Dissertations / Theses on the topic 'Hitting time of random walk'
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BORRELLO, DAVIDE. "Interacting particle systems: stochastic order, attractiveness and random walks on small world graphs." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2009. http://hdl.handle.net/10281/7467.
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The main subject of the thesis is concerned with interacting particle systems, which are
classes of spatio-temporal stochastic processes describing the evolution of particles in interaction with each other.
The particles move on a finite or infinite discrete space and on each element of this space the state of the configuration is integer valued. Configurations of particles evolve in continuous
time according to a Markov process. Here the space is either the infinite deterministic
d-dimensional lattice or a random graph given by the finite d-dimensional torus with random matchings. In Part I we investigate the stochastic order in a particle system with multiple births, deaths and jumps on the d-dimensional lattice: stochastic order is a key tool to understand the ergodic properties of a system. We give applications on biological models of spread of epidemics and metapopulation dynamics systems. In Part II we analyse the coalescing random walk in a class of finite random graphs modeling social networks, the small world graphs. We derive the law of the meeting time of two random walks on small world graphs and we use this result to understand the role
of random connections in meeting time of random walks and to investigate the behavior
of coalescing random walks.
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Santos, Raqueline Azevedo Medeiros. "Cadeias de Markov Quânticas." Laboratório Nacional de Computação Científica, 2010. http://www.lncc.br/tdmc/tde_busca/arquivo.php?codArquivo=199.
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Em Ciência da Computação, os caminhos aleatórios são utilizados em algoritmos randômicos, especialmente em algoritmos de busca, quando desejamos encontrar um estado marcado numa cadeia de Markov. Nesse tipo de algoritmo é interessante estudar o Tempo de Alcance, que está associado a sua complexidade computacional. Nesse contexto, descrevemos a teoria clássica de cadeias de Markov e caminhos aleatórios, assim como o seu análogo quântico. Dessa forma, definimos o Tempo de Alcance sob o escopo das cadeias de Markov quânticas. Além disso, expressões analíticas calculadas para o tempo de Alcance quântico e para a probabilidade de encontrarmos um elemento marcado num grafo completo são apresentadas como os novos resultados dessa dissertação.In Computer Science, random walks are used in randomized algorithms, specially in search algorithms, where we desire to find a marked state in a Markov chain.In this type of algorithm,it is interesting to study the Hitting Time, which is associated to its computational complexity. In this context, we describe the classical theory of Markov chains and random walks,as well as their quantum analogue.In this way,we define the Hitting Time under the scope of quantum Markov chains. Moreover, analytical expressions calculated for the quantum Hitting Time and for the probability of finding a marked element on the complete graph are presented as the new results of this dissertation.
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Gubiec, Tomasz, and Ryszard Kutner. "Two-step memory within Continuous Time Random Walk." Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-183316.
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Gubiec, Tomasz, and Ryszard Kutner. "Two-step memory within Continuous Time Random Walk." Diffusion fundamentals 20 (2013) 64, S. 1, 2013. https://ul.qucosa.de/id/qucosa%3A13643.
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Chang, Qiang. "Continuous-time random-walk simulation of surface kinetics." The Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=osu1166592142.
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Katzenbeisser, Walter, and Wolfgang Panny. "Simple Random Walk Statistics. Part I: Discrete Time Results." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 1990. http://epub.wu.ac.at/1078/1/document.pdf.
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In a famous paper Dwass [I9671 proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows to extend Dwass's results in several ways, viz. arbitrary endpoints, horizontal steps, and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process. (author's abstract)Series: Forschungsberichte / Institut für Statistik
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Li, Chao. "Option pricing with generalized continuous time random walk models." Thesis, Queen Mary, University of London, 2016. http://qmro.qmul.ac.uk/xmlui/handle/123456789/23202.
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The pricing of options is one of the key problems in mathematical finance. In recent years, pricing models that are based on the continuous time random walk (CTRW), an anomalous diffusive random walk model widely used in physics, have been introduced. In this thesis, we investigate the pricing of European call options with CTRW and generalized CTRW models within the Black-Scholes framework. Here, the non-Markovian character of the underlying pricing model is manifest in Black-Scholes PDEs with fractional time derivatives containing memory terms. The inclusion of non-zero interest rates leads to a distinction between different types of \forward" and \backward" options, which are easily mapped onto each other in the standard Markovian framework, but exhibit significant dfferences in the non-Markovian case. The backward-type options require us in particular to include the multi-point statistics of the non-Markovian pricing model. Using a representation of the CTRW in terms of a subordination (time change) of a normal diffusive process with an inverse L evy-stable process, analytical results can be obtained. The extension of the formalism to arbitrary waiting time distributions and general payoff functions is discussed. The pricing of path-dependent Asian options leads to further distinctions between different variants of the subordination. We obtain analytical results that relate the option price to the solution of generalized Feynman-Kac equations containing non-local time derivatives such as the fractional substantial derivative. Results for L evy-stable and tempered L evy-stable subordinators, power options, arithmetic and geometric Asian options are presented.
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Niemann, Markus. "From Anomalous Deterministic Diffusion to the Continuous-Time Random Walk." Wuppertal Universitätsbibliothek Wuppertal, 2010. http://d-nb.info/1000127621/34.
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Lau, Hon Wai. "Random walk in networks : first passage time and speed analysis /." View abstract or full-text, 2009. http://library.ust.hk/cgi/db/thesis.pl?PHYS%202009%20LAU.
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Allen, Andrew. "A Random Walk Version of Robbins' Problem." Thesis, University of North Texas, 2018. https://digital.library.unt.edu/ark:/67531/metadc1404568/.
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Robbins' problem is an optimal stopping problem where one seeks to minimize the expected rank of their observations among all observations. We examine random walk analogs to Robbins' problem in both discrete and continuous time. In discrete time, we consider full information and relative ranks versions of this problem. For three step walks, we give the optimal stopping rule and the expected rank for both versions. We also give asymptotic upper bounds for the expected rank in discrete time. Finally, we give upper and lower bounds for the expected rank in continuous time, and we show that the expected rank in the continuous time problem is at least as large as the normalized asymptotic expected rank in the full information discrete time version.
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Niemann, Markus [Verfasser]. "From Anomalous Deterministic Diffusion to the Continuous-Time Random Walk / Markus Niemann." Wuppertal : Universitätsbibliothek Wuppertal, 2010. http://d-nb.info/1000127621/34.
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Helfferich, Julian [Verfasser], and Alexander [Akademischer Betreuer] Blumen. "Glass dynamics in the continuous-time random walk framework = Glasdynamik als Zufallsprozess." Freiburg : Universität, 2015. http://d-nb.info/1125885513/34.
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Yilmaz, Emre, and Shakir Husain. "Hitting a BRIC Wall : MIST countries becoming the new BRICs?" Thesis, Södertörns högskola, Institutionen för samhällsvetenskaper, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:sh:diva-18374.
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The purpose of this study is to examine a completely new phenomenon called the MIST, by two portfolios: the Goldman Sachs Next 11 equity fund, and the Goldman Sachs BRIC fund, in order to establish whether or not the MIST countries are a better investment decision in terms of risk, return and growth. Furthermore, the study examines in which form these emerging markets lies in terms of market efficiency, and if the random walk theory is present. The opportunities and challenges for Mexico, Indonesia, South Korea and Turkey are also brought upon to determine whether these countries have the potential to exhibit the same success as the BRIC countries did for a decade. Since the growth of the BRIC countries are slowing down, Jim O’Neill, the same founder of the term BRIC, coined the nations MIST. The BRIC countries are facing several difficulties and have led investors to draw out from these countries stocks. Investors that were pouring in money to the BRIC countries during the period 2001-2009, have from 2011, withdrawn 15 billion dollars from the BRIC stocks. Mexico, Indonesia, South Korea and Turkey. Derived from the next eleven countries, these countries have a major effect on the global economy due to their economical and political circumstances. For many investors, the MIST countries that are growing faster than the BRIC are regarded to be the new biggest emerging markets. Investing in BRIC funds are stated to be a disaster today, while on the other hand, the MIST countries are growing and outpacing the BRIC fund. The methodology used was to compare two different portfolios, Goldman Sachs N-11 equity fund in the period 2011-2013 against the Goldman Sachs BRIC fund in two different periods, 2011-2013 and 2006-2008 with S&P 500 as the market index. In addition, a hypothesis test was carried out for this period to observe whether or not to reject the null hypothesis. The results of this study shows that the null hypothesis was rejected and that the N-11 equity fund is a better investment decision, in terms of risk, return and growth today. These emerging markets are under the weak form market efficiency and the random walk theory is present in the N-11 equity fund. This makes the authors’ results more of a speculation than a definite conclusion about the future, as one cannot "beat the market".
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Telcs, Andras, and h197tel@ella hu. "Volume and Time Doubling of Graphs and Random Walks, the Strongly." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1016.ps.
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Bernergård, Zandra. "Connection between discrete time random walks and stochastic processes by Donsker's Theorem." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-48719.
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In this paper we will investigate the connection between a random walk and a continuous time stochastic process. Donsker's Theorem states that a random walk under certain conditions will converge to a Wiener process. We will provide a detailed proof of this theorem which will be used to prove that a geometric random walk converges to a geometric Brownian motion.
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Kanzler, Ludwig. "A study of the efficiency of the foreign exchange market through analysis of ultra-high frequency data." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.297525.
