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Dissertations / Theses on the topic 'Random walk'
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Renlund, Henrik. "Reinforced Random Walk." Thesis, Uppsala University, Department of Mathematics, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121389.
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Yip, Siu Kwan. "On the Pfaffian property of annihilating random walk and coalescing random walk." Thesis, University of Warwick, 2014. http://wrap.warwick.ac.uk/63894/.
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In this thesis we are to investigate two discrete interacting particle systems, namely annihilating random walk and coalescing random walk. By mapping the annihilating random walk to Glauber model and employing empty interval method respectively, we prove there is a similar structure behind them albeit their apparent differences, that is, they are both Pfaffian point process under a special initial condition. Then we extend the result to investigate whether the Pfaffian property preserves in the case of multi-time correlation function, which is called extended Pfaffian property. And we also investigate the case which the initial condition is changed from independent particles to another peculiar one-sided initial condition and proved it also preserved the Pfaffian property.
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nl, jvdberg@cwi. "Randomly Coalescing Random Walk in Dimension $ge$ 3." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1051.ps.
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Lo, Chak Hei. "On some random walk problems." Thesis, Durham University, 2017. http://etheses.dur.ac.uk/12498/.
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We consider several random walk related problems in this thesis. In the first part, we study a Markov chain on R₊ x S, where R₊ is the non-negative real numbers and S is a finite set, in which when the R₊-coordinate is large, the S-coordinate of the process is approximately Markov with stationary distribution πi on S. Denoting by μi(x) the mean drift of the R₊-coordinate of the process at (x,i) Ε R₊ x S, we give an exhaustive recurrence classification in the case where Σiπiμi(x) → 0, which is the critical regime for the recurrence-transience phase transition. If μi(x) → 0 for all i, it is natural to study the Lamperti case where μi(x) = O(1/x); in that case the recurrence classification is known, but we prove new results on existence and non-existence of moments of return times. If μi(x) → di for di ≠0 for at least some i, then it is natural to study the generalized Lamperti case where μi(x) = di + O(1/x). By exploiting a transformation which maps the generalized Lamperti case to the Lamperti case, we obtain a recurrence classification and an existence of moments result for the former. The generalized Lamperti case is seen to be more subtle, as the recurrence classification depends on correlation terms between the two coordinates of the process. In the second part of the thesis, for a random walk Sn on R^d we study the asymptotic behaviour of the associated centre of mass process Gn = n⁻¹Σ^n i=1 Si. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, Gn is recurrent if d=1 and transient if d≥2. In the transient case we show that Gn has diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which Gn is transient in d=1.
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Zhao, Kai. "QUANTUM RANDOM WALK ON FRACTALS." Diss., Temple University Libraries, 2018. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/490714.
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MathematicsPh.D.
Quantum walks are the quantum mechanical analogue of classical random walks. Discrete-time quantum walks have been introduced and studied mostly on the line Z or higher dimensional space Z d but rarely defined on graphs with fractal dimensions because the coin operator depends on the position and the Fourier transform on the fractals is not defined. Inspired by its nature of classical walks, different quantum walks will be defined by choosing different shift and coin operators. When the coin operator is uniform, the results of classical walks will be obtained upon measurement at each step. Moreover, with measurement at each step, our results reveal more information about the classical random walks. In this dissertation, two graphs with fractal dimensions will be considered. The first one is Sierpinski gasket, a degree-4 regular graph with Hausdorff di- mension of df = ln 3/ ln 2. The second is the Cantor graph derived like Cantor set, with Hausdorff dimension of df = ln 2/ ln 3. The definitions and amplitude functions of the quantum walks will be introduced. The main part of this dissertation is to derive a recursive formula to compute the amplitude Green function. The exiting probability will be computed and compared with the classical results. When the generation of graphs goes to infinity, the recursion of the walks will be investigated and the convergence rates will be obtained and compared with the classical counterparts.
Temple University--Theses
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Zhang, Zhihan. "Random walk on simplicial complexes." Electronic Thesis or Diss., université Paris-Saclay, 2020. http://www.theses.fr/2020UPASM010.
