Estimation of the self-similarity exponent has attracted growing interest in recent decades and became a research subject in various fields and disciplines. Real-world data exhibiting self-similar behavior and/or parametrized by self-similarity exponent (in particular Hurst exponent) have been collected in different fields ranging from finance and human sciencies to hydrologic and traffic networks. Such rich classes of possible applications obligates researchers to investigate qualitatively new methods for estimation of the self-similarity exponent as well as identification of long-range dependencies (or long memory). In this thesis I present the Bayesian estimation of the Hurst exponent. In contrast to previous methods, the Bayesian approach allows the possibility to calculate the point estimator and confidence intervals at the same time, bringing significant advantages in data-analysis as discussed in this thesis. Moreover, it is also applicable to short data and unevenly sampled data, thus broadening the range of systems where the estimation of the Hurst exponent is possible. Taking into account that one of the substantial classes of great interest in modeling is the class of Gaussian self-similar processes, this thesis considers the realizations of the processes of fractional Brownian motion and fractional Gaussian noise. Additionally, applications to real-world data, such as the data of water level of the Nile River and fixational eye movements are also discussed. / Die Abschätzung des Selbstähnlichkeitsexponenten hat in den letzten Jahr-zehnten an Aufmerksamkeit gewonnen und ist in vielen wissenschaftlichen Gebieten und Disziplinen zu einem intensiven Forschungsthema geworden. Reelle Daten, die selbsähnliches Verhalten zeigen und/oder durch den Selbstähnlichkeitsexponenten (insbesondere durch den Hurst-Exponenten) parametrisiert werden, wurden in verschiedenen Gebieten gesammelt, die von Finanzwissenschaften über Humanwissenschaften bis zu Netzwerken in der Hydrologie und dem Verkehr reichen. Diese reiche Anzahl an möglichen Anwendungen verlangt von Forschern, neue Methoden zu entwickeln, um den Selbstähnlichkeitsexponenten abzuschätzen, sowie großskalige Abhängigkeiten zu erkennen. In dieser Arbeit stelle ich die Bayessche Schätzung des Hurst-Exponenten vor. Im Unterschied zu früheren Methoden, erlaubt die Bayessche Herangehensweise die Berechnung von Punktschätzungen zusammen mit Konfidenzintervallen, was von bedeutendem Vorteil in der Datenanalyse ist, wie in der Arbeit diskutiert wird. Zudem ist diese Methode anwendbar auf kurze und unregelmäßig verteilte Datensätze, wodurch die Auswahl der möglichen Anwendung, wo der Hurst-Exponent geschätzt werden soll, stark erweitert wird. Unter Berücksichtigung der Tatsache, dass der Gauß'sche selbstähnliche Prozess von bedeutender Interesse in der Modellierung ist, werden in dieser Arbeit Realisierungen der Prozesse der fraktionalen Brown'schen Bewegung und des fraktionalen Gauß'schen Rauschens untersucht. Zusätzlich werden Anwendungen auf reelle Daten, wie Wasserstände des Nil und fixierte Augenbewegungen, diskutiert.

