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The U.S. Coast Guard maintains a network structure to connect its nation-wide assets. This paper analyzes and models four highly aggregate traces of the traffic to/from the Coast Guard Data Network ship-shore nodes, so that the models may be used to predict future system demand. These internet traces (polled at 5â 40â intervals) are shown to adhere to a Gaussian distribution upon detrending, which imposes limits to the exponential distribution of higher time-resolution traces. Wavelet estimation of the Hurst-parameter is shown to outperform estimation by another common method (Sample-Variances). The First Differences method of detrending proved problematic to this analysis and is shown to decorrelate AR(1) processes where 0.65< phi1 <1.35 and correlate AR(1) processes with phi1 <-0.25. The Hannan-Rissanen method for estimating (phi,theta) is employed to analyze this series and a one-step ahead forecast is generated. / Master of Science

A utilização de modelos de teletrafego que contemplem caractersticas tais como autossimilaridade e dependencia de longa duraçao tem se mostrado cada vez mais como sendo ponto-chave na correta caracterizaçao do teletrafego Local Area Network (LAN) e Wide Area Network (WAN) [1, 2]. Tal caracterizaçao e necessaria para o monitoramento e controle de teletrafego em redes convergentes [3]. Nesse contexto, a questão da estimaçao precisa do parâmetro de autossimilaridade, denominado de parâmetro de Hurst, torna-se essencial. Entretanto, estudos comprovam que, alem da dependência de longa duraçao, redes WAN podem, não raramente, apresentar caractersticas mistas de dependência de longa e de curta duraçao [4, 5]. Enquanto vasta literatura cientca, tanto teorica como pratica, tem abordado com anco a questão da acuracia de diversos estimadores para o parâmetro de Hurst [6, 7, 8, 9], pouca atenção tem sido dada a questão da estimação deste parâmetro na presenca de dependência de curta duração. O presente trabalho de pesquisa concentrou-se no estudo dos metodos de estimaçao do parametro de Hurst baseados no espectro wavelet, em particular atraves do metodo de Abry-Veitch [10] { baseado na transformada Discrete Wavelet Transform (DWT) { e atraves do espectro obtido atraves da transformada Discrete Wavelet Packet Transform (DWPT). Os resultados baseados no metodo de Abry-Veitch demonstram que, atraves de um ajuste apropriado dos par^ametros de estimaçao, tal metodo permite uma estimaçao robusta na presenca de componentes com dependencia de curta duraçao, mesmo em situaçoes de mudanca de regime de tal componente, caracterstica desejavel para a estimaçao em tempo real do parametro de Hurst. Entretanto, a dispersao consideravel apresentada, em alguns casos, pelas estimativas do metodo de Abry-Veitch, motivou o estudo da utilizaçao do espectro wavelet obtido via transformada DWPT para realizaçao da estimaçao do parametro de Hurst. Os resultados indicam que a utilizaçao de tal transformada gera um espectro wavelet tal que e possvel detectar a presenca ou não de componentes com dependencia de curta duraçao. Ao final, os resultados da pesquisa realizada são sumarizados e utilizados em uma proposta de mecanismo de estimaçao do parametro de Hurst em tempo real, na presenca simultanea de componentes de dependencia de longa e curta duracão. / The use of network trac models that hold self-similar and long-range dependence characteristics have shown to be a key element on the correct characterization of Local Area Network (LAN) and Wide Area Network (WAN) network trac [1, 2]. Such characterization is necessary to monitor and control the network trac in converged networks [3]. In this context, the accurate estimation of the selfsimilarity parameter, named Hurst parameter, is a major issue. However, studies show that, besides the long-range dependence, WAN network trac may, not uncommonly, present mixed long and short-range dependence characteristics [4, 5]. While great part of either theoretical or practical scientic literature has been focused on the issue of Hurst parameter estimator accuracy [6, 7, 8, 9], little attention has been given to the estimation of such parameter in the presence of short-range dependence. This research work has focused on the study of the Hurst parameter estimation methods based on the wavelet spectrum, specially through the Abry-Veitch method [10] { which is based on the Discrete Wavelet Transform (DWT) transform { and through the wavelet spectrum based on the Discrete Wavelet Packet Transform (DWPT) transform. The results based on the Abry-Veitch method show that, through a suitable adjustment of the estimation parameters, such method yields a robust estimation in the presence of short-range dependence components, even in changing conditions of such component, a desirable characteristic for the real-time estimation of the Hurst parameter. However, the signi cant dispersion presented, occasionally, by the Abry-Veitch method estimates motivated the research of the usage of the wavelet spectrum obtained via DWPT transform to estimate the Hurst parameter. The results show that the usage of such transform generates such a wavelet spectrum that it is possible to detect whether short-range dependence components are present, or not, in the analyzed series. At the end, the research results are summarized and used to propose a realtime Hurst parameter estimation mechanism, in the presence of simultaneous long- and short-range dependence components.

