the latter is asymptotically normal, with variance equal to the spectral density operator. An application of this result is the detection of periodic patterns in a functional time series. We develop a test to detect such patterns, which is valid for an arbitrary frequency. We show that the asymptotic distribution of the norm of the discrete Fourier transform belongs to the attraction domain of the Gumbel distribution. In a second part, we work on the functional generalization of the GARCH model. This model is used to describe the dynamics of volatility, i.e. conditional variance, in financial data. We propose an estimation method inspired by the quasi-maximum likelihood estimator, although the proper likelihood function does not exist in infinite dimension. We show that this estimator is convergent, asymptotic normal and we evaluate its performances on simulated and real data.

- / -

- / -

Ο σκοπός αυτής της μεταπτυχιακής εργασίας είναι διπλός και συγκεκριμένα αφορά στη μελέτη του απλού / γενικευμένου πολλαπλού μοντέλου παλινδρόμησης όταν σε αυτό παραβιάζεται μια από τις συνθήκες των Gauss-Markov και πιο συγκεκριμένα όταν, Cov{ε_i,ε_j }≠0, ∀ i≠j και στην ανάλυση χρονοσειρών. Αρχικά, γίνεται συνοπτική αναφορά στο απλό και στο πολλαπλό γραμμικό μοντέλο παλινδρόμησης, στις ιδιότητες καθώς και στις εκτιμήσεις των συντελεστών παλινδρόμησης. Περιγράφονται οι ιδιότητες των τυχαίων όρων όπως μέση τιμή, διασπορά, συντελεστές συσχέτισης κ.α., εφόσον υπάρχει παραβίαση της ιδιότητας της συνδιασποράς αυτών. Τέλος, περιγράφεται ο έλεγχος για αυτοσυσχέτιση των τυχαίων όρων των Durbin-Watson καθώς και μια ποικιλία διορθωτικών μέτρων με σκοπό την εξάλειψή της. Στο δεύτερο μέρος, αρχικά αναφέρονται βασικές έννοιες της θεωρίας των χρονοσειρών. Στη συνέχεια, γίνεται ανάλυση διαφόρων στάσιμων χρονοσειρών και συγκεκριμένα, ξεκινώντας από το λευκό θόρυβο, παρουσιάζονται οι χρονοσειρές κινητού μέσου (ΜΑ), οι αυτοπαλινδρομικές χρονοσειρές (ΑR), οι χρονοσειρές ARMA, καθώς και η γενική περίπτωση μη στάσιμων χρονοσειρών, των ΑRΙΜΑ χρονοσειρών και παρατίθενται συνοπτικά τα πρώτα στάδια ανάλυσης μιας χρονοσειράς για κάθε μια από τις περιπτώσεις αυτές. Η εργασία αυτή βασίστηκε σε δύο σημαντικά βιβλία διακεκριμένων επιστημόνων, του κ. Γεώργιου Κ. Χρήστου, Εισαγωγή στην Οικονομετρία και στο βιβλίο των John Neter, Michael H. Kutner, Christofer J. Nachtsheim και William Wasserman, Applied Linear Regression Models. / The purpose of this thesis is twofold, namely concerns the study of the simple / generalized multiple regression model when this violated one of the conditions of Gauss-Markov specifically when, Cov {e_i, e_j} ≠ 0, ∀ i ≠ j and time series analysis. Initially, there is a brief reference to the simple and multiple linear regression model, the properties and estimates of regression coefficients. Describe the properties of random terms such as mean, variance, correlation coefficients, etc., if there is a breach of the status of their covariance. Finally, described the test for autocorrelation of random terms of the Durbin-Watson and a variety of corrective measures to eliminate it. In the second part, first mentioned basic concepts of the theory of time series. Then, various stationary time series analyzes and specifically, starting from the white noise, the time series moving average presented (MA), the aftopalindromikes time series (AR) time series ARMA, and the general case of non-stationary time series of ARIMA time series and briefly presents the first analysis steps in a time series for each of these cases. This work was based on two important books of distinguished scientists, Mr. George K. Christou, Introduction to Econometrics, and in the book of John Neter, Michael H. Kutner, Christofer J. Nachtsheim and William Wasserman, Applied Linear Regression Models.