The field of the dissertation research is features of computer network packet traffic, the impact of network node features on traffic service, methods of real-time analysis of network traffic features and their application for dynamic prognostication of computer network packet traffic variance. The object of the research is the features of computer network packet traffic, the impact of network node features on computer network traffic service, methods of real-time network traffic features analysis and their application for dynamic prognostication of network traffic variances. The aim of work is to investigate fractal processes in computer networks, grounding on the results obtained to select methods suitable for real-time analysis of network traffic and to work out methods for real-time measurement of self-similarity as well as to apply it for perfection of computer networks service quality. Possibilities for mathematical modelling of network components, computer network packet traffic models and models using service theory instruments have been analysed. The package of network traffic features analysis has been worked out; it was used for analysis, assessment and comparison of methods for computer networks fractality and self-similarity research. For assessment of self-similarity of the network traffic time lines analysis, frequency/wave feature estimates, self-similarity analysis methods based on time line stability parameters estimators and assessed by the chaos theory... [to full text] / Disertacijos tyrimų sritis – kompiuterių tinklo paketinio srauto savybės, tinklo mazgo savybių įtaka srauto aptarnavimui, tinklo srauto savybių realaus laiku analizės metodai ir jų taikymas kompiuterių tinklo srauto kaitos dinaminiam prognozavimui. Tyrimų objektas – kompiuterių tinklo paketinio srauto savybės, tinklo mazgo savybių įtaka paketinio kompiuterių tinklo srauto aptarnavimui, realaus laiko tinklo srauto savybių analizės metodai ir jų taikymas tinklo srauto kaitos dinaminiam prognozavimui. Darbo tikslas – ištirti fraktalinius procesus kompiuterių tinkluose, remiantis gautais rezultatais parinkti metodus, tinkamus tinklo srauto analizei realiu laiku, ir sukurti savastingumo matavimo realiu laiku metodiką bei ją pritaikyti kompiuterių tinklų aptarnavimo kokybei gerinti. Išanalizuotos tinklo komponentų matematinio modeliavimo galimybės, kompiuterių tinklo paketinio srauto modeliai ir modeliai, naudojantys aptarnavimo teorijos instrumentus. Parengtas tinklo srauto savybių analizės paketas, panaudotas kompiuterių tinklų fraktališkumo ir savastingumo tyrimo metodams analizuoti, vertinti ir palyginti. Ištirti paketinio kompiuterių tinklo srauto laiko eilučių analizės, dažninių/banginių savybių įvertinimo, laiko eilutės stabilumo parametrų įverčiais grindžiami bei chaoso teorijos priemonėmis įvertinami savastingumo analizės metodai. Sudarytas tinklo srauto savastingumo realiu laiku analizės paketas, kurį naudojant savastingumo matavimui realiu laiku atrinktas robastinis... [toliau žr. visą tekstą]

Nowadays there has been a growing interest in anomalous diffusion: the super difusive and sub-difusive processes. The problem about normal diffusion already well established whereas many problems still exist in anomalous diffusion. Several mathematical models and computational techniques have been developed to model such processes. In this work we studied a non-Markovian Random Walk (RW), in one dimension in which the development of the process is governed by decisions taken in the distant past. We used as tool of analysis, analytical and numerical procedures (Monte Carlo method). In this problem, the walker takes its decisions (go right or left) at a given time t, based on the decisions taken in the past, namely in a fraction f of the total time. As far as the decision making process is considered only the distant past is taken into account. This loss of recent memory leads the probability density function of the position to change from Gaussian to non-Gaussian and leads to the emergence of log-periodic oscillations in position, besides producing a change in the behavior of non-persistent to persistent, causing anomalous diffusion. This change is characterized by the Hurst exponent, and is found, surprisingly, in a region where there is negative feedback. The diagram of phases depending on the parameters f and p (fraction of old memory and feedback), shows the following phases: classical non persistence, classical persistence, log-periodic non persistence, log-periodic persistence, Gaussian and non Gaussian with respect to the position of the walker. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Atualmente tem crescido o interesse por processos de difusão anômala, i.e., os super difusivos e sub-difusivos. O problema voltado para difusão normal já é bem conhecido, enquanto para difusões anômalas ainda existem vários problemas em abertos. Várias técnicas computacionais e modelos matemáticos têm sido desenvolvidos para modelar tais processos. Estudamos neste trabalho uma caminhada aleatória, não Markoviana em uma dimensão, em que o desenvolvimento do processo é regido por decisões tomadas em relação ao passado distante. Utilizamos como ferramenta de análise uma abordagem analítica e numérica (via método de Monte Carlo). Nesse problema, o caminhante toma suas decisões (entre ir para a direita ou para a esquerda), num determinado tempo t, com base nas decisões tomadas no passado, numa fração f do tempo transcorrido. Quando f<1 o passado recente é esquecido e apenas o passado distante é considerado. Essa perda de memória recente induz a função densidade de probabilidade da posição a passar de um regime Gaussiano para não Gaussiano e leva ao surgimento de oscilações log-periódicas na posição, além de produzir uma mudança no comportamento, de não persistente para persistente, ocasionando difusão anômala. Essa mudança é caracterizada pelo expoente de Hurst e ocorre também, surpreendentemente, numa região de feedback negativo. O diagrama de fases em função dos parâmetros f e p (fração de memória antiga e feedback), mostra as seguintes regiões: não persistência clássica; persistência clássica; não persistência log-periódica e persistência log-periódica; região Gaussiana e não Gaussiana da posição.