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La notion de laplacien d’un graphe peut être généralisée aux complexes simpliciaux et aux hypergraphes. Cette notion contient des informations sur la topologie de ces structures. Dans la première partie de cette thèse,nous définissons une nouvelle chaîne de Markov sur les complexes simpliciaux. Pour un degré donné k de simplexes, l’espace d’états n’est pas les k-simplexes comme dans les articles précédents sur ce sujet mais plutôt l’ensemble des k-chaines ou k-co-chaines. Ce nouveau cadre est la généralisation naturelle sur les chaînes de Markov canoniques sur des graphes. Nous montrons que le générateur de notre chaîne de Markov est lié au Laplacien supérieur défini dans le contexte de la topologie algébrique pour structure discrète. Nous établissons plusieurs propriétés clés de ce nouveau procédé. Nous montrons que lorsque les complexes simpliciaux examinés sont une séquence de triangulation du tore plat, les chaînes de Markov tendent vers une forme différentielle valorisée processus continu.Dans la deuxième partie de cette thèse, nous explorons quelques applications de la marche aléatoire, i.e. la détection de trous basée sur la marche aléatoire et les noyaux complexes simpliciaux. Pour la détection de trous basée sur la marche aléatoire, nous introduisons un algorithme pour faire des simulations effectuées pour la marche aléatoire à valeur cyclique (k = 1) sur un complexe simplicial avec trous. Pour les noyaux de complexes simpliciaux, nous étendons la définition des noyaux de graphes basés sur la marche aléatoire afin de mesurer la similitude entre deux complexes simpliciauxThe notion of Laplacian of a graph can be generalized to simplicial complexes and hypergraphs. This notion contains information on the topology of these structures. In the first part of this thesis, we define a new Markov chain on simplicial complexes. For a given degree k of simplices, the state space is not thek-simplices as in previous papers about this subject but rather the set of k-chains or k-cochains.This new framework is the natural generalization on the canonical Markov chains on graphs.We show that the generator of our Markov chainis related to the upper Laplacian defined in the context of algebraic topology for discrete structure. We establish several key properties of this new process. We show that when the simplicial complexes under scrutiny are a sequence of ever refining triangulation of the flat torus, the Markov chains tend to a differential form valued continuous process.In the second part of this thesis, we explore some applications of the random walk, i.e., random walk based hole detection and simplicial complexes kernels. For the random walk based hole detection, we introduce an algorithm tomake simulations carried for the cycle-valuedrandom walk (k = 1) on a simplicial complex with holes. For the simplicial complexes kernels,we extend the definition of random walk based graph kernels in order to measure the similarity between two simplicial complexes
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Shiraishi, Daisuke. "Random walk on non-intersecting two-sided random walk trace is subdiffusive in low dimensions." 京都大学 (Kyoto University), 2012. http://hdl.handle.net/2433/157839.
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Phetpradap, Parkpoom. "Intersections of random walks." Thesis, University of Bath, 2011. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.548100.
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We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk. We call this the range of the random walk. We derive a large deviation principle for the probability that the range of simple random walk deviates from its mean. Our result describes the behaviour for deviation below the typical value. This is a result analogous to that obtained by van den Berg, Bolthausen, and den Hollander for the volume of the Wiener sausage. In the second part of the thesis, we are interested in the number of lattice sites visited by two independent simple random walks starting at the origin. We call this the intersection of ranges. We derive a large deviation principle for the probability that the intersection of ranges by time n exceeds a multiple of n. This is also an analogous result of the intersection volume of two independent Wiener sausages.
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Bui, Hoai Thang Computer Science & Engineering Faculty of Engineering UNSW. "Guided random-walk based model checking." Awarded by:University of New South Wales. Computer Science & Engineering, 2009. http://handle.unsw.edu.au/1959.4/44829.