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Fan, Rong. "Random Walk With Absorbing Barriers Modeled by Telegraph Equation With Absorbing Boundaries." OpenSIUC, 2018. https://opensiuc.lib.siu.edu/dissertations/1588.
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Organisms have movements that are usually modeled by particles’ random walks. Under some mathematical technical assumptions the movements are described by diffusion equations. However, empirical data often show that the movements are not simple random walks. Instead, they are correlated random walks and are described by telegraph equations. This thesis considers telegraph equations with and without bias corresponding to correlated random walks with and without bias. Analytical solutions to the equations with absorbing boundary conditions and their mean passage times are obtained. Numerical simulations of the corresponding correlated random walks are also performed. The simulation results show that the solutions are approximated very well by the corresponding correlated random walks and the mean first passage times are highly consistent with those from simulations on the corresponding random walks. This suggests that telegraph equations can be a good model for organisms with the movement pattern of correlated random walks. Furthermore, utilizing the consistency of mean first passage times, we can estimate the parameters of telegraph equations through the mean first passage time, which can be estimated through experimental observation. This provides biologists an easy way to obtain parameter values. Finally, this thesis analyzes the velocity distribution and correlations of movement steps of amoebas, leaving fitting the movement data to telegraph equations as future work.
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Stoor, Daniel. "Solution of the Stefan problem with general time-dependent boundary conditions using a random walk method." Thesis, Örebro universitet, Institutionen för naturvetenskap och teknik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-385147.
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This work deals with the one-dimensional Stefan problem with a general time- dependent boundary condition at the fixed boundary. The solution will be obtained using a discrete random walk method and the results will be compared qualitatively with analytical- and finite difference method solutions. A critical part has been to model the moving boundary with the random walk method. The results show that the random walk method is competitive in relation to the finite difference method and has its advantages in generality and low effort to implement. The finite difference method has, on the other hand, higher accuracy for the same computational time with the here chosen step lengths. For the random walk method to increase the accuracy, longer execution times are required, but since the method is generally easily adapted for parallel computing, it is possible to speed up. Regarding applications for the Stefan problem, there are a large range of examples such as climate models, the diffusion of lithium-ions in lithium-ion batteries and modelling steam chambers for oil extraction using steam assisted gravity drainage.
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Nakamura, Chikara. "Asymptotic behaviors of random walks; application of heat kernel estimates." Kyoto University, 2018. http://hdl.handle.net/2433/232222.
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Ahlquist, Blair 1979. "Probability on graphs: A comparison of sampling via random walks and a result for the reconstruction problem." Thesis, University of Oregon, 2010. http://hdl.handle.net/1794/11144.
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vi, 48 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.We compare the relaxation times of two random walks - the simple random walk and the metropolis walk - on an arbitrary finite multigraph G. We apply this result to the random graph with n vertices, where each edge is included with probability p = [Special characters omitted.] where λ > 1 is a constant and also to the Newman-Watts small world model. We give a bound for the reconstruction problem for general trees and general 2 × 2 matrices in terms of the branching number of the tree and some function of the matrix. Specifically, if the transition probabilities between the two states in the state space are a and b , we show that we do not have reconstruction if Br( T ) [straight theta] < 1, where [Special characters omitted.] and Br( T ) is the branching number of the tree in question. This bound agrees with a result obtained by Martin for regular trees and is obtained by more elementary methods. We prove an inequality closely related to this problem.
Committee in charge: David Levin, Chairperson, Mathematics; Christopher Sinclair, Member, Mathematics; Marcin Bownik, Member, Mathematics; Hao Wang, Member, Mathematics; Van Kolpin, Outside Member, Economics
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Abe, Yoshihiro. "Cover times and extrema of local times for random walks on graphs." 京都大学 (Kyoto University), 2016. http://hdl.handle.net/2433/215283.
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Gatsios, Rafael Confetti. "Reavaliação da superioridade dos analistas na previsão de resultado futuro das empresas brasileiras de capital aberto." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/96/96133/tde-28032018-163418/.
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A pesquisa apresenta um estudo sobre a superioridade dos analistas de mercado com relação aos modelos random walk na previsão de resultados futuros das empresas brasileiras de capital aberto a curto e longo prazo. A literatura tradicional indica superioridade irrestrita dos analistas de mercado sobre os modelos de séries temporais por conta das vantagens de tempo e informação desses agentes. No entanto, estudos recentes da literatura internacional apontam para a necessidade de reavaliação dessa superioridade indicando que, para determinadas características da empresa e principalmente para estimativas de longo prazo, não se verifica superioridade dos analistas com relação aos modelos de séries temporais. Partindo desses achados, essa pesquisa defende a TESE de que para o caso brasileiro a superioridade dos analistas não é irrestrita. Este trabalho avalia as previsões de lucro dos analistas e dos modelos random walk, simples e com crescimento, a curto e longo prazo, para as empresas brasileiras de capital aberto no período de 2010 a 2015. Os dados foram obtidos via plataforma da Thomson Reuters®, nas bases de dados do I/B/E/S® e Thomson Financial. Seguindo a literatura, foram utilizados testes de diferença de média. Como diferencial da pesquisa, foi realizada uma análise de dados em painel no sentido de permitir uma avaliação mais precisa sobre os determinantes da superioridade dos analistas para o caso brasileiro. Ainda, foi proposto um modelo de regressão linear simples para avaliar o conteúdo informacional das previsões dos analistas de mercado e dos modelos random walk. Os resultados indicam: i) maior acurácia de previsão paras os modelos random walk simples quando comparados com os modelos de random walk com crescimento; ii) para a amostra total, nota-se maior acurácia da previsão dos modelos random walk a curto e longo prazo, com superioridade dos analistas apenas para previsões com três meses de defasagem; iii) além da defasagem de previsão, a variabilidade dos lucros, a quantidade de analistas, a dispersão das estimativas dos analistas, o tamanho da empresa, o resultado positivo ou negativo, a listagem em índice de mercado e a idade da empresa no mercado de capitais são fatores que alteram a superioridade dos analistas para o caso brasileiro; iv) maior conteúdo informacional das previsões random walk para previsão de lucros futuros das empresas. Esses resultados são importantes nas decisões de investimento. Ainda, os achados são relevantes para pesquisas da área de finanças e contabilidade que utilizam essa variável para responder a diferentes questões de pesquisa, uma vez que, ao contrário do apontado pela literatura internacional, as evidências sugerem superioridade de previsão dos modelos random walk quando comparados às previsões dos analistas de mercadoThe research presents a study regarding the superiority of market analysts in relation to the random walk models in the forecast of future results of Brazilian companies in the short and long term. The traditional literature indicates unrestricted superiority of market analysts on time series models because of the time and information advantages of these agents. However, recent studies in the international literature point to the need for a reassessment of this superiority, indicating that, for certain company characteristics and especially for long-term estimates, there is no superiority of analysts with respect to time series models. Based on these findings, this research advocates that in the case of Brazil, the superiority of analysts is not unrestricted. This paper evaluates the analysts\' forecasts and the random walk models, both simple and with growth, in the short and long term, for Brazilian publicly traded companies during the period from 2010 to 2015. Data was obtained via the Thomson Reuters® platform, in the I/B/E/S® and Thomson Financial®databases. Following the literature, mean-comparison tests (t-test) were used. As a research differential, a panel data analysis was carried out in order to allow a more precise evaluation of the determinants of analysts\' superiority for the case of Brazil. Furthermore, a simple linear regression model was proposed to evaluate the informational content of market analysts\' forecasts and random walk models. The results indicate: i) greater accuracy of prediction for the simple random walk models, when compared to the random walk models with growth ii) that for the total sample, we can see a greater accuracy of the forecast of random walk models in the short and long term, with analyst superiority only for forecasts with a 3-month lag; (iii) in addition to forecast lag, profit variability, analyst size, dispersion of analysts\' estimates, company size, positive or negative result, market index listing and age of the company in the capital market are factors that alter the superiority of the analysts in the case of Brazil; iv) greater informational content of the random walk forecasts for the prediction of future companies\' profits. These results are important for investment decisions. Moreover, the findings are relevant for research in the field of finance and accounting that use this variable to answer different research questions, since, contrary to the international literature, the evidence suggests forecasting superiority of the random walk models when compared to the market analysts\' forecasts.
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Merschjann, Christoph [Verfasser]. "Optically generated small polarons : Time-resolved pump-multiprobe experiments in lithium niobate vs. random-walk charge-transport simulations in oxide crystals / Christoph Merschjann." Aachen : Shaker, 2007. http://d-nb.info/1164339370/34.
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Puyguiraud, Alexandre. "Upscaling transport in heterogeneous media : from pore to Darcy scale through Continuous Time Random Walks." Thesis, Montpellier, 2019. http://www.theses.fr/2019MONTG016/document.