La mémoire longue, aussi appelée la dépendance à long terme (LRD), est couramment détectée dans l’analyse de séries chronologiques dans de nombreux domaines, par exemple,en finance, en économétrie, en hydrologie, etc. Donc l’étude des séries temporelles à mémoire longue est d’une grande valeur. L’introduction du processus ARFIMA (fractionally autoregressive integrated moving average) établit une relation entre l’intégration fractionnaire et la mémoire longue, et ce modèle a trouvé son pouvoir de prévision à long terme, d’où il est devenu l’un des modèles à mémoire longue plus populaires dans la littérature statistique. Précisément, un processus à longue mémoire ARFIMA (p, d, q) est défini comme suit : Φ(B)(I − B)d (Xt − µ) = Θ(B)εt, t ∈ Z, où Φ(z) = 1 − ϕ1z − · · · − ϕpzp et Θ(z) = 1 + · · · + θ1zθpzq sont des polynômes d’ordre p et q, respectivement, avec des racines en dehors du cercle unité; εt est un bruit blanc Gaussien avec une variance constante σ2ε. Lorsque d ∈ (−1/2,1/2), {Xt} est stationnaire et inversible. Cependant, l’hypothèse a priori de la stationnarité des données réelles n’est pas raisonnable. Par conséquent, de nombreux auteurs ont fait leurs efforts pour proposer des estimateurs applicables au cas non-stationnaire. Ensuite, quelques questions se lèvent : quel estimateurs doit être choisi pour applications, et à quoi on doit faire attention lors de l’utilisation de ces estimateurs. Donc à l’aide de la simulation de Monte Carlo à échantillon fini, nous effectuons une comparaison complète des estimateurs semi-paramétriques, y compris les estimateurs de Fourier et les estimateurs d’ondelettes, dans le cadre des séries non-stationnaires. À la suite de cette étude comparative, nous avons que (i) sans bonnes échelles taillées, les estimateurs d’ondelettes sont fortement biaisés et ils ont généralement une performance inférieure à ceux de Fourier; (ii) tous les estimateurs étudiés sont robustes à la présence d’une tendance linéaire en temps dans le niveau de {Xt} et des effets GARCH dans la variance de {Xt}; (iii) dans une situation où le probabilité de transition est bas, la consistance des estimateurs quand même tient aux changements de régime dans le niveau de {Xt}, mais les changements ont une contamination au résultat d’estimation; encore, l’estimateur d’ondelettes de log-regression fonctionne mal dans ce cas; et (iv) en général, l’estimateur complètement étendu de Whittle avec un polynôme locale (fully-extended local polynomial Whittle Fourier estimator) est préféré pour une utilisation pratique, et cet estimateur nécessite une bande (i.e. un nombre de fréquences utilisés dans l’estimation) plus grande que les autres estimateurs de Fourier considérés dans ces travaux. / Long memory, also called long range dependence (LRD), is commonly detected in the analysis of real-life time series data in many areas; for example, in finance, in econometrics, in hydrology, etc. Therefore the study of long-memory time series is of great value. The introduction of ARFIMA (fractionally autoregressive integrated moving average) process established a relationship between the fractional integration and long memory, and this model has found its power in long-term forecasting, hence it has become one of the most popular long-memory models in the statistical literature. Specifically, an ARFIMA(p,d,q) process X, is defined as follows: cD(B)(I - B)d X, = 8(B)c, , where cD(z)=l-~lz-•••-~pzP and 8(z)=1-B1z- .. •-Bqzq are polynomials of order $p$ and $q$, respectively, with roots outside the unit circle; and c, is Gaussian white noise with a constant variance a2 . When c" X, is stationary and invertible. However, the a priori assumption on stationarity of real-life data is not reasonable. Therefore many statisticians have made their efforts to propose estimators applicable to the non-stationary case. Then questions arise that which estimator should be chosen for applications; and what we should pay attention to when using these estimators. Therefore we make a comprehensive finite sample comparison of semi-parametric Fourier and wavelet estimators under the non-stationary ARFIMA setting. ln light of this comparison study, we have that (i) without proper scale trimming the wavelet estimators are heavily biased and the y generally have an inferior performance to the Fourier ones; (ii) ail the estimators under investigation are robust to the presence of a linear time trend in levels of XI and the GARCH effects in variance of XI; (iii) the consistency of the estimators still holds in the presence of regime switches in levels of XI , however, it tangibly contaminates the estimation results. Moreover, the log-regression wavelet estimator works badly in this situation with small and medium sample sizes; and (iv) fully-extended local polynomial Whittle Fourier (fextLPWF) estimator is preferred for a practical utilization, and the fextLPWF estimator requires a wider bandwidth than the other Fourier estimators.