Cette thèse porte sur l'existence de champs browniens fractionnaires indexés par des variétés riemanniennes. Ces objets héritent des propriétés qui font le succès du mouvement brownien fractionnaire classique (H-autosimilarité des trajectoires ajustable, accroissements stationnaires), mais autorisent à considérer des applications où les données sont portées par un espace qui peut par exemple être courbé ou troué. L'existence de ces champs n'est assurée que lorsque la quantité 2H est inférieure à l'indice fractionnaire de la variété, qui n'est connu que dans un petit nombre d'exemples. Dans un premier temps nous donnons une condition nécessaire pour l'existence de champ brownien fractionnaire. Dans le cas du champ brownien (correspondant à H=1/2) indexé par des variétés qui ont des géodésiques fermées minimales, cette condition s'avère très contraignante : nous donnons des résultats de non-existence dans ce cadre, et montrons notamment qu'il n'existe pas de champ brownien indexé par une variété compacte non simplement connexe. La condition nécessaire donne également une preuve courte d'un fait attendu qui est la non-dégénérescence du champ brownien indexé par les espaces hyperboliques réels. Dans un second temps nous montrons que l'indice fractionnaire du cylindre est nul, ce qui constitue un exemple totalement dégénéré. Nous en déduisons que l'indice fractionnaire d'un espace métrique n'est pas continu par rapport à la convergence de Gromov-Hausdorff. Nous généralisons ce résultat sur le cylindre à un produit cartésien qui possède une géodésique fermée minimale, et donnons une majoration de l'indice fractionnaire de surfaces asymptotiquement proches du cylindre au voisinage d'une géodésique fermée minimale. / The aim of the thesis is the study of the existence of fractional Brownian fields indexed by Riemannian manifolds. Those fields inherit key properties of the classical fractional Brownian motion (sample paths with self-similarity of adjustable parameter H, stationary increments), while allowing to consider applications with data indexed by a space which can be for example curved or with a hole. The existence of those fields is only insured when the quantity 2H is inferior or equal to the fractional index of the manifold, which is known only in a few cases. In a first part we give a necessary condition for the fractional Brownian field to exist. In the case of the Brownian field (corresponding to H=1/2) indexed by a manifold with minimal closed geodesics this condition happens to be very restrictive. We give several nonexistence results in this situation. In particular we show that there exists no Brownian field indexed by a nonsimply connected compact manifold. Our necessary condition also gives a short proof of an expected result: we prove the nondegeneracy of fractional Brownian fields indexed by the real hyperbolic spaces. In a second part we show that the fractional index of the cylinder is null, which gives a totally degenerate case. We deduce from this result that the fractional index of a metric space is noncontinuous with respect to the Gromov-Hausdorff convergence. We generalise this result about the cylinder to a Cartesian product with a closed minimal geodesic. Furthermore we give a bound of the fractional index of surfaces asymptotically close to the cylinder in the neighbourhood of a closed minimal geodesic.