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Les mécanismes responsables du transport hydrodynamique anormal (non-Fickéen) peuvent être rattachés à la complexité de la géométrie du milieu à l'échelle des pores. Dans cette thèse, nous étudions la dynamique des vitesses de particules à l'échelle des pores. À l'aide de simulations de suivi de particules effectuées sur un échantillon numérisé de grès de Berea, nous présentons une analyse détaillée de l'évolution Lagrangienne et Eulérienne et de leur dépendance aux conditions initiales. Le long de leur ligne de courant, la vitesse des particules montre un signal intermittent complexe, alors que leur sériede vitesses spatiales présente des fluctuations régulières. La distribution spatiale des vitesses des particules converge rapidementvers l'état stationnaire. Ces résultats dénotent un processus Markovienqui permet de prédire les fluctuations de vitesse dans le réseau poral.Ces processus, associés à la tortuosité et à la distance de corrélation de vitesse permettent de paramétrer un modèle de marche aléatoire dans le temps (CTRW) et de réaliser le changement d’échelle pour simuler le transport à l’échelle de Darcy. Le modèle, comme tout modèle issu d’un changement d'échelle, repose sur la définition d'un volume élémentaire représentatif (VER). Nous montrons qu’un VER basé sur les statistiques de vitesse permet de définir un support pertinent pour la modélisation du transport hydrodynamique pré-asymptotique à asymptotique, et ainsi d’éviter les limitations associées à l’équation d’advection-dispersionFickéenne. Cette approche est utilisée pour étudier l’impact de l’hétérogénéité du réseau poral sur le volume de mélange et la masse du produit d’une réaction bimoléculaireThe mechanisms responsible for anomalous (non-Fickian) hydrodynamictransport can be traced back to the complexity of the medium geometry atthe pore-scale. In this thesis, we investigate the dynamics of pore-scaleparticle velocities. Using particle tracking simulations performed on adigitized Berea sandstone sample, we present a detailed analysis of theevolution of the Lagrangian and Eulerian evolution and their dependenceon the initial conditions. The particles experience a complexintermittent temporal velocity signal along their streamline while theirspatial velocity series exhibit regular fluctuations. The spatialvelocity distribution of the particles converges quickly to thesteady-state. These results lead naturally to Markov processes for theprediction of these velocity series.These processes, together with the tortuosity and the velocitycorrelation distance that are properties of the medium, allow theparameterization of a continuous time random walk (CTRW) for theupscaling of the transport. The model, like any upscaled model, relieson the definition of a representative elementary volume (REV). We showthat an REV based on the velocity statistics allows defining a pertinentsupport for modeling pre-asymptotic to asymptotic hydrodynamictransport at Darcy scale using, for instance, CTRW, thus overcomingthe limitations associated with the Fickian advection dispersionequation. Finally, we investigate the impact of pore-scale heterogeneityon a bimolecular reaction and explore a methodology for the predictionof the mixing volume and the chemical mass produced
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Erich, Roger Alan. "Regression Modeling of Time to Event Data Using the Ornstein-Uhlenbeck Process." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1342796812.
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Kosowski, Adrian. "Time and Space-Efficient Algorithms for Mobile Agents in an Anonymous Network." Habilitation à diriger des recherches, Université Sciences et Technologies - Bordeaux I, 2013. http://tel.archives-ouvertes.fr/tel-00867765.
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Computing with mobile agents is rapidly becoming a topic of mainstream research in the theory of distributed computing. The main research questions undertaken in this study concern the feasibility of solving fundamental tasks in an anonymous network, subject to limitations on the resources available to the agent. The considered challenges include: exploring a graph by means of an agent with limited memory, discovery of the network topology, and attempting to meet with another agent in another network (rendezvous). The constraints imposed on the agent include the number of moves which the agent is allowed to perform in the network, the amount of state memory available to the agent, the ability of the agent to communicate with other agents, as well as its a priori knowledge of the network topology or of global parameters.
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Shapira, Assaf. "Bootstrap percolation and kinetically constrained models in homogeneous and random environments." Thesis, Sorbonne Paris Cité, 2019. http://www.theses.fr/2019USPCC066.
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Cette thèse est consacrée à l'étude des modèles aux contraintes cinétiques et de la percolation bootstrap, dans l'intersection entre les probabilités, la combinatoire et la physique statistique. Les modèles aux contraintes cinétiques ont été introduits dans les années 80 pour modéliser la transition liquide-verre, dont la compréhension reste toujours un des plus grands problèmes de la physique de la matière condensée. Ils ont été depuis profondément étudiés par des physiciens dans l'espoir d'éclaircir ce problème et la communauté mathématique s'en intéresse de plus en plus lors de la dernière décennie. Ces modèles sont des systèmes de particules en interaction dont la théorie générale est maintenant bien établie. Leur analyse rencontre tout de même des difficultés qui nécessitent le développement de nouveaux outils mathématiques.La percolation bootstrap est une classe d'automates cellulaires, i.e. déterministes en temps discret. Elle a été considérée pour la première fois en 1979 et son étude est depuis devenue un domaine actif en combinatoire et en probabilités.Les modéles aux contraintes cinétiques et la percolation bootstrap ont été introduits séparément mais sont fortement reliés – on verra que la percolation bootstrap est une version déterministe des modèles aux contraintes cinétiques et que ces derniers sont une version stochastique de la percolation bootstrap.On se concentrera sur les échelles de temps de ces deux modèles dans le but de comprendre le comportement des matériaux vitreuxThis thesis concerns with Kinetically Constrained Models and Bootstrap Percolation, two topics in the intersection between probability, combinatorics and statistical mechanics. Kinetically constrained models were introduced by physicists in the 1980's to model the liquid-glass transition, whose understanding is still one of the big open questions in condensed matter physics. They have been studied extensively in the physics literature in the hope to shed some light on this problem, and in the last decade they have also received an increasing attention in the probability community. We will see that even though they belong to the well established field of interacting particle systems with stochastic dynamics, kinetically constrained models pose challenging and interesting problems requiring the development of new mathematical tools.Bootstrap percolation, on the other hand, is a class of monotone cellular automata, namely discrete in time and deterministic dynamics, the first example being the r-neighbor bootstrap percolation introduced in 1979. Since then, the study of bootstrap percolation has been an active domain in both the combinatorial and probabilistic communities, with several breakthroughs in the recent years.Though introduced in different contexts, kinetically constrained models and the bootstrap percolation, as we will see, are intimately related; and one may think of bootstrap percolation as a deterministic counterpart of kinetically constrained models, and of kinetically constrained models as the natural stochastic version of bootstrap percolation
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Terçariol, César Augusto Sangaletti. "Caminhadas deterministas parcialmente auto-repulsivas: resultados analíticos para o efeito da memória do turista na exploração de meios desordenados." Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/59/59135/tde-18122009-153031/.
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Considere um meio desordenado constituído por $N$ pontos cujas coordenadas são geradas aleatoriamente de maneira uniforme e independente nas arestas unitárias de um hipercubo $d$-dimensional. As probabilidades de vizinhança entre os pares de pontos deste meio são expressas através da fórmula de Cox. Um caminhante parte de um dado ponto deste meio desordenado e se movimenta obedecendo à regra determinista de ir para o ponto mais próximo que não tenha sido visitado nos últimos $\\mu$ passos. Este processo foi denominado de caminhada determinista do turista. Cada trajetória gerada por esta dinâmica possui uma parte inicial não-periódica de $t$ passos (transiente) e uma parte final periódica de $p$ passos (atrator). Neste trabalho, obtemos analiticamente algumas distribuições estatísticas para a caminhada determinista do turista com memória $\\mu$ arbitrária em sistemas unidimensionais e com memória $\\mu=2$ no modelo Random Link (que corresponde ao limite $d ightarrow 1$). Estes resultados nos permitiram compreender o papel da memória no comportamento exploratório do turista e explicar a equivalência não-trivial entre o modelo Random Link e o modelo Random Map (que é um caso limite das redes de Kauffman). Enfatizamos que o número de pontos explorados pelo turista é a grandeza fundamental nos problemas considerados. As distribuições analíticas obtidas foram validadas através de experimentos numéricos. Também obtivemos uma dedução alternativa para a fórmula de Cox, apresentando os resultados finais em termos de distribuições estatísticas elementares.Consider a medium characterized by $N$ points whose coordinates are randomly and independently generated by a uniform distribution along the unitary edges of a $d$-dimensional hypercube. The neighborhood probabilities between any pair of points in this medium are given by the Cox formula. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go the nearest point which has not been visited in the preceding $\\mu$ steps. This process has been called the deterministic tourist walk. Each trajectory generated by this dynamics has an initial non-periodic part of $t$ steps (transient) and a final periodic part of $p$ steps (attractor). In this work, we obtain analytically some statistical distributions for the deterministic tourist walk with arbitrary memory $\\mu$ in one-dimensional systems and with memory $\\mu=2$ in the random link model (which corresponds to $d ightarrow 1$ limit). These results enable us to understand the main role played by the memory on the tourist\'s exploratory behavior and explain the non-trivial equivalence between the random link model and the random map model (which is a limiting case of the Kauffman model). We stress that the number of explored points is the fundamental quantity in the considered problems. The obtained distributions have been validated by numerical experiments. We also obtain an alternative derivation for the Cox formula, writing the final results in terms of known statistical distributions.
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Levernier, Nicolas. "Temps de premier passage de processus non-markoviens." Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066118/document.