A utilização de modelos de teletrafego que contemplem caractersticas tais como autossimilaridade e dependencia de longa duraçao tem se mostrado cada vez mais como sendo ponto-chave na correta caracterizaçao do teletrafego Local Area Network (LAN) e Wide Area Network (WAN) [1, 2]. Tal caracterizaçao e necessaria para o monitoramento e controle de teletrafego em redes convergentes [3]. Nesse contexto, a questão da estimaçao precisa do parâmetro de autossimilaridade, denominado de parâmetro de Hurst, torna-se essencial. Entretanto, estudos comprovam que, alem da dependência de longa duraçao, redes WAN podem, não raramente, apresentar caractersticas mistas de dependência de longa e de curta duraçao [4, 5]. Enquanto vasta literatura cientca, tanto teorica como pratica, tem abordado com anco a questão da acuracia de diversos estimadores para o parâmetro de Hurst [6, 7, 8, 9], pouca atenção tem sido dada a questão da estimação deste parâmetro na presenca de dependência de curta duração. O presente trabalho de pesquisa concentrou-se no estudo dos metodos de estimaçao do parametro de Hurst baseados no espectro wavelet, em particular atraves do metodo de Abry-Veitch [10] { baseado na transformada Discrete Wavelet Transform (DWT) { e atraves do espectro obtido atraves da transformada Discrete Wavelet Packet Transform (DWPT). Os resultados baseados no metodo de Abry-Veitch demonstram que, atraves de um ajuste apropriado dos par^ametros de estimaçao, tal metodo permite uma estimaçao robusta na presenca de componentes com dependencia de curta duraçao, mesmo em situaçoes de mudanca de regime de tal componente, caracterstica desejavel para a estimaçao em tempo real do parametro de Hurst. Entretanto, a dispersao consideravel apresentada, em alguns casos, pelas estimativas do metodo de Abry-Veitch, motivou o estudo da utilizaçao do espectro wavelet obtido via transformada DWPT para realizaçao da estimaçao do parametro de Hurst. Os resultados indicam que a utilizaçao de tal transformada gera um espectro wavelet tal que e possvel detectar a presenca ou não de componentes com dependencia de curta duraçao. Ao final, os resultados da pesquisa realizada são sumarizados e utilizados em uma proposta de mecanismo de estimaçao do parametro de Hurst em tempo real, na presenca simultanea de componentes de dependencia de longa e curta duracão. / The use of network trac models that hold self-similar and long-range dependence characteristics have shown to be a key element on the correct characterization of Local Area Network (LAN) and Wide Area Network (WAN) network trac [1, 2]. Such characterization is necessary to monitor and control the network trac in converged networks [3]. In this context, the accurate estimation of the selfsimilarity parameter, named Hurst parameter, is a major issue. However, studies show that, besides the long-range dependence, WAN network trac may, not uncommonly, present mixed long and short-range dependence characteristics [4, 5]. While great part of either theoretical or practical scientic literature has been focused on the issue of Hurst parameter estimator accuracy [6, 7, 8, 9], little attention has been given to the estimation of such parameter in the presence of short-range dependence. This research work has focused on the study of the Hurst parameter estimation methods based on the wavelet spectrum, specially through the Abry-Veitch method [10] { which is based on the Discrete Wavelet Transform (DWT) transform { and through the wavelet spectrum based on the Discrete Wavelet Packet Transform (DWPT) transform. The results based on the Abry-Veitch method show that, through a suitable adjustment of the estimation parameters, such method yields a robust estimation in the presence of short-range dependence components, even in changing conditions of such component, a desirable characteristic for the real-time estimation of the Hurst parameter. However, the signi cant dispersion presented, occasionally, by the Abry-Veitch method estimates motivated the research of the usage of the wavelet spectrum obtained via DWPT transform to estimate the Hurst parameter. The results show that the usage of such transform generates such a wavelet spectrum that it is possible to detect whether short-range dependence components are present, or not, in the analyzed series. At the end, the research results are summarized and used to propose a realtime Hurst parameter estimation mechanism, in the presence of simultaneous long- and short-range dependence components.