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The ever increasing use of computer systems in society brings emergent challenges to companies and system designers. The reliability of software and hardware can be financially critical, and lives can depend on it. The growth in size and complexity of software, and increasing concurrency, compounds the problem. The potential for errors is greater than ever before, and the stakes are higher than ever before. Formal methods, particularly model checking, is an approach that attempts to prove mathematically that a model of the behaviour of a product is correct with respect to certain properties. Certain errors can therefore be proven never to occur in the model. This approach has tremendous potential in system development to provide guarantees of correctness. Unfortunately, in practice, model checking cannot handle the enormous sizes of the models of real-world systems. The reason is that the approach requires an exhaustive search of the model to be conducted. While there are exceptions, in general model checkers are said not to scale well. In this thesis, we deal with this scaling issue by using a guiding technique that avoids searching areas of the model, which are unlikely to contain errors. This technique is based on a process of model abstraction in which a new, much smaller model is generated that retains certain important model information but discards the rest. This new model is called a heuristic. While model checking using a heuristic as a guide can be extremely effective, in the worst case (when the guide is of no help), it performs the same as exhaustive search, and hence it also does not scale well in all cases. A second technique is employed to deal with the scaling issue. This technique is based on the concept of random walks. A random walk is simply a `walk' through the model of the system, carried out by selecting states in the model randomly. Such a walk may encounter an error, or it may not. It is a non-exhaustive technique in the sense that only a manageable number of walks are carried out before the search is terminated. This technique cannot replace the conventional model checking as it can never guarantee the correctness of a model. It can however, be a very useful debugging tool because it scales well. From this point of view, it relieves the system designer from the difficult task of dealing with the problem of size in model checking. Using random walks, the effort goes instead into looking for errors. The effectiveness of model checking can be greatly enhanced if the above two techniques are combined: a random walk is used to search for errors, but the walk is guided by a heuristic. This in a nutshell is the focus of this work. We should emphasise that the random walk approach uses the same formal model as model checking. Furthermore, the same heuristic technique is used to guide the random walk as a guided model checker. Together, guidance and random walks are shown in this work to result in vastly improved performance over conventional model checking. Verification has been sacrificed of course, but the new technique is able to find errors far more quickly, and deal with much larger models.
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Thomson, D. J. "Random walk models of turbulent dispersion." Thesis, Brunel University, 1988. http://bura.brunel.ac.uk/handle/2438/5549.
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An understanding of the dispersion of contaminants in turbulent flows is important in many fields ranging from air pollution to chemical engineering, and random walk models provide one approach to understanding and calculating aspects of dispersion. Two types of random walk model are investigated in this thesis. The first type, so-called "one-particle models", are capable of predicting only mean concentrations while the second type, "two-particle models", are able to give some information on the fluctuations in concentration as well. Many different one-particle random walk models have been proposed previously and several criteria have emerged to distinguish good models from bad. In this thesis, the relationships between the various criteria are examined and it is shown that most of the criteria are equivalent. It is also shown how a model can be designed to (i) satisfy the criteria exactly and (ii) be consistent with inertial subrange theory. Some examples of models which obey the criteria are described. The theory developed for one-particle models is then extended to the two-particle case and used to design a two-particle model suitable for modelling dispersion in high Reynolds number isotropic turbulence. The properties of this model are investigated in detail and compared with previous models. In contrast to most previous models, the new model is three-dimensional and leads to a prediction for the particle separation probability density function which is in agreement with inertial subrange theory. The values of concentration variance from the new model are compared with experimental data and show encouraging agreement.
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Zimmermann, Jochen [Verfasser], and Andreas [Akademischer Betreuer] Buchleitner. "Random walks with nonlinear interactions on heterogeneous networks = Random Walk mit nichtlinearen Wechselwirkungen auf heterogenen Netzwerken." Freiburg : Universität, 2015. http://d-nb.info/1123482381/34.
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Stewart, Jeffrey A. D. "The random-walk model and the distribution of latencies, a test of the random-walk theory of decision." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0012/MQ31254.pdf.
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Antal, Peter. "Trapping problems for the simple random walk /." [S.l.] : [s.n.], 1994. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=10759.
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Li, Dan. "Shortest paths through a reinforced random walk." Thesis, Uppsala universitet, Analys och tillämpad matematik, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-153802.
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Helmuth, Tyler. "Heaps, graphical models, and random walk representations." Thesis, University of British Columbia, 2014. http://hdl.handle.net/2429/51774.