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Cette thèse cherche à quantifier le temps de premier passage (FPT) d'un marcheur non-markovien sur une cible. La première partie est consacrée au calcul du temps moyen de premier passage (MFPT) pour différents processus non-markoviens confinés, pour lesquels les variables cachées sont connues. Notre méthode, qui adapte un formalisme existant, repose sur la détermination de la distribution des variables cachées au moment du FPT. Nous étendons ensuite ces idées à processus non-markoviens confinés généraux, sans introduire les variables cachées - en général inconnues. Nous montrons que le MFPT est entièrement déterminé par la position du marcheur dans le futur du FPT. Pour des processus gaussiens à incréments stationnaires, cette position est très proche d'une processus gaussien, hypothèse qui permet de déterminer ce processus de manière auto-cohérente, et donc de calculer le MFPT. Nous appliquons cette théorie à différents exemples en dimension variée, obtenant des résultats très précis quantitativement. Nous montrons également que notre théorie est exacte perturbativement autour d'une marche markovienne. Dans une troisième partie, nous explorons l'influence du vieillissement sur le FPT en confinement, et prédisons la dépendance en les paramètres géométriques de la distribution de ce FPT, prédictions vérifiées sur maints exemples. Nous montrons en particulier qu'une non-linéarité du MFPT avec le volume confinant est une caractéristique d'un processus vieillissant. Enfin, nous étudions les liens entre les problèmes avec et sans confinement. Notre travail permet entre autre de d'estimer l'exposant de persistance associé à des processus gaussiens non-markoviens vieillissantThe aim of this thesis is the evaluation of the first-passage time (FPT) of a non-markovian walker over a target. The first part is devoted to the computation of the mean first-passage time (MFPT) for different non-markovien confined processes, for which hidden variables are explicitly known. Our methodology, which adapts an existing formalism, relies on the determination of the distribution of the hidden variables at the instant of FPT. Then, we extend these ideas to the case of general non-markovian confined processes, without introducing the -often unkown- hidden variables. We show that the MFPT is entirely determined by the position of the walker in the future of the FPT. For gaussian walks with stationary increments, this position can be accurately described by a gaussian process, which enable to determine it self-consistently, and thus to find the MFPT. We apply this theory on many examples, in various dimensions. We show moreover that this theory is exact perturbatively around markovian processes. In the third part, we explore the influence of aging properties on the the FPT in confinement, and we predict the dependence of its statistic on geometric parameters. We verify these predictions on many examples. We show in particular that the non-linearity of the MFPT with the confinement is a hallmark of aging. Finally, we study some links between confined and unconfined problems. Our work suggests a promising way to evaluate the persistence exponent of non-markovian gaussian aging processes
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Terçariol, César Augusto Sangaletti. ""Resultados analíticos para as distribuições estatísticas relacionadas à caminhada determinista do turista sem memória: efeito da dimensionalidade do sistema e modelos de campo médio"." Universidade de São Paulo, 2004. http://www.teses.usp.br/teses/disponiveis/59/59135/tde-29092005-191004/.
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Considere um meio caracterizado por $N$ pontos cujas coordenadas são geradas aleatoriamente de maneira uniforme nas arestas unitárias de um hipercubo $d$-dimensional. Um caminhante parte de cada ponto deste meio desordenado e se movimenta obedecendo à regra determinista de ir para o ponto mais próximo que não tenha sido visitado nos últimos $mu$ passos. Este processo foi denominado de caminhada determinista do turista. Cada trajetória gerada por esta dinâmica possui uma parte inicial não-periódica de $t$ passos (transiente) e uma parte final periódica de $p$ passos (atrator). As probabilidades de vizinhança são expressas através da fórmula de Cox, que é parametrizada pela função beta incompleta normalizada $I_d = I_{1/4}[1/2,(d+1)/2]$. Enfati-zamos aqui que a distribuição relevante é $S_{mu,d}^{(N)}(t,p)$, a distribuição conjunta de $t$ e $p$, que tem como casos particulares as distribuições marginais, previamente estudadas. O objetivo deste estudo é obter analiticamente estas distribuições para a caminhada determinista do turista sem memória no espaço euclideano, no modelo de distâncias aleatórias (que corresponde ao limite $d
ightarrow infty$) e no modelo de mapeamento aleatório (que é um caso limite das redes de Kauffman). As distribuições analíticas obtidas foram validadas através de experimentos numéricos. A distribuição conjunta de tempos de transiente e período de atratores, no limite termodinâmico para uma dimensionalidade arbitrária vale: $S_{1,d}^{(infty)}(t,p) = [Gamma(1+I_d^{-1}) cdot (t+I_d^{-1})/Gamma(t+p+I_d^{-1})] cdot delta_{p,2}$, onde $t=0,1,2,ldots,infty$; $Gamma(z)$ é a função gama e $delta_{i,j}$ é o delta de Kronecker. A caminhada determinista do turista sem memória no modelo de mapeamento aleatório produz uma distribuição de períodos não-trivial ($S_{0,rm}^{(N)}(p) propto p^{-1}$), que é obtida de $S_{0,rm}^{(N)}(t,p) = Gamma(N)/{Gamma[N+1-(t+p)]N^{t+p}}$, onde enfatizamos que o número de pontos explorados $n_e=t+p$ é a grandeza fundamental nos problemas considerados.Consider a medium characterized by $N$ points whose coordinates are randomly generated by a uniform distribution along the unitary edges of a $d$-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go the nearest point which has not been visited in the preceding $mu$ steps. This process has been called the deterministic tourist walk. Each trajectory generated by this dynamics has an initial non-periodic part of $t$ steps (transient) and a final periodic part of $p$ steps (attractor). The neighborhood probabilities are given by the Cox formula, which is parameterized by the normalized incomplete beta function $I_d = I_{1/4}[1/2,(d+1)/2]$. Here we stress that the relevant distribution is the joint $t$ and $p$ distribution $S_{mu,d}^{(N)}(t,p)$, which has as particular cases, the marginal distributions previously studied. The objective of this study is to obtain analytically these distributions for the memoryless deterministic tourist walk in the euclidean space, random link model (which corresponds to $d ightarrow infty$ limit) and random map model (which is a limiting case of the Kauffman model). The obtained distributions have been validated by numerical experiments. The joint transient time and attractor period distribution in the thermodynamic limit for an arbitrary dimensionality is: $S_{1,d}^{(infty)}(t,p) = [Gamma(1+I_d^{-1}) cdot (t+I_d^{-1})/Gamma(t+p+I_d^{-1})] cdot delta_{p,2}$, where $t=0,1,2,ldots,infty$; $Gamma(z)$ is the gamma function and $delta_{i,j}$ is the Kronecker's delta. The memoryless deterministic tourist walk in the random map leads to a non-trivial cycle distribution ($S_{0,rm}^{(N)}(p) propto p^{-1}$), which is obtained from $S_{0,rm}^{(N)}(t,p) = Gamma(N)/{Gamma[N+1-(t+p)]N^{t+p}}$, where we stress that the number of explored points $n_e=t+p$ is the fundamental quantity in the considered problems.
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Ung, Hervé. "Quasi real-time model for security of water distribution network." Thesis, Bordeaux, 2016. http://www.theses.fr/2016BORD0015/document.
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Le but de cette thèse est de modéliser la propagation d’un contaminant au sein d’un réseau d’eau potable muni de capteurs temps réel. Elle comporte les trois axes de développement suivant: la résolution des équations de transport, celle du problème d’identification des sources de contamination et le placement des capteurs.Le transport d’un produit chimique est modélisé dans un réseau d’eau potable par l’équation de transport réaction 1-D avec l’hypothèse de mélange parfait aux noeuds. Il est proposé d’améliorer sa prédiction par l’ajout d’un modèle de mélange imparfait aux jonctions double T et d’un modèle de dispersion prenant en compte un profil de vitesse 3-D et la diffusion radiale. Le premier modèle est créé à l’aide d’un plan d’expériences avec triangulation de Delaunay, de simulations CFD, et de la méthode d’interpolation krigeage. Le second utilise les équations adjointes du problème de transport avec l’ajout de particules évoluant à l’aide d’une marche aléatoire, cette dernière modélisant la diffusion radiale dans la surface droite du tuyau.Le problème d’identification des sources consiste, à l’aide de réponses positives ou négatives à la contamination des noeuds capteurs, à trouver l’origine, le temps d’injection et la durée de la contamination. La résolution de ce problème inverse est faite par la résolution des équations de transport adjointes par formulation backtracking. La méthode donne la liste des sources potentielles ainsi que le classement de celles-ci selon leur probabilité d’être la vraie source de contamination. Elle s’exprime en fonction de combien, en pourcentage, cette source potentielle peut expliquer les réponses positives aux capteurs.Le placement des capteurs est optimisé pour l’identification des sources. L’objectif est la maximisation du potentiel de détection de la véritable source de contamination. Deux résolutions sont testées. La première utilise un algorithme glouton combiné à une méthode de Monte Carlo.La seconde utilise une méthode de recherche locale sur graphe.Finalement les méthodes sont appliquées à un cas test réel avec dans l’ordre : le placement des capteurs, l’identification de la source de contamination et l’estimation de sa propagationThe aim of this thesis is to model the propagation of a contaminant inside a water distribution network equipped with real time sensors. There are three research directions: the solving of the transport equations, the source identification and the sensor placement. Classical model for transport of a chemical product in a water distribution network isusing 1D-advection-reaction equations with the hypothesis of perfect mixing at junctions. It isproposed to improve the predictions by adding a model of imperfect mixing at double T-junctions and by considering dispersion effect in pipes which takes into account a 3-D velocity profile. The first enhancement is created with the help of a design of experiment based on the Delaunay triangulation, CFD simulations and the interpolation method Kriging. The second one uses the adjoint formulation of the transport equations applied with an algorithm of particle backtracking and a random walk, which models the radial diffusion in the cross-section of a pipe.The source identification problem consists in finding the contamination origin, itsinjection time and its duration from positive and negative responses given by the sensors. The solution to this inverse problem is computed by solving the adjoint transport equations with a backtracking formulation. The method gives a list of potential sources and the ranking of thosemore likely to be the real sources of contamination. It is function of how much, in percentage, they can explain the positive responses of the sensors.The sensor placement is chosen in order to maximize the ranking of the real source of contamination among the potential sources. Two solutions are proposed. The first one uses agreedy algorithm combined with a Monte Carlo method. The second one uses a local search method on graphs. Finally the methods are applied to a real test case in the following order: the sensor placement, the source identification and the estimation of the contamination propagation
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Bodin, Jacques. "Transport de solutés dans des réseaux de fractures discrets : nouvelle approche Lagrangienne dans le domaine des temps." Paris 6, 2001. http://www.theses.fr/2001PA066394.