Submitted by (ana.araujo@ufrpe.br) on 2016-07-06T14:22:48Z No. of bitstreams: 1 Heliovanio Torres Bandeira.pdf: 959027 bytes, checksum: 9873348980adb3c73410a63f86c250d6 (MD5) / Made available in DSpace on 2016-07-06T14:22:48Z (GMT). No. of bitstreams: 1 Heliovanio Torres Bandeira.pdf: 959027 bytes, checksum: 9873348980adb3c73410a63f86c250d6 (MD5) Previous issue date: 2006-06-19 / Ionic channels are formed by one or few protein molecules found in biological membranes and constitute one of the possible ways for the transport of ions through these membranes. These proteins can assume different conformational open and closed states, phenomenon named ion channel kinetics. The transitions from one state to another are dependent on the potential energy barrier that separates them and can be controlled by electric field, ions, chemical substances and other physical agents. The dwell times in which the proteinchannel stays in one these conformational states have been modeled assuming that the process is Markovian. A chaotic model also was proposed for modeling the ion channel kinetics (LIEBOVITCH e TÓTH., 1991).In this work we use the R/S Hurst analysis to test the long-range correlation found in calcium-activated potassium channel kinetics in Leydig cells. The Hurst coefficient H, a parameter that show the memory existent in a kinetic process (NOGUEIRA et al., 1995), was calculated to a calcium-activated potassium channel in Leydig cells recording and it was equal to H = 0,66±0,044 (n=4), disclosing that the system presents a persistent memory. The R/S analysis when applied to the opening and closing dwell time series obtained from ion channel simulated data using a chaotic model was inadequate to describe the long-term correlation previously found in the experimental data. As conclusion, this work shows that: (i) really, opening and closing dwell times for the single calciumactivated potassium channel of Leydig cells present long-term correlation and (ii) the chaotic model, proposed by Liebovitch and Thót (1991), is not adequate to describe the memory found in the kinetic of this channel. / Canais iônicos são compostos de uma ou poucas moléculas de proteínas que se encontram nas membranas biológicas e constituem uma das vias possíveis para o transporte de íons através dessas membranas. Essas proteínas podem assumir diferentes estados conformacionais, abertos e fechados, fenômeno denominado de cinética de canais iônicos. As transições entre os estados cinéticos dos canais dependem das barreiras de energias potenciais que separam esses estados e, que podem ser controladas por campo elétrico, íons, substâncias químicas e outros agentes. Os tempos de permanências dos canais em cada um dos estados conformacionais têm sido modelados assumindo-se que este processo é markoviano. Um modelo caótico também foi proposto para modelar a cinética de canal iônico (LIEBOVITCH e TÓTH, 1991). Neste trabalho utilizamos a análise R/S de Hurst para testar a correlação de longo alcance (memória) na cinética de um canal para potássio ativado por cálcio em células de Leydig. O coeficiente de Hurst H, um parâmetro que mostra a memória existente em um processo cinético (NOGUEIRA et al., 1995), foi calculado para um registro de um canal para potássio ativado por cálcio e foi encontrado um valor de H = 0,66 ± 0,044 (n=4), evidenciando que o sistema apresenta uma memória persistente. A análise R/S aplicada à seqüência temporal de aberturas e fechamentos obtida para um canal iônico simulado por um modelo caótico mostrou que esse modelo é inadequado para descrever a correlação de longo alcance encontrada nos dados experimentais. Como conclusão, este trabalho mostra que: (i) tempos de permanência para aberturas e fechamentos do canal para potássio ativado por cálcio em células de Leydig apresentam correlação de longo alcance (memória);(ii) o modelo caótico, proposto por Liebovitch e Tóth (1991), é inadequado para descrever a memória encontrada na cinética do canal.