Doctor of Philosophy (PhD) / This thesis mainly builds on the Variance Gamma (VG) model for financial assets over time of Madan & Seneta (1990) and Madan, Carr & Chang (1998), although the model based on the t distribution championed in Heyde & Leonenko (2005) is also given attention. The primary contribution of the thesis is the development of VG models, and the extension of t models, which accommodate a dependence structure in asset price returns. In particular it has become increasingly clear that while returns (log price increments) of historical financial asset time series appear as a reasonable approximation of independent and identically distributed data, squared and absolute returns do not. In fact squared and absolute returns show evidence of being long range dependent through time, with autocorrelation functions that are still significant after 50 to 100 lags. Given this evidence against the assumption of independent returns, it is important that models for financial assets be able to accommodate a dependence structure.

Doctor of Philosophy (PhD) / This thesis mainly builds on the Variance Gamma (VG) model for financial assets over time of Madan & Seneta (1990) and Madan, Carr & Chang (1998), although the model based on the t distribution championed in Heyde & Leonenko (2005) is also given attention. The primary contribution of the thesis is the development of VG models, and the extension of t models, which accommodate a dependence structure in asset price returns. In particular it has become increasingly clear that while returns (log price increments) of historical financial asset time series appear as a reasonable approximation of independent and identically distributed data, squared and absolute returns do not. In fact squared and absolute returns show evidence of being long range dependent through time, with autocorrelation functions that are still significant after 50 to 100 lags. Given this evidence against the assumption of independent returns, it is important that models for financial assets be able to accommodate a dependence structure.

Title: Long range dependence in time series Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Abstract: The diploma thesis demonstrates the necessity of a study of long range dependence, introduces fractional Gaussian noise and discusses possible definitions of long memory. It is done by notions of ergodic theory and by second moment characteristics and spectral density. These definitions are confronted with the model of fractional Gaussian noise and with intuitive understanding of long range memory. Relations and connections between these criteria are studied as well. The work is restricted to the study of discrete time processes. 1

Title: Long range dependence in time series Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Abstract: The diploma thesis demonstrates the necessity of a study of long range dependence, introduces fractional Gaussian noise and discusses possi- ble definitions of long memory. It is done by notions of ergodic theory and by second moment characteristics and spectral density. These definitions are confronted with the model of fractional Gaussian noise and with intuitive un- derstanding of long range memory. Relations and connections between these criteria are studied as well. The work is restricted to the study of discrete time processes. Method for Hurst index estimation for fractional Gaussian noise and it's application on logarithmic returns of shares of selected produ- cers of beer are included in this work. 1

<div>In spatio-temporal data analysis, the problem of non-separable space-time covariance functions is important and hard to deal with. Most of the famous constructions of these covariance functions are fully symmetric, which is inappropriate in many spatiotemporal processes. The Non-Fully Symmetric Space-Time (NFSST) Matern model by Zhang, T. and Zhang, H. (2015) provides a way to construct a non-fully symmetric non-separable space-time correlation function from marginal spatial and temporal Matern correlation functions.</div><div>In this work we use the relationship between the spatial Matern and temporal Cauchy correlation functions and their spectral densities, and provide a modification to their Bochner’s representation by including a space-time interaction term. Thus we can construct a non-fully symmetric space-time Matern-Cauchy model, from any given marginal spatial Matern and marginal temporal Cauchy correlation functions. We are able to perform computation and parameter estimate on this family, using the Taylor expansion of the correlation functions. This model has attractive properties: it has much faster estimation compared with NFSST Matern model when the spatio-temporal data is large; it enables the existence of temporal long-range dependence (LRD), adding substantially to the flexibility of marginal correlation function in the time domain. Several spatio-temporal meteorological data sets are studied using our model, including one case with temporal LRD.</div>