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Albers, Tony. "Weak nonergodicity in anomalous diffusion processes." Doctoral thesis, Universitätsbibliothek Chemnitz, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-214327.
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Anomale Diffusion ist ein weitverbreiteter Transportmechanismus, welcher für gewöhnlich mit ensemble-basierten Methoden experimentell untersucht wird. Motiviert durch den Fortschritt in der Einzelteilchenverfolgung, wo typischerweise Zeitmittelwerte bestimmt werden, entsteht die Frage nach der Ergodizität. Stimmen ensemble-gemittelte Größen und zeitgemittelte Größen überein, und wenn nicht, wie unterscheiden sie sich? In dieser Arbeit studieren wir verschiedene stochastische Modelle für anomale Diffusion bezüglich ihres ergodischen oder nicht-ergodischen Verhaltens hinsichtlich der mittleren quadratischen Verschiebung. Wir beginnen unsere Untersuchung mit integrierter Brownscher Bewegung, welche von großer Bedeutung für alle Systeme mit Impulsdiffusion ist. Für diesen Prozess stellen wir die ensemble-gemittelte quadratische Verschiebung und die zeitgemittelte quadratische Verschiebung gegenüber und charakterisieren insbesondere die Zufälligkeit letzterer. Im zweiten Teil bilden wir integrierte Brownsche Bewegung auf andere Modelle ab, um einen tieferen Einblick in den Ursprung des nicht-ergodischen Verhaltens zu bekommen. Dabei werden wir auf einen verallgemeinerten Lévy-Lauf geführt. Dieser offenbart interessante Phänomene, welche in der Literatur noch nicht beobachtet worden sind. Schließlich führen wir eine neue Größe für die Analyse anomaler Diffusionsprozesse ein, die Verteilung der verallgemeinerten Diffusivitäten, welche über die mittlere quadratische Verschiebung hinausgeht, und analysieren mit dieser ein oft verwendetes Modell der anomalen Diffusion, den subdiffusiven zeitkontinuierlichen ZufallslaufAnomalous diffusion is a widespread transport mechanism, which is usually experimentally investigated by ensemble-based methods. Motivated by the progress in single-particle tracking, where time averages are typically determined, the question of ergodicity arises. Do ensemble-averaged quantities and time-averaged quantities coincide, and if not, in what way do they differ? In this thesis, we study different stochastic models for anomalous diffusion with respect to their ergodic or nonergodic behavior concerning the mean-squared displacement. We start our study with integrated Brownian motion, which is of high importance for all systems showing momentum diffusion. For this process, we contrast the ensemble-averaged squared displacement with the time-averaged squared displacement and, in particular, characterize the randomness of the latter. In the second part, we map integrated Brownian motion to other models in order to get a deeper insight into the origin of the nonergodic behavior. In doing so, we are led to a generalized Lévy walk. The latter reveals interesting phenomena, which have never been observed in the literature before. Finally, we introduce a new tool for analyzing anomalous diffusion processes, the distribution of generalized diffusivities, which goes beyond the mean-squared displacement, and we analyze with this tool an often used model of anomalous diffusion, the subdiffusive continuous time random walk
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Guerre, Emmanuel. "Méthode non paramétriques d'analyse des séries temporelles multivariées : estimation de mesures de dépendances." Paris 6, 1993. http://www.theses.fr/1993PA066110.
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Dans un premier chapitre, on presente differentes hypotheses permettant, si elles sont verifiees, d'obtenir de meilleures vitesses de convergence pour des estimateurs utilisant ces proprietes. Les deux chapitres suivants s'interessent a l'estimation de mesures caracterisant ces hypotheses de dependance: on etudie la convergence presque sure et la loi limite d'estimateurs non parametriques de contrastes de kullback. Le dernier chapitre s'interesse a un probleme different, de choix de modeles. On propose des tests pour determiner si une marche aleatoire est de type geometrique ou arithmetique
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Cheng, Teddy Man Lai. "Application of filtering theory for optimum strategies in stock market investment." Thesis, Queensland University of Technology, 1997.
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Pöschke, Patrick. "Influence of Molecular Diffusion on the Transport of Passive Tracers in 2D Laminar Flows." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/19526.
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In dieser Arbeit betrachten wir das Strömungs-Diffusions-(Reaktions)-Problem für
passive Markerteilchen, die in zweidimensionalen laminaren Strömungsmustern mit
geringem thermischem Rauschen gelöst sind. Der deterministische Fluss umfasst Zellen
in Form von Quadraten oder Katzenaugen. In ihnen tritt Rotationsbewegung auf.
Einige der Strömungen bestehen aus wellenförmigen Bereichen mit gerader Vorwärtsbewegung.
Alle Systeme sind entweder periodisch oder durch Wände begrenzt. Eine
untersuchte Familie von Strömungen interpoliert kontinuierlich zwischen Reihen von
Wirbeln und Scherflüssen. Wir analysieren zahlreiche numerische Simulationen, die
bisherige theoretische Vorhersagen bestätigen und neue Phänomene offenbaren. Ohne
Rauschen sind die Teilchen in einzelnen Bestandteilen des Flusses für immer gefangen.
Durch Hinzufügen von schwachem thermischen Rauschen wird die normale Diffusion
für lange Zeiten stark verstärkt und führt zu verschiedenen Diffusionsarten für
mittlere Zeiten. Mit Continuous-Time-Random-Walk-Modellen leiten wir analytische
Ausdrücke in Übereinstimmung mit den numerischen Ergebnissen her, die je nach
Parametern, Anfangsbedingungen und Alterungszeiten von subdiffusiver bis superballistischer
anomaler Diffusion für mittlere Zeiten reichen. Wir sehen deutlich, dass
einige der früheren Vorhersagen nur für Teilchen gelten, die an der Separatrix des
Flusses starten - der einzige Fall, der in der Vergangenheit ausführlich betrachtet wurde
- und dass das System zu vollkommen anderem Verhalten in anderen Situationen
führen kann, einschließlich einem Schwingenden beim Start im Zentrum einesWirbels
nach einer gewissen Alterungszeit. Darüber hinaus enthüllen die Simulationen, dass
Teilchenreaktionen dort häufiger auftreten, wo sich die Geschwindigkeit der Strömung
stark ändert, was dazu führt, dass langsame Teilchen von schnelleren getroffen werden,
die ihnen folgen.
Die umfangreichen numerischen Simulationen, die für diese Arbeit durchgeführt
wurden, mussten jetzt durchgeführt werden, da wir die Rechenleistung dafür besitzen.In this thesis, we consider the advection-diffusion-(reaction) problem for passive tracer particles suspended in two-dimensional laminar flow patterns with small thermal noise. The deterministic flow comprises cells in the shape of either squares or cat’s eyes. Rotational motion occurs inside them. Some of the flows consist of sinusoidal regions of straight forward motion. All systems are either periodic or are bounded by walls. One examined family of flows continuously interpolates between arrays of eddies and shear flows. We analyse extensive numerical simulations, which confirm previous theoretical predictions as well as reveal new phenomena. Without noise, particles are trapped forever in single building blocks of the flow. Adding small thermal noise, leads to largely enhanced normal diffusion for long times and several kinds of diffusion for intermediate times. Using continuous time random walk models, we derive analytical expressions in accordance with numerical results, ranging from subdiffusive to superballistic anomalous diffusion for intermediate times depending on parameters, initial conditions and aging time. We clearly see, that some of the previous predictions are only true for particles starting at the separatrix of the flow - the only case considered in depth in the past - and that the system might show a vastly different behavior in other situations, including an oscillatory one, when starting in the center of an eddy after a certain aging time. Furthermore, simulations reveal that particle reactions occur more frequently at positions where the velocity of the flow changes the most, resulting in slow particles being hit by faster ones following them. The extensive numerical simulations performed for this thesis had to be done now that we have the computational means to do so. Machines are powerful tools in order to gain a deeper and more detailed insight into the dynamics of many complicated dynamical and stochastic systems.
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Cupertino, Thiago Henrique. "Machine learning via dynamical processes on complex networks." Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/55/55134/tde-25032014-154520/.
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Extracting useful knowledge from data sets is a key concept in modern information systems. Consequently, the need of efficient techniques to extract the desired knowledge has been growing over time. Machine learning is a research field dedicated to the development of techniques capable of enabling a machine to \"learn\" from data. Many techniques have been proposed so far, but there are still issues to be unveiled specially in interdisciplinary research. In this thesis, we explore the advantages of network data representation to develop machine learning techniques based on dynamical processes on networks. The network representation unifies the structure, dynamics and functions of the system it represents, and thus is capable of capturing the spatial, topological and functional relations of the data sets under analysis. We develop network-based techniques for the three machine learning paradigms: supervised, semi-supervised and unsupervised. The random walk dynamical process is used to characterize the access of unlabeled data to data classes, configuring a new heuristic we call ease of access in the supervised paradigm. We also propose a classification technique which combines the high-level view of the data, via network topological characterization, and the low-level relations, via similarity measures, in a general framework. Still in the supervised setting, the modularity and Katz centrality network measures are applied to classify multiple observation sets, and an evolving network construction method is applied to the dimensionality reduction problem. The semi-supervised paradigm is covered by extending the ease of access heuristic to the cases in which just a few labeled data samples and many unlabeled samples are available. A semi-supervised technique based on interacting forces is also proposed, for which we provide parameter heuristics and stability analysis via a Lyapunov function. Finally, an unsupervised network-based technique uses the concepts of pinning control and consensus time from dynamical processes to derive a similarity measure used to cluster data. The data is represented by a connected and sparse network in which nodes are dynamical elements. Simulations on benchmark data sets and comparisons to well-known machine learning techniques are provided for all proposed techniques. Advantages of network data representation and dynamical processes for machine learning are highlighted in all casesA extração de conhecimento útil a partir de conjuntos de dados é um conceito chave em sistemas de informação modernos. Por conseguinte, a necessidade de técnicas eficientes para extrair o conhecimento desejado vem crescendo ao longo do tempo. Aprendizado de máquina é uma área de pesquisa dedicada ao desenvolvimento de técnicas capazes de permitir que uma máquina \"aprenda\" a partir de conjuntos de dados. Muitas técnicas já foram propostas, mas ainda há questões a serem reveladas especialmente em pesquisas interdisciplinares. Nesta tese, exploramos as vantagens da representação de dados em rede para desenvolver técnicas de aprendizado de máquina baseadas em processos dinâmicos em redes. A representação em rede unifica a estrutura, a dinâmica e as funções do sistema representado e, portanto, é capaz de capturar as relações espaciais, topológicas e funcionais dos conjuntos de dados sob análise. Desenvolvemos técnicas baseadas em rede para os três paradigmas de aprendizado de máquina: supervisionado, semissupervisionado e não supervisionado. O processo dinâmico de passeio aleatório é utilizado para caracterizar o acesso de dados não rotulados às classes de dados configurando uma nova heurística no paradigma supervisionado, a qual chamamos de facilidade de acesso. Também propomos uma técnica de classificação de dados que combina a visão de alto nível dos dados, por meio da caracterização topológica de rede, com relações de baixo nível, por meio de medidas de similaridade, em uma estrutura geral. Ainda no aprendizado supervisionado, as medidas de rede modularidade e centralidade Katz são aplicadas para classificar conjuntos de múltiplas observações, e um método de construção evolutiva de rede é aplicado ao problema de redução de dimensionalidade. O paradigma semissupervisionado é abordado por meio da extensão da heurística de facilidade de acesso para os casos em que apenas algumas amostras de dados rotuladas e muitas amostras não rotuladas estão disponíveis. É também proposta uma técnica semissupervisionada baseada em forças de interação, para a qual fornecemos heurísticas para selecionar parâmetros e uma análise de estabilidade mediante uma função de Lyapunov. Finalmente, uma técnica não supervisionada baseada em rede utiliza os conceitos de controle pontual e tempo de consenso de processos dinâmicos para derivar uma medida de similaridade usada para agrupar dados. Os dados são representados por uma rede conectada e esparsa na qual os vértices são elementos dinâmicos. Simulações com dados de referência e comparações com técnicas de aprendizado de máquina conhecidas são fornecidos para todas as técnicas propostas. As vantagens da representação de dados em rede e de processos dinâmicos para o aprendizado de máquina são evidenciadas em todos os casos
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Bicout, Dominique. "Diffusion multiple de la lumière dans des écoulements." Grenoble 1, 1992. http://www.theses.fr/1992GRE10192.
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Ce manuscrit presente une etude theorique et experimentale sur la possibilite d'etendre la spectroscopie de diffusion multiple de la lumiere aux ecoulements laminaires et stationnaires, et notamment a l'etude des instabilites dans les ecoulements. Cette etude porte sur l'analyse de la fonction de correlations temporelle de l'intensite diffusee. En regime de diffusion multiple, la fonction de correlation temporelle c#1(t) est sensible au module des gradients de vitesse et non a la vitesse moyenne de l'ecoulement. Cette fonction de correlation est caracterisee par le temps 1/kl* en reflexion, et par le temps l*/l1/kl en transmission. Ce temps est le temps necessaire pour que deux diffuseurs initialement distants de l* se deplacent d'une distance relative de la longueur d'onde en presence du gradient de vitesse. Dans les systemes homogenes, ou le gradient de vitesse est le meme partout, la fonction de correlation decroit aux temps courts comme: c#1(t)1t/ en reflexion, et c#1(t)1(lt/l*)#2 en transmission. L'etude des systemes inhomogenes, dans lesquels les gradients de vitesse ne sont pas invariants par translation, nous conduit a introduire la distribution locale de la densite des chemins de diffusion multiple de taille nl* donnee: #n(r#1,r#2,r), ou r#1 et r#2 sont respectivement le point de depart et le point final des chemins de diffusion multiple, et r le point courant ou la densite des chemins de diffusion est consideree. Cette distribution pondere la contribution des gradients de vitesse locaux au gradient de vitesse global #e#f#f. Lorsque la taille caracteristique (la portee) des gradients de vitesse coincide avec la taille l du systeme, la fonction de correlation c#1(t) a le meme comportement que celui obtenu dans un systeme homogene: il suffit de remplacer, dans le temps de correlation, par #e#f#f. Toutes ces considerations ont ete confirmees par les experiences de diffusion multiple de la lumiere dans les ecoulements de couette plan (systeme homogene) et de poiseuille plan (systeme inhomogene). Ces experiences montrent que l'on peut utiliser la diffusion multiple pour caracteriser certaines valeurs moyennes des gradients de vitesse dans des ecoulements laminaires. Enfin, en utilisant cette technique, nous avons determine le seuil et le nombre d'onde de l'instabilite de taylor-couette. Les resultats experimentaux montrent que, au-dessus du seuil d'instabilite, le gradient de vitesse augmente avec le nombre de taylor suivant une loi de landau. De plus, nous obtenons a travers la fonction de correlation temporelle l'image de la structure periodique des rouleaux de taylor, ce qui demontre qu'il est possible d'obtenir dans certaines conditions une representation fidele des gradients de vitesse locaux
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Golder, Jacques. "Modélisation d'un phénomène pluvieux local et analyse de son transfert vers la nappe phréatique." Phd thesis, Université d'Avignon, 2013. http://tel.archives-ouvertes.fr/tel-01057725.
Full textAbstract:
Dans le cadre des recherches de la qualité des ressources en eau, l'étude du processus de transfert de masse du sol vers la nappe phréatique constitue un élément primordial pour la compréhension de la pollution de cette dernière. En effet, les éléments polluants solubles à la surface (produits liés aux activités humaines tels engrais, pesticides...) peuvent transiter vers la nappe à travers le milieu poreux qu'est le sol. Ce scénario de transfert de pollution repose sur deux phénomènes : la pluie qui génère la masse d'eau à la surface et la dispersion de celle-ci à travers le milieu poreux. La dispersion de masse dans un milieu poreux naturel comme le sol forme un sujet de recherche vaste et difficile aussi bien au plan expérimental que théorique. Sa modélisation constitue une préoccupation du laboratoire EMMAH, en particulier dans le cadre du projet Sol Virtuel dans lequel un modèle de transfert (modèle PASTIS) a été développé. Le couplage de ce modèle de transfert avec en entrée un modèle décrivant la dynamique aléatoire de la pluie est un des objectifs de la présente thèse. Ce travail de thèse aborde cet objectif en s'appuyant d'une part sur des résultats d'observations expérimentaux et d'autre part sur de la modélisation inspirée par l'analyse des données d'observation. La première partie du travail est consacrée à l'élaboration d'un modèle stochastique de pluie. Le choix et la nature du modèle sont basés sur les caractéristiques obtenus à partir de l'analyse de données de hauteur de pluie recueillies sur 40 ans (1968-2008) sur le Centre de Recherche de l'INRA d'Avignon. Pour cela, la représentation cumulée des précipitations sera assimilée à une marche aléatoire dans laquelle les sauts et les temps d'attente entre les sauts sont respectivement les amplitudes et les durées aléatoires entre deux occurrences d'événements de pluie. Ainsi, la loi de probabilité des sauts (loi log-normale) et celle des temps d'attente entre les sauts (loi alpha-stable) sont obtenus en analysant les lois de probabilité des amplitudes et des occurrences des événements de pluie. Nous montrons alors que ce modèle de marche aléatoire tend vers un mouvement brownien géométrique subordonné en temps (quand les pas d'espace et de temps de la marche tendent simultanément vers zéro tout en gardant un rapport constant) dont la loi de densité de probabilité est régie par une équation de Fokker Planck fractionnaire (FFPE). Deux approches sont ensuite utilisées pour la mise en œuvre du modèle. La première approche est de type stochastique et repose sur le lien existant entre le processus stochastique issu de l'équation différentielle d'Itô et la FFPE. La deuxième approche utilise une résolution numérique directe par discrétisation de la FFPE. Conformément à l'objectif principal de la thèse, la seconde partie du travail est consacrée à l'analyse de la contribution de la pluie aux fluctuations de la nappe phréatique. Cette analyse est faite sur la base de deux relevés simultanées d'observations de hauteurs de pluie et de la nappe phréatique sur 14 mois (février 2005-mars 2006). Une étude statistique des liens entre les signaux de pluie et de fluctuations de la nappe est menée comme suit : Les données de variations de hauteur de nappe sont analysées et traitées pour isoler les fluctuations cohérentes avec les événements de pluie. Par ailleurs, afin de tenir compte de la dispersion de masse dans le sol, le transport de la masse d'eau pluviale dans le sol sera modélisé par un code de calcul de transfert (modèle PASTIS) auquel nous appliquons en entrée les données de hauteurs de pluie mesurées. Les résultats du modèle permettent entre autre d'estimer l'état hydrique du sol à une profondeur donnée (ici fixée à 1.6m). Une étude de la corrélation entre cet état hydrique et les fluctuations de la nappe sera ensuite effectuée en complément à celle décrite ci-dessus pour illustrer la possibilité de modéliser l'impact de la pluie sur les fluctuations de la nappe
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Liou, Zong-Yi, and 劉宗宜. "On the Cover Time of a Random Walk." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/77544057860307492010.
Full textAbstract:
碩士國立中山大學
應用數學系研究所
101
Imagine that a particle starts from the origin of the x-axis and moves at times t = 0, 1, . . . one step to the right with probability p or one step to the left with probability q = 1 − p. If p = q = 1/2, it is usually called a simple symmetric random walk and if p ≠ q, it is usually called a simple asymmetric random walk. For a simple random walk and a given positive integer n, define the cover time to be the time when the number of points visited has just increased to the given number n. In this thesis, we first review the cover time of a simple random walk on Z starting from 0. Then we study the cover time in a simple random walk starting from 0 with a reflection barrier.
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"Simple Random Walk Statistics. Part I: Discrete Time Results." Department of Statistics and Mathematics, 1990. http://epub.wu-wien.ac.at/dyn/dl/wp/epub-wu-01_a00.
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Yeh, Ming-Kai, and 葉銘凱. "The study of using continuous time random walk model in contaminant transport modeling." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/76223852580153147071.
Full textAbstract:
碩士國立成功大學
資源工程學系碩博士班
95
Due to the heterogeneities of physical and chemical properties of porous media, the plumes of contamination are often shown non-Gaussian behaviors. We proposed a numerical model to simulate non-Gaussian distribution by using the continuous time random walk(CTRW) method. The parameters required in the CTRW model are defined by fitting results of CTRW and these of traditional advection-dispersion model. The proposed model is applied to Borden site. Results show that CTRW well describes the non-Gaussian behavior of solute transport and is a better model to simulate the contaminate transport.
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Chang, Po-Tzu, and 張博詞. "Combining probabilistic model with graph-based random walk to improve search quality through exploiting time-sensitive query information." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/85158393904916978116.
Full textAbstract:
碩士國立臺灣大學
資訊工程學研究所
99
Search Engine services provide platforms for users to search their intent using query. The intent of query may vary in different time period. Time related information should be taking into consideration when search engine return search results. In this paper, we present new re-ranking methods based on time information to improve search result quality. This paper aims at re-ranking search result depending on time sensitive information to improve the following situation: 1. Existed Queries dataset: URLs clicked by queries have sufficient time click information in training data. 2. Rare Queries dataset: URLs clicked by queries have on clicks information in training data and bad search results dataset. We propose SVM Regression using time related features to effectively re-rank the search result of each query depending on click number in each time periods. And propose useful features generated from three methodologies on Existed Query dataset: (a) Probabilistic Prior, (b) Probabilistic Model using Language Model and KL-divergence, and (c) Page Rank approach based on Time click. Besides, without click information on rare query dataset, we also propose features on rare queries dataset (a) Extract clicks from related query (b) Time based Page Rank. Then combine some features for SVM Regression to predict. In my experiment results show that the proposed approach gains 10.28% improve over the original ranking in the AOL query log on Existed Query dataset. In rare query dataset, SVM Regression gains 1.14% improvement on Existed queries and 12.9% improvement on Non-Existed queries. In the end, we analysis the improvement of each methods and discuss the pros and cons between these methods.
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Lukovic, Mirko. "Anomalous Diffusion in Ecology." Doctoral thesis, 2014. http://hdl.handle.net/11858/00-1735-0000-0023-9984-1.
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Sousa, Ângelo Miguel Raposo Almeida e. "Forecasting key performance indicator of mobile networks: application to mobile cellular networks." Master's thesis, 2019. http://hdl.handle.net/10773/29563.
Full textAbstract:
The increase of data trafic in the world has increased the need for mobile
network operators to take greater care in planning and managing theirs
infrastructures. This work explores the performance of several statistical
forecasting models aplied in voice and data trafic. This data was obtained
from an European mobile network, and, regarding the predictive models, it
was applied classic models like exponential smoothing, Holt-Winters, ARIMA,
Random-Walk; as well as two more recent model proposals. Regarding
the daily data, the proposed model could predict values with higher precision
compared to the other models. For hourly data, depending on the time
zone where the models were tested, the models with higher performance were
Random-Walk and the second proposed model. In summary, this dissertation
shows the performance of several classic statistical models, and how these
compare to recently proposed models. It also shows that mobile network
operator can use statistical forecasting methods to try to get information on
how their network might react in future, giving valueable insights to perform
a better managment of their network.O aumento do tráfego de dados no mundo, aumentou a necessidade dos operadores de redes móveis terem um maior cuidado a planear e gerir as suas infraestruturas. Este trabalho explora o desempenho de vários modelos estatísticos de previsão aplicados a tráfego de voz e de dados. Os dados têm origem numa rede móvel Europeia. Relativamente aos modelos preditivos, foram aplicados modelos clássicos como alisamento exponencial, Holt-Winters, ARIMA, Random-Walk; bem como duas propostas de modelos mais recentes. Em suma, esta dissertação mostra o desempenho de alguns modelos estatísticos clássicos, e como estes se comparam a modelos recentemente propostos. Também mostra que operadores de redes podem usar métodos preditivos estatísticos para tentar obter informações de como a sua rede pode reagir no futuro, dando assim informações valiosas para que estes efetuem uma melhor gestão da sua rede.
Mestrado em Engenharia Eletrónica e Telecomunicações
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Rajani, Vishaal. "Quantitative analysis of single particle tracking experiments: applying ecological methods in cellular biology." Master's thesis, 2010. http://hdl.handle.net/10048/1316.
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Single-particle tracking (SPT) is a method used to study the diffusion of various molecules within the cell. SPT involves tagging proteins with optical labels and observing their individual two-dimensional trajectories with a microscope. The analysis of this data provides important information about protein movement and mechanism, and is used to create multistate biological models. One of the challenges in SPT analysis is the variety of complex environments that contribute to heterogeneity within movement paths. In this thesis, we explore the limitations of current methods used to analyze molecular movement, and adapt analytical methods used in animal movement analysis, such as correlated random walks and first-passage time variance, to SPT data of leukocyte function-associated antigen-1 (LFA-1) integral membrane proteins. We discuss the consequences of these methods in understanding different types of heterogeneity in protein movement behaviour, and provide support to results from current experimental work.Applied Mathematics
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Turcotte, Jean-Sébastien. "Temps de Branchement du Mouvement Brownien Branchant Inhomogène." Thèse, 2014. http://hdl.handle.net/1866/10636.
Full textAbstract:
Ce mémoire étudie le comportement des particules dont la position est maximale
au temps t dans la marche aléatoire branchante et le mouvement brownien
branchant sur R, pour des valeurs de t grandes. Plus exactement, on regarde le
comportement du maximum d’une marche aléatoire branchante dans un environnement
inhomogène en temps, au sens où la loi des accroissements varie en fonction
du temps. On compare avec des modèles connus ou simplifiés, en particulier
le modèle i.i.d., où l’on observe des marches aléatoires indépendantes et le modèle
de la marche aléatoire homogène. On s’intéresse par la suite aux corrélations entre
les particules maximales d’un mouvement brownien branchant. Plus précisément,
on étudie le temps de branchement entre deux particules maximales. Finalement,
on applique les méthodes et les résultats des premiers chapitres afin d’étudier
les corrélations dans un mouvement brownien branchant dans un environnement
inhomogène. Le résultat principal du mémoire stipule qu’il y a existence de temps
de branchement au centre de l’intervalle [0, t] dans le mouvement brownien branchant
inhomogène, ce qui n’est pas le cas pour le mouvement brownien branchant
standard. On présentera également certaines simulations numériques afin de corroborer
les résultats numériques et pour établir des hypothèses pour une recherche
future.This thesis studies the behavior of particles that are maximal at time t in branching random walk and branching Brownian motion on R, for large values of t. Precisely, we look at the behavior of the maximum in a branching random walk in a time-inhomogeneous environment, where the law of the increments varies with respect to time. We compare with known or simplified models such as the model where random walks are taken to be i.i.d. and the branching random walk in a time-homogeneous environment model. We then take a look at the correlations between maximal particles in a branching brownian motion. Specifically, we look at the branching time between those maximal particles. Finally, we apply results and methods from the first chapters to study those same correlations in branching Brownian motion in a inhomogeneous environment. The thesis’ main result establishes existence of branching time at the center of the interval [0, t] for the branching Brownian motion in a inhomogeneous environment, which is not the case for standard branching brownian motion.We also present results of simulations that agree with theoretical results and help establishing new hypotheses for future research.
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CIALLELLA, ALESSANDRO. "Particle-based modeling of dynamics in presence of obstacles." Doctoral thesis, 2018. http://hdl.handle.net/11573/1272803.
Full textAbstract:
The aim of this dissertation is to study the effects of the presence of obstacles in the dynamics of some particle-based models. The interest is due to the existence of non-trivial phenomena observed in systems modeling different contexts, from biological scenarios to pedestrian dynamics. Indeed, obstacles can interfere with the motion of particles producing opposite effects, both “slowing down” and “speeding up” the dynamics, depending on the model and on some features of the obstacles, such as position, size, shape. In many different contexts it is required to pay great attention to this matter. In cells, for instance, it is possible to observed anomalous diffusion: due to the presence of macromolecules playing the role of obstacles, the mean-square displacement of biomolecules scales as a power law with exponent smaller than one. Different situations in grain and pedestrian dynamics are known where the presence of an obstacle favour the egress of the agents of the system.
In the first part of this work we introduce the linear Boltzmann equation in our domain: a 2D rectangular strip with two open sides and two reflective sides with large reflective obstacles. We state our results concerning the characterization of the stationary solution. We construct a Monte Carlo algorithm to simulate the dynamics and we use it to construct numerically the stationary states and to evaluate the residence time in presence of obstacles, i.e. the time needed by particles to cross the strip. Finally, we prove the results on the stationary solution of the equation. We find that the residence time is not monotonic with respect to the size and the location of the obstacles, since the obstacle can force those particles that eventually cross the strip to spend a smaller time in the strip itself. We prove that the stationary state of the linear Boltzmann dynamics, in the diffusive regime, converges to the solution of the Laplace equation with Dirichlet boundary conditions on the open sides and homogeneous Neumann boundary conditions on the other sides and on the obstacle boundaries.
In the second part we consider a 2D finite rectangular lattice where a rectangular fixed obstacle is present. We consider particles performing a simple symmetric random walk and we study numerically the residence time behavior. The residence time does not depend monotonically on the geometric properties of the obstacle, such as its width, length, and position. In some cases, due to the presence of the obstacle, the mean residence time is shorter with respect to the one measured for the obstacle-free strip. We give a complete interpretation of the results on the residence times, even by constructing the reduced 1D picture of the lattice: a simple symmetric random walk on an interval with two singular sites mimicking the 2D case. We calculate the residence time for this 1D model via Monte Carlo simulations, finding good correspondence between the results of the 1D and the 2D model. Finally, we produce analytical calculation of the residence time for the 1D model.
In the third part we introduce a probabilistic cellular automaton model to study the motion of pedestrians: a lattice model without exclusion based on a variation of the simple symmetric random walk on the square lattice. The model, despite the basic rules of its dynamics, captures some interesting features of the difficulty in obtaining an optimal strategy of evacuation in a very difficult situation (no visibility, no external lead to the exits). We introduce an interpersonal attraction parameter for the evacuation of confined and darkened spaces: a "buddying" threshold (of no-exclusion per site) mimicking the tendency of pedestrians to form groups (herding effect) and to cooperate. We examine how the dynamics of the crowd is influenced by the tendency to form big groups. The effect of group sizes on outgoing fluxes, evacuation times and wall effects is carefully studied with a Monte Carlo framework accounting also for the presence of an internal obstacle. A strong asymmetry emerges in the effect of the same obstacle placed in different position, opening the possibility to optimize the evacuation by adding a suitable barrier positioned appropriately.
The last part of this thesis is devoted to the Lorentz model. Strategies to study the convergence of the particle density in the dynamics of the Lorentz model to the solution of the linear Boltzmann equation in a suitable low-density limit are discussed in the geometry of the rectangular strip with two open side and two reflective side in presence of obstacles.
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Rakotobe, Joss. "Théorème Central Limite pour les marches aléatoires biaisées sur les arbres de Galton-Watson avec feuilles." Thèse, 2016. http://hdl.handle.net/1866/18777.
Full textAbstract:
L’objectif en arrière-plan est de montrer que plusieurs modèles de marches aléatoires en milieux aléatoires (MAMA) sont reliés à un modèle-jouet appelé le modèle de piège de Bouchaud. Le domaine des MAMA est très vaste, mais nous nous intéressons particulièrement à une classe de modèle où la marche est réversible et directionnellement transiente.
En particulier, nous verrons pourquoi on pense que ces modèles se ressemblent et quel genre de similarités on s’attend à obtenir, une fois qu’on aura présenté le modèle de Bouchaud. Nous verrons aussi quelques techniques de base utilisés de ce domaine, telles que les temps de régénérations.
Comme contribution, nous allons démontrer un théorème central limite pour la marche aléatoire β-biaisée sur un arbre de Galton-Watson.This Master thesis is part of a larger project of linking the behaviours of a certain type of random walks in random environments (RWRE) with those of a toy model called the Bouchaud’s trap model. The domain of RWRE is very wide but our interest will be on a particular kind of models which are reversible and directionally transient. More specifically, we will see why those models have similar behaviours and what kind of results we could expect once we have reviewed the Bouchaud’s trap model. We will also present some basic technic used in this field, such as regeneration times. As a contribution, we will demonstrate a central limit theorem for the β-biased random walk on a Galton-Watson tree.
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Denys, Mateusz. "Analysis of interevent times by methods of statistical physics." Doctoral thesis, 2017. https://depotuw.ceon.pl/handle/item/2469.
Full textAbstract:
The problem of excessive losses is significant in economic theory and investment practice, as well as for random processes analysis. One of the recently discovered characteristics of losses and profits on financial markets, which was discovered by Bogachev, Bunde, Ludescher, and Tsallis and is not entirely explained yet, is the universality of the distribution of times between losses (or profits) exceeding a given threshold value. In other words, this distribution does not depend on the underlying asset type or the time resolution of data, but rather only depends on the particular threshold height. Interestingly, similar results were obtained, e.g., in geophysical time series (describing the earthquakes exceeding a particular magnitude). In this thesis I present a thorough description of this universality, employing two complementary approaches: (i) an analytical approach, based on the extreme value theory (EVT) and the continuous-time random walk (CTRW) model, and (ii) a numerical approach, based on the Potts model from statistical mechanics. The thesis makes original contributions to the field of knowledge with the following: (i) an analytical model of the aforementioned universality, with a thorough empirical verification for losses and some proposed applications to value-at-risk simulation, profits description, and geophysical data description, and (ii) an agent-based spin model of financial markets with the novel interpretation of the spin variable (as regards financial-market models), reproducing the main empirical stylized facts, such as the shape of a usual and an absolute-value autocorrelation function of the returns as well as the distribution of times between superthreshold losses. These results extend the knowledge and understanding of stochastic processes, agent-based modeling, financial markets and geophysical systems.Zagadnienie dużych strat jest kluczowe zarówno z teoretycznego punktu widzenia (w tym w analizie stochastycznej), jak i z punktu widzenia inwestorów giełdowych. Jedną z ostatnio odkrytych (przez Bogacheva, Bundego, Ludeschera i Tsallisa) i dotąd w pełni niewyjaśnionych własności strat i zysków w finansowych szeregach czasowych jest uniwersalność rozkładu czasów pomiędzy kolejnymi stratami (lub zyskami) przekraczającymi zadaną wysokość progową. Innymi słowy, rozkład ten nie zależy od rodzaju rozpatrywanego instrumentu finansowego czy skali czasowej danych, które bierzemy pod uwagę, a jedynie od zadanej wysokości progu. Podobne wyniki uzyskano także np. dla geofizycznych szeregów czasowych (dla trzęsień ziemi przekraczających zadaną magnitudę). W niniejszej rozprawie przedstawiam gruntowny opis tej uniwersalności, wykorzystujący dwa komplementarne podejścia: (i) podejście analityczne, oparte na teorii wartości ekstremalnych (extreme value theory, EVT) i na modelu błądzenia losowego w czasie ciągłym (continuous-time random walk, CTRW) oraz (ii) podejście numeryczne, oparte na modelu Pottsa z mechaniki statystycznej. Oryginalny wkład w rozwój wiedzy z dziedziny, której praca jest poświęcona, stanowią: (i) analityczny model wyżej wymienionej uniwersalności wraz z dokładnym sprawdzeniem jego poprawności za pomocą różnorodnych danych empirycznych dla strat i propozycjami zastosowań do symulacji zmian wielkości value at risk (VaR), do opisu zysków i do opisu danych geofizycznych oraz (ii) agentowy spinowy model rynków finansowych z nowatorską (jeśli chodzi o modele rynków finansowych) interpretacją zmiennej spinowej, odtwarzający szereg empirycznych faktów stylizowanych, takich jak kształt funkcji autokorelacji zwykłych i absolutnych stóp zwrotu oraz wspomniany rozkład czasów pomiędzy ponadprogowymi stratami. Wyniki przedstawione w pracy poszerzają wiedzę na temat procesów stochastycznych, modelowania agentowego, rynków finansowych oraz zjawisk geofizycznych.