This thesis investigates the statistical properties of Kernel Smoothed Empirical Likelihood (KSEL, e.g. Smith, 1997 and 2004) estimator and various associated inference procedures in weakly dependent data. New tests for structural stability are proposed and analysed. Asymptotic analyses and Monte Carlo experiments are applied to assess these new tests, theoretically and empirically. Chapter 1 reviews and discusses some estimation and inferential properties of Empirical Likelihood (EL, Owen, 1988) for identically and independently distributed data and compares it with Generalised EL (GEL), GMM and other estimators. KSEL is extensively treated, by specialising kernel-smoothed GEL in the working paper of Smith (2004), some of whose results and proofs are extended and refined in Chapter 2. Asymptotic properties of some tests in Smith (2004) are also analysed under local alternatives. These special treatments on KSEL lay the foundation for analyses in Chapters 3 and 4, which would not otherwise follow straightforwardly. In Chapters 3 and 4, subsample KSEL estimators are proposed to assist the development of KSEL structural stability tests to diagnose for a given breakpoint and for an unknown breakpoint, respectively, based on relevant work using GMM (e.g. Hall and Sen, 1999; Andrews and Fair, 1988; Andrews and Ploberger, 1994). It is also original in these two chapters that moment functions are allowed to be kernel-smoothed after or before the sample split, and it is rigorously proved that these two smoothing orders are asymptotically equivalent. The overall null hypothesis of structural stability is decomposed according to the identifying and overidentifying restrictions, as Hall and Sen (1999) advocate in GMM, leading to a more practical and precise structural stability diagnosis procedure. In this framework, these KSEL structural stability tests are also proved via asymptotic analysis to be capable of identifying different sources of instability, arising from parameter value change or violation of overidentifying restrictions. The analyses show that these KSEL tests follow the same limit distributions as their counterparts using GMM. To examine the finite-sample performance of KSEL structural stability tests in comparison to GMM's, Monte Carlo simulations are conducted in Chapter 5 using a simple linear model considered by Hall and Sen (1999). This chapter details some relevant computational algorithms and permits different smoothing order, kernel type and prewhitening options. In general, simulation evidence seems to suggest that compared to GMM's tests, these newly proposed KSEL tests often perform comparably. However, in some cases, the sizes of these can be slightly larger, and the false null hypotheses are rejected with much higher frequencies. Thus, these KSEL based tests are valid theoretical and practical alternatives to GMM's.

Η μελέτη αυτή ασχολείται με την ανάλυση των χρονολογικών σειρών ως αντικείμενο κατανόησης του παρελθόντος και πρόβλεψης του μέλλοντος. Στα πρώτα κεφάλαια γίνεται μια εισαγωγή στις χρονολογικές σειρές, ποια η χρησιμότητα τους και τι μπορούν αυτές να περιγράψουν, καθώς αναλύονται ορισμένες βασικές έννοιες αυτών, όπως διάφορα μέτρα και στασιμότητα, και αναλύονται χαρακτηριστικά όπως η τάση, η περιοδικότητα κ.α. Έπειτα εξετάζονται συγκεκριμένες κατηγορίες χρονολογικών σειρών, όπως είναι οι στάσιμες και μη στάσιμες, και γίνεται εκτίμηση των παραμέτρων των παραπάνω σειρών με διαφορές μεθόδους. Τέλος, παρουσιάζονται διάφοροι μέθοδοι πρόβλεψης με την βοήθεια των χρονολογικών σειρών και γίνεται εφαρμογή αυτών των μεθόδων. / This work deals with time series analysis in a way to understand the past and predict the future. In the firsts chapters an introduction to time series was presented, in order to figure out their usage and what they can describe such as, main concerns’ analysis, (e.g. measures, stationary) and various other analysis characteristics as trend, variation, etc. In addition specific categories of time series were examined, like stationary and non stationary time series, and estimate same of their parameters with various methods. In the final chapters, various estimation methods were presented with the help of time series and who these methods are applied in practice.