Sabe-se que, na literatura, existem muitos modelos para se fazer previsão para séries temporais financeiras. Sabe-se também que não há um modelo perfeito e que os mais utilizados atualmente são os modelos de redes neurais recorrentes e os da família GARCH. Referências internacionais apontam que existe uma técnica de medição de uma janela temporal para se identificar o tipo de comportamento existente em uma série temporal; tal técnica é conhecida como Expoente de Hurst. É uma medida que qualifica a série como persistente ou anti-persistente. Este trabalho analisou se o Expoente de Hurst, interfere na qualidade das previsões feitas com o modelo de redes neurais recorrentes com e sem o uso do filtro de ondaletas, utilizando os preços diários das principais commodities, ações negociadas no mercado e a taxa de câmbio. no período de janeiro de 1998 a dezembro de 2010. Com a pesquisa observa-se, na maioria dos casos, há uma possível melhora na qualidade das previsões para as séries antipersistentes. / It is known that there are a lot of models to forecast financial time series. It is known, also, that there is not a perfect model and the most used nowadays are the Recurrent Neural Network models and those from the GARCH family. International references point to a technique of measurement using windowing in order to identify the kind of behavior that is present in time series. This technique is known as Hurst Exponent. It is a measure that qualifies the time series as persistent or anti-persistent. This work analyzed if the Hurst Exponent interferes in the quality of the forecasts made with the Neural Network models with and without the wavelet filter, using the main commodities, stock prices, Ibovespa index and the Dollar/Real exchange rate in the period ranging from January 1998 to December 2010. The initial conclusions concerning the models worked out are positives.

El presente trabajo comprueba la hipótesis de eficiencia de mercado (EMH) a través de una medida de persistencia temporal conocida como exponente de Hurst. Esta aproximación además de estar relacionada con la dimensión fractal, permite expandir el análisis de la hipótesis de mercado eficiente, propuesta por Eugene Fama en 1970. El cálculo del exponente de Hurst se realiza en base al método de rango reescalado; y se extiende su aplicación a una estimación dinámica entre el periodo 2006-2021. Este indicador sirve como índice de eficiencia de mercado, y se estima para las series de retornos diarios de los mercados de valores del MILA, conformado por Chile, Colombia, México y Perú. Los resultados demuestran que Perú es el mercado menos eficiente, y con mayor número de ciclos de ineficiencia para el periodo calculado. Por otro lado, México resulta ser el único mercado del MILA que estuvo dentro de la región de eficiencia. / The present work tests the market efficiency hypothesis (EMH) through a measure of time persistence known as the Hurst exponent. This approach, in addition to being related to the fractal dimension, allows us to expand the analysis of the efficient market hypothesis proposed by Eugene Fama in 1970. The calculation of the Hurst exponent is carried out based on the rescaled range method; and its application is extended to a dynamic estimation between the 2006-2021 period. This indicator serves as a market efficiency index and is estimated for the series of daily returns of the MILA securities markets, made up of Chile, Colombia, Mexico, and Peru. The results show that Peru is the least efficient market, and with the highest number of inefficiency cycles for the calculated period. On the other hand, Mexico turns out to be the only MILA market which has been in the efficient region. / Trabajo de investigación

Turbochargers are used on many automotive internal combustion engines to increase power density. The broad operating range of the engine also requires a wide range of the turbocharger compressor. At low mass flows, however, turbo compressor operation becomes unstable and eventually enters surge. Surge is characterized by large oscillations in mass flow and pressure. Due to the associated noise, control problems, and possibility of mechanical component damage, this has to be avoided. Different indicators exist to classify compressor operation as stable or unstable on a gas stand. They are based on pressure oscillations, speed oscillations, or inlet temperature increase. In this thesis, a new stability indicator is proposed based on the Hurst exponent of the pressure signal. The Hurst exponent is a number between zero and one that describes what kind of long-term correlations are present in a time