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Spin systems such as the Ising model are central topics in statistical mechanics and probability theory. In the late 1960s Symanzik made the important discovery that properties of spin systems could be expressed in terms of the behaviour of random walks. This thesis contributes to the understanding of these connections by developing and analyzing random walk representations of graphical models arising in statistical mechanics. Concretely, the results of this thesis can be divided into two parts. The first part is a lace expansion analysis of a model called loop-weighted walk. Loop-weighted walk is a non-Markovian model of random walks that are discouraged (or encouraged), depending on the value of a parameter ⋋ ≥ 0, from completing loops. The model arises naturally as a random walk representation of correlations in a statistical mechanics model called the cycle gas. A challenging aspect of this model is that it is not repulsive, meaning the weight of the future of a walk may either increase or decrease if the past is forgotten. The second part of this thesis is an essentially elementary derivation of a random walk representation for the partition function of the Ising model on any finite graph. Such representations have a long history for planar graphs. For non-planar graphs the additional ingredient needed is a way to compute the intersection numbers of curves on surfaces. The representations for non-planar graphs lead to random walk representations of spin-spin correlation functions that were previously unknown.Science, Faculty of
Mathematics, Department of
Graduate
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Luhar, Ashok Kumar. "Random walk modelling of air pollution dispersion." Thesis, University of Cambridge, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.387006.
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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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Ampuero, Fernanda. "Networks: a random walk in degree space." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/100/100132/tde-25072018-200138/.
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The present work aims to contribute to the study of networks by mapping the temporal evolution of the degree to a random walk in degree space. We analyzed how and when the degree approximates a pre-established value through a parallel with the first-passage problem of random walks. The mean time for the first-passage was calculated for the dynamical versions the Watts-Strogatz and Erdos-Renyi models. We also analyzed the degree variance for the random recursive tree and Barabasi-Albert modelsO presente trabalho visa contribuir com a pesquisa na área de redes através do mapeamento da evolução temporal do grau com um passeio aleatório no espaço do mesmo. Para tanto, foi feita uma análise de quando e como a quantidade de ligações do vértice se aproxima de um valor pré-estabelecido, mediante um paralelo com o problema da primeira passagem de passeios aleatórios. O tempo médio para a primeira passagem para as versões dinâmicas dos modelos Watts-Strogatz e Erdos-Rényi foram calculados. Além disso, foi realizado um estudo da variância do grau para os modelos da árvore recursiva aleatória e Barabási-Albert
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Sinibaldi, Alessandro. "Quantum Random Walk e localizzazione di Anderson." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23647/.
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I due nuclei tematici principali del presente elaborato sono il modello del Quantum Random Walk (QRW) e il fenomeno della localizzazione di Anderson, i quali vengono prima trattati separatamente nei loro aspetti più rilevanti e in seguito ne viene esaminata la relazione reciproca. In particolare, dopo una breve introduzione sul Random Walk classico, si espone la formalizzazione teorica del QRW e si procede ad un'analisi dettagliata della distribuzione di probabilità del processo, con maggiore attenzione rivolta al QRW di Hadamard. La localizzazione di Anderson viene illustrata a partire dal modello tight-binding di un elettrone in un reticolo, di cui si affronta sia la versione in assenza di disordine che quella 1D disordinata, con inizialmente una descrizione di natura teorica e dopo uno studio di tipo numerico basato su diversi metodi computazionali. L'ultima parte è dedicata alla connessione tra i due argomenti e rappresenta il punto di arrivo dell'intero elaborato, dal momento che essa verte sulla localizzazione di Anderson per un QRW 1D con disordine. Viene presentato il modello e si ricavano risultati analitici che caratterizzano le proprietà di localizzazione del sistema, con approfondimento dei casi di disordine completo e di disordine debole. Anche nel capitolo finale è riservato spazio alle simulazioni, che vengono impiegate per studiare numericamente la localizzazione e per verificare la correttezza degli esiti dei calcoli teorici.
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Celikkanat, Abdulkadir. "Graph Representation Learning with Random Walk Diffusions." Electronic Thesis or Diss., université Paris-Saclay, 2021. http://www.theses.fr/2021UPASG030.
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L'objectif principal de l'Apprentissage de Représentations sur Graphes est de plonger les nœuds dans un espace vectoriel de petite dimension. Dans cette thèse, nous abordons plusieurs enjeux dans le domaine. Tout d'abord, nous étudions comment exploiter l'existence de communautés structurelles locales inhérentes aux graphes tout en apprenant les représentations. Nous apprenons des représentations améliorées de la communauté en combinant les informations latentes avec les représentations. De plus, nous nous concentrons sur l'expressivité des représentations. Nous mettons l'accent sur les distributions de familles exponentielles