<p>This thesis, consisting of three papers and a summary, studies topics in the theory of stochastic processes related to long-range dependence. Much recent interest in such probabilistic models has its origin in measurements of Internet traffic data, where typical characteristics of long memory have been observed. As a macroscopic feature, long-range dependence can be mathematically studied using certain scaling limit theorems. </p><p>Using such limit results, two different scaling regimes for Internet traffic models have been identified earlier. In one of these regimes traffic at large scales can be approximated by long-range dependent Gaussian or stable processes, while in the other regime the rescaled traffic fluctuates according to stable ``memoryless'' processes with independent increments. In Paper I a similar limit result is proved for a third scaling scheme, emerging as an intermediate case of the other two. The limit process here turns out to be a non-Gaussian and non-stable process with long-range dependence.</p><p>In Paper II we derive a representation for the latter limit process as a stochastic integral of a deterministic function with respect to a certain compensated Poisson random measure. This representation enables us to study some further properties of the process. In particular, we prove that the process at small scales behaves like a Gaussian process with long-range dependence, while at large scales it is close to a stable process with independent increments. Hence, the process can be regarded as a link between these two processes of completely different nature.</p><p>In Paper III we construct a class of processes locally behaving as Gaussian and globally as stable processes and including the limit process obtained in Paper I. These processes can be chosen to be long-range dependent and are potentially suitable as models in applications with distinct local and global behaviour. They are defined using stochastic integrals with respect to the same compensated Poisson random measure as used in Paper II.</p>

This thesis, consisting of three papers and a summary, studies topics in the theory of stochastic processes related to long-range dependence. Much recent interest in such probabilistic models has its origin in measurements of Internet traffic data, where typical characteristics of long memory have been observed. As a macroscopic feature, long-range dependence can be mathematically studied using certain scaling limit theorems. Using such limit results, two different scaling regimes for Internet traffic models have been identified earlier. In one of these regimes traffic at large scales can be approximated by long-range dependent Gaussian or stable processes, while in the other regime the rescaled traffic fluctuates according to stable ``memoryless'' processes with independent increments. In Paper I a similar limit result is proved for a third scaling scheme, emerging as an intermediate case of the other two. The limit process here turns out to be a non-Gaussian and non-stable process with long-range dependence. In Paper II we derive a representation for the latter limit process as a stochastic integral of a deterministic function with respect to a certain compensated Poisson random measure. This representation enables us to study some further properties of the process. In particular, we prove that the process at small scales behaves like a Gaussian process with long-range dependence, while at large scales it is close to a stable process with independent increments. Hence, the process can be regarded as a link between these two processes of completely different nature. In Paper III we construct a class of processes locally behaving as Gaussian and globally as stable processes and including the limit process obtained in Paper I. These processes can be chosen to be long-range dependent and are potentially suitable as models in applications with distinct local and global behaviour. They are defined using stochastic integrals with respect to the same compensated Poisson random measure as used in Paper II.

This thesis examines the concept of image self-similarity and provides solutions to various associated inverse problems such as resolution enhancement and missing fractal codes. In general, many real-world inverse problems are ill-posed, mainly because of the lack of existence of a unique solution. The procedure of providing acceptable unique solutions to such problems is known as regularization. The concept of image prior, which has been of crucial importance in image modelling and processing, has also been important in solving inverse problems since it algebraically translates to the regularization procedure. Indeed, much recent progress in imaging has been due to advances in the formulation and practice of regularization. This, coupled with progress in optimization and numerical analysis, has yielded much improvement in computational methods of solving inverse imaging problems. Historically, the idea of self-similarity was important in the development of fractal image coding. Here we show that the self-similarity properties of natural images may be used to construct image priors for the purpose of addressing certain inverse problems. Indeed, new trends in the area of non-local image processing have provided a rejuvenated appreciation of image self-similarity and opportunities to explore novel self-similarity-based priors. We first revisit the concept of fractal-based methods and address some open theoretical problems in the area. This includes formulating a necessary and sufficient condition for the contractivity of the block fractal transform operator. We shall also provide some more generalized formulations of fractal-based self-similarity constraints of an image. These formulations can be developed algebraically and also in terms of the set-based method of Projection Onto Convex Sets (POCS). We then revisit the traditional inverse problems of single frame image zooming and multi-frame resolution enhancement, also known as super-resolution. Some ideas will be borrowed from newly developed non-local denoising algorithms in order to formulate self-similarity priors. Understanding the role of scale and choice of examples/samples is also important in these proposed models. For this purpose, we perform an extensive series of numerical experiments and analyze the results. These ideas naturally lead to the method of self-examples, which relies on the regularity properties of natural images at different scales, as a means of solving the single-frame image zooming problem. Furthermore, we propose and investigate a multi-frame super-resolution counterpart which does not require explicit motion estimation among video sequences.

This thesis examines the concept of image self-similarity and provides solutions to various associated inverse problems such as resolution enhancement and missing fractal codes. In general, many real-world inverse problems are ill-posed, mainly because of the lack of existence of a unique solution. The procedure of providing acceptable unique solutions to such problems is known as regularization. The concept of image prior, which has been of crucial importance in image modelling and processing, has also been important in solving inverse problems since it algebraically translates to the regularization procedure. Indeed, much recent progress in imaging has been due to advances in the formulation and practice of regularization. This, coupled with progress in optimization and numerical analysis, has yielded much improvement in computational methods of solving inverse imaging problems. Historically, the idea of self-similarity was important in the development of fractal image coding. Here we show that the self-similarity properties of natural images may be used to construct image priors for the purpose of addressing certain inverse problems. Indeed, new trends in the area of non-local image processing have provided a rejuvenated appreciation of image self-similarity and opportunities to explore novel self-similarity-based priors. We first revisit the concept of fractal-based methods and address some open theoretical problems in the area. This includes formulating a necessary and sufficient condition for the contractivity of the block fractal transform operator. We shall also provide some more generalized formulations of fractal-based self-similarity constraints of an image. These formulations can be developed algebraically and also in terms of the set-based method of Projection Onto Convex Sets (POCS). We then revisit the traditional inverse problems of single frame image zooming and multi-frame resolution enhancement, also known as super-resolution. Some ideas will be borrowed from newly developed non-local denoising algorithms in order to formulate self-similarity priors. Understanding the role of scale and choice of examples/samples is also important in these proposed models. For this purpose, we perform an extensive series of numerical experiments and analyze the results. These ideas naturally lead to the method of self-examples, which relies on the regularity properties of natural images at different scales, as a means of solving the single-frame image zooming problem. Furthermore, we propose and investigate a multi-frame super-resolution counterpart which does not require explicit motion estimation among video sequences.

Made available in DSpace on 2015-03-13T17:08:20Z (GMT). No. of bitstreams: 1 IrenaldoPCJ.pdf: 3773351 bytes, checksum: 66d713ebc6c47b7533bafff9f32c98e7 (MD5) Previous issue date: 2009-04-27 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this dissertation it was studied the rupture characteristic of earthquakes of the Town of Cascavel CE, Northeastern Brazil. Located on the border of the Potiguar Basin, the Town of Cascavel is one of the most seismically active intraplate areas in the country. In this town, on November 20th, 1980 a 5,2mb earthquake occurred. This was the largest earthquake ever reported in Northeast Brazil. Studies of this region using instruments were possible after 1989, with several campaigns being done using seismographic networks. From the beginning of the monitoring to April 2008 more than 55,000 events were recorded. With the data collected by a network with six 3-components digital seismographic stations during the campaigns done from September 29th, 1997 to March 5th, 1998, estimates of source parameters were found fitting the displacement spectra in the frequency domain for each event. From the fitting of the displacement spectra it was possible to obtain the corner frequency ( ) c f and long period amplitude ( ) W0 . Source parameters were determined following Brune (1970) and Madariaga (1976) models. Twenty-one seismic events were analyzed (0.7 ? ? 2.1) b m in order to estimate the source dimension (r ), seismic moment ( ) M0 , static stress drop (Ds ), apparent stress ( ) a s , seismic energy ( ) S E and moment magnitude ( ) W M for each of the events. It was observed that the ratio between radiated seismic energy and moment seismic (apparent stress) increases with increasing moment and hence magnitude at the observed range. As suggested by Abercrombie (1995), also in this work there is a breakdown in the scaling for earthquakes with magnitudes smaller than three ( < 3.0) W M , so that the rupture physics is different for larger events. If this assumption is valid, the earthquakes analyzed in this work are not selfsimilar. Thus, larger events tend to radiated more energy per unit area than smaller ones. / Nesta disserta??o foi estudada a caracter?stica de ruptura dos sismos da cidade de Cascavel CE, Nordeste Brasileiro. Localizada na borda da Bacia Potiguar, a cidade de Cascavel ? uma das ?reas intraplaca mais sismicamente ativa do Brasil. Neste munic?pio, no dia 20 de novembro de 1980, ocorreu o maior sismo de que se tem not?cia no Nordeste, com magnitude igual 5,2mb . A partir de 1989, essa regi?o tem sido estudada instrumentalmente, sendo realizadas diversas campanhas com redes sismogr?ficas. Desde o in?cio do monitoramento at? abril de 2008, foram registrados mais de 55.000 eventos. Com os dados coletados por uma rede de seis esta??es digitais triaxiais em uma campanha realizada entre 29 de setembro de 1997 e 05 de mar?o de 1998, foi realizado um estudo para determinar os par?metros de fonte, ajustando-se os espectros de deslocamento de cada sismo no dom?nio da frequ?ncia. A partir dos ajustes dos espectros de deslocamento, foi poss?vel obter os valores da frequ?ncia de corte ( ) c f e da amplitude de longo per?odo ( ) W0 . Os par?metros foram determinados a partir dos modelos de fonte propostos por Brune (1970) e Madariaga (1976) para 21 sismos (0,7 ? ? 2,1) b m , obtendo-se as estimativas do raio da fonte (r ), momento s?smico ( ) M0 , stress drop est?tico (Ds ), stress aparente ( ) a s , energia s?smica irradiada ( ) S E e magnitude momento ( ) W M de cada evento. Foi observado que o stress drop e a raz?o entre a energia irradiada e o momento s?smico (stress aparente) aumentam com o incremento do momento e, consequentemente, com o valor da magnitude para a escala investigada. Assim como sugerido por Abercrombie (1995), neste trabalho tamb?m parece haver um quebra na rela??o de escala para sismos com magnitudes menores que tr?s ( < 3,0) W M , o que implica em um processo de ruptura diferente para terremotos grandes e pequenos. Caso esta hip?tese seja v?lida, os sismos analisados neste trabalho n?o s?o autosimilares. Assim, os eventos maiores tendem a irradiar mais energia por unidade de ?rea que os menores.

Notre thèse est une contribution à la compréhension de la structure du prix du quota carbone, à l’analyse de sa fluctuation, et aux interactions pouvant exister entre d’une part le système communautaire d’échange de quota d’émission (SCEQE) et le mécanisme pour le développement propre (MDP) et d’autre part entre le marché du quota carbone et ceux du secteur de l’énergie. Pour réaliser cet objectif, nous faisons appel à l’approche temps-fréquence (ondelettes) qui permet d’analyser le comportement local d’un signal, en isolant les composantes relatives aux fréquences de trading des agents. Cette méthode permet de décomposer la variance d’une série chronologique en différentes composantes fréquentielles dont on peut suivre les évolutions au cours du temps. Les horizons temporels de placement sont affectés à chaque bande de fréquences liés à un degré de risque du marché. Les bandes à haute fréquence obtenues par la décomposition en ondelettes renferment l’information relative à la structure de la série chronologique à court terme et non; celle à moyenne et à basse fréquence captent respectivement les structures de la série chronologique à moyen et long terme. Pour concrétiser notre apport nous proposons un développement de notre recherche en 3 chapitres. Le premier présente le marché carbone, sa création, son fonctionnement, et les différents acteurs qui l’animent. La formation du prix du CO2, son évolution et ses facteurs déterminants. Le deuxième développe une analyse multidimensionnelle du comouvement entre le prix du quota (SCEQE) et du crédit carbone (MDP) selon deux hypothèses : la première est l’homogénéité du comportement des agents. La relation de comouvement entre le prix du quota (SCEQE) et du crédit carbone (MDP) suppose que les intervenants sur le marché disposent d’une même stratégie d’investissement. On utilise dans ce cas les outils économétriques standards (Analyse de la Cointégration, de la causalité, modélisation vecteur Autorégressif). Nous mettons en évidence l’existence d’une causalité unidirectionnelle du CER vers l’EUA conforme aux faits observés de l’accélération du mécanisme pour le développement propre (MDP). En effet les industriels portent un grand intérêt sur le MDP, et cela a eu un impact direct sur le prix de l’EUA sur le marché européen du carbone. Nous observons aussi une dynamique d’interconnexion par l’intermédiaire du VAR(1) entre l’EUA et le CER. La deuxième hypothèse est celle de l’hétérogénéité du comportement des agents. Nous étudions la relation de comouvement entre le prix du quota (SCEQE) et celui crédit carbone (MDP) par une analyse multi-échelle dérivée de la théorie des ondelettes. Nous montrons que quels que soient les différents horizons d’investissement retenus (court, moyen et long terme), il existe une relation positive significative entre les deux séries de prix. De plus La causalité dynamique par ondelettes sur chaque paire de bandes de fréquence, confirme notre hypothèse d’une relation instable entre le EUA et le CER. Le troisième chapitre complète le précédent par l’analyse du comouvement multi-échelle entre le prix du quota carbone issu du SCEQE et ceux des marchés énergétiques (pétrole, le Charbon et le Gaz). La cohérence par ondelettes pierre angulaire de cette étude peut être interprétée comme une mesure de corrélation locale calculée de façon non-paramétrique. Cette première tentative de l’analyse multi-échelle de la relation de comouvement entre le marché du CO2, du pétrole, du Gaz et du Charbon fondée sur l’hypothèse d’hétérogénéité des agents montre qu’il est possible aux intervenants sur le marché carbone d’avoir une grande palette de choix de stratégies leurs permettant de mieux se prémunir contre les risques liés aux fortes volatilités du prix du carbone. / The purpose of this thesis is to contribute to the understanding of the carbon price quota structure, the analysis of its fluctuation, and the interactions that may exist between on the one hand the EU system of emission quota exchange (ETS) and the mechanism for clean development (CDM) and on the other hand between carbon quota market and the quota in the energy sector. For this study, we introduced the hypothesis of heterogeneity of agents' behavior on the carbon market where coexist multiple scales of investments.To achieve this, we used a time-frequency approach (wavelets) that can analyze the local behavior of a signal by isolating the components related to the agents’ trading frequency. This method allows variance decomposition of a time series into different frequency components. The time investment horizons are assigned to each frequency band associated with a degree of market risk. High frequency bands obtained by wavelet decomposition contain information pertaining to the short-term time series structure; the medium and the low frequencies respectively capture the structures of the time series in the medium and long term.To complete this project we proposed developing of our research in 3 chapters:The first chapter presents the carbon market, its creation, its operation, and the various actors who animate it. The formation of the CO2 price, its evolution and its determinants were also analyzed during this chapter.The second chapter develops a multi-scale analysis of the co-movement between price of quota (ETS) and carbon credit (CDM). This particular study was based on two assumptions:- The first assumption was the homogeneity of the various agents’ behaviors. The co-movement relationship between the price of quota (ETS) and carbon credit (CDM) assumes that market participants have the same investment strategy especially since all stakeholders agents invest in these markets on the same horizons. This relationship is studied using standard econometric tools such as Analysis of Co-integration of causality and Vector Autoregressive modeling.The main result of this specific part highlights the existence of a unidirectional causality from the ERC to the EUA with an observed acceleration of the mechanism for clean development (CDM). Manufacturers have a strong interest in the CDM which has had a direct impact on the price of EUAs on the European carbon market. We also observed a dynamic interconnection through the VAR (1) between EUA and CER.- The second assumption is the heterogeneity of agents' behavior. We highlighted the relationship of co-movement between price of quota (ETS) and carbon credit (CDM) to a multi-scale analysis derived from wavelet theory. We show that whatever the different investment horizons retained (short, medium and long term), there is a significant positive relationship between the two sets of prices. The more dynamic causal wavelet on each pair of frequency bands detects an unstable relationship between EUA and CER which confirms our working hypothesis.And finally, a third chapter that complements the previous analysis by the co-movement between multi-scale carbon allowance prices resulting from the ETS to those energy markets (oil, coal and gas). Using coherence wavelet, it examined the simultaneous dependence (co-movement) between two price series in time and frequency. It can be interpreted as a local measurement of correlation calculated non-parametrically. Overall, this study is the first attempt at a multi-scale analysis of the co-movement relationship between the CO2 market, Oil, Gas and Coal which is based on the hypothesis of heterogeneity of agents using a template from wavelet algorithm. The model used in this study will allow stakeholders agents on the carbon market to have a great range of choices for their strategies to be able to anticipate wisely because of the high volatility of carbon prices on their different investment horizons.

Le but de cette thèse est de définir puis d'étudier différentes notions d'irrégularité uniforme ou ponctuelle permettant de traduire le fait qu'une fonction peut avoir des 'grands accroissements' à toutes les échelles. Pour cela on 'inverse' les notions de régularité Höldérienne usuelles. L'objectif principal du travail est ensuite de relier ces différentes notions à la théorie des ondelettes. Les critères ondelettes établis vont ainsi permettre de définir des fonctions ou des champs aléatoires dont le comportement est différent suivant la gamme d'échelles considérée. Par ailleurs, si on se place du point de vue ponctuel, une question naturelle est celle de la définition d'une analyse multifractale -dite faible- liée à la notion d'irrégularité ponctuelle. Les ondelettes vont alors permettre de définir des séries d'ondelettes multifractales pour l'irrégularité ponctuelle. Enfin, nous étudions des exemples de champs aléatoires où des propriétés de régularité directionelle apparaissent. Nous nous sommes ainsi centré sur l'étude d'un modèle de champ aléatoire gaussien particulier vérifiant une relation d'autosimilarité matricielle. Nous avons ensuite généralisé ce modèle et introduit des champs gaussiens autosimilaires par rapport à un groupe / The main purpose of this thesis is the definition and the study of different concepts of uniform or pointwise irregularity which enable one to account for the fact that a function may have 'large increments' at any scales. To this end, we 'invert' the usual notions of Hölderian regularity. The main goal is then to relate these different concepts to wavelet theory. The wavelet criteria supplied enable to define functions or random fields the behavior of which differ with respect the family of scales chosen. Moreover, if we consider the pointwise point of view, a natural question is that of the definition of a weak multifractal analysis related to pointwise irregularity. Finally, we study examples of random fields with some properties of directional regularity. Thus we focus on the study of a special model of operator scaling Gaussian field. We then extend this model and introduced group self-similar Gaussian fields

This work focuses on a possibility of using current computer vision alghoritms and methods for automatic similarity matching of so called graffiti tags. Those are such graffiti, that are used as a fast and simple signature of their authors. The process of development and implementation of CBIR system, which is created for this task, is described. For the purposes of finding images similarity, local features are used, most notably self-similarity features.

This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.

In der vorliegenden Arbeit wurden Strömungen räumlich begrenzter Strahlen untersucht. Zum einen wurde die Ausströmung eines runden Strahls in ein Rohr betrachtet, der sog. begrenzte Strahl. Zum anderen wurde die Ausströmung von 7 hexagonal angeordneten runden Strahlen in ein Rohr betrachtet, das sog. hexagonale Strahlbündel. Die Motivation zur vorliegenden Arbeit ergab sich aus der Entwicklung von Durchflussmessgeräten, die als Bypassapparaturen ausgeführt sind und stromabwärts des Staudruckkörpers ein Strömungsgebiet mit begrenzten Strahlen aufweisen. Dafür wurden mit Hilfe der Ähnlichkeitstheorie die zugrundeliegenden Kennzahlen bestimmt. Besonderes Augenmerk lag auf der Charakterisierung der instationären bzw. turbulenten Strömungseigenschaften für Reynolds-Zahlen zwischen 1.000 und 20.000. Es wurden die selbstähnlichen Eigenschaften der Strömungen untersucht, wobei sich insbesondere für den begrenzten Strahl wichtige Erkenntnisse ergaben. Für das hexagonale Strahlbündel wurden mit Hilfe der numerischen Strömungssimulation die grundlegenden Eigenschaften des Strömungsfeldes untersucht. Dabei weisen die Geometriekennzahlen einen dominierenden Einfluss auf. So konnten in Abhängigkeit zum Durchmesserverhältnis und Strahlabstand drei Strömungsformen identifiziert und experimentell mittels Laser-Doppler Anemometrie nachgewiesen werden. Eine wesentliche Fragestellung bestand in der Anwendung der numerischen Strömungssimulation, des RANS-Ansatzes und des SST-Turbulenzmodells. Dazu wurde die Anpassung der Modellkoeffizienten untersucht, wobei für den begrenzten Strahl ein allgemein gültiger Satz gefunden wurde. / In the present work flows of spatially limited radiation were investigated. On the one hand, the outflow of a round jet into a pipe was considered, the so-called confined jet. On the other hand, the outflow of 7 hexagonal arranged round jets into a pipe was con-sidered, the so-called hexagonal jet array. The motivation for the present work arose from the development of flowmeters which are designed as bypass apparatures that have a jet array flow downstream of the dynamic pressure body. For this purpose the underlying similarity parameters were determined. Special attention was paid to the cha-racterization of transient and turbulent flow properties for Reynolds numbers between 1,000 and 20,000. The self-similar properties of the flows were investigated with im-portant findings in particular for the confined jet. For the hexagonal jet array the basic properties of the flow field were investigated by using computational fluid dynamics. The geometric similarity parameters have a dominant influence. Thus, depending on the di-ameter ratio and jet distance ratio, three flow patterns could be identified and experimen-tally detected by the use of laser Doppler anemometry. An important question was the application of the computational fluid dynamic method, the RANS approach and the SST turbulence model. For this purpose a generally valid set of model coefficients was found for the confined jet flow.