series. Data from three cold gas stand experiments are analyzed using this criterion. Results show that the Hurst exponent of the compressor outlet pressure signal has good characteristics. Stable operation is being indicated by values larger than 0.5. As compressor operation moves towards the surge line, the Hurst exponent decreases towards zero. An additional distinction between the long-term correlations of small and large amplitude fluctuations by means of higher order Hurst exponents can be used as an early warning indicator. Further tests using compressor housing accelerometers show that the Hurst exponent is not a good choice for real-time surge detection on the engine. Reasons are the long required sampling time compared to competing methods, and the fact that other periodically repeating oscillations lead to Hurst exponents close to zero independent of compressor operation. / Turboladdare används ofta på förbränningsmotorer för att öka motorns effekttäthet. Motorns breda driftområde ställer krav på ett brett driftområde för turboladdarens kompressor. Vid låga massflöden blir kompressordriften dock mindre stabil, och surge kan uppträda. Surge innebär stora oscillationer i tryck och massflöde genom kompressorn. På grund av oljud, reglerproblem och risken för mekaniska skador vill man undvika surge. Det finns indikatorer för att bedöma kompressorns stabilitet på ett gas stand. Indikatorerna är baserade på tryckoscillationer, varvtalsoscillationer, eller temperaturökning i gasen i kompressorinloppet. I denna avhandling presenteras en ny indikator baserad på Hurst-exponenten, beräknad på trycksignalen. Hurst-exponenten är ett tal mellan noll och ett som beskriver vilka typer av långtidskorrelationer det finns i signalen. Mätningar från tre gas-stand-experiment har analyserats på detta sätt. Analyserna visar att Hurst-exponenten baserad på kompressorutloppstrycket fungerar bra som som surgeindikator. Stabil drift av kompressorn indikeras av att Hurst-exponenten är större än 0.5. När kompressordriftpunkten närmar sig surgelinjen faller Hurst-exponenten mot noll. En distinktion mellan oscillationer med små och stora amplituder kan används för att få en tidig varning. Analyser av vibrationsmätningar på kompressorhuset vid motorapplikation visar att Hurst-exponenten inte är lämplig som realtidsindikator på en motor. Detta kommer sig dels av att data behöver samlas in under en längre tid än med andra tänkbara indikatorer, dels av att andra periodiska oscillationer i signalen kopplade till motorns naturliga beteende leder till Hurst-exponenter nära noll även vid stabil kompressordrift. / <p>QC 20170510</p> / CCGEx - Compressor off-Design

The aim of this thesis is to provide an empirical evidence of multifractality in financial time series and to discuss the relevance of this concept for the current financial theory. We have applied two methods, the Multifractal Detrended Fluctuation analysis and the Generalized Hurst exponent method, on components of the Dow Jones Industrial Average. We analyzed daily data of 30 companies traded on U.S. stock markets from 2002 to 2012. We present results supporting presence of multiscaling in open-close returns. Contrary to published literature, we were not able to find any significant multiscaling in volatility. Moreover based on our analysis, multiscaling is not present in standardized returns and as multifractality requires relatively complicated models, this is our most valuable result. 1

Over the last decades, multifractality has become a downright stylized fact in financial markets. However, its presence has not been adequately statistically proved. The main aim of this thesis is to contribute to the discussion by an ex- tensive statistical analysis of the problem. We investigate returns and volatility of the collection of the four stock indices employing the three popular methods: the GHE, the MF-DFA, and the MF-DMA method. By comparing the results of the original series to those for simulated monofractal series, we conclude that stock market returns as well as volatility exhibit a multifractal nature. Additionally, in order to understand the origin of underlying multifractality, we study vari- ous surrogate series. We found that a fat-tailed distribution significantly affects multifractality. On the other, we were not able to confirm the impact of time correlations as the results strongly depend on the applied model. JEL Classification F12, G02, G10, C12, C22, C49, C58 Keywords econophysics, multifractality, financial markets, Hurst exponent Author's e-mail jana.sedlarikova@gmail.com Supervisor's e-mail kristoufek@ies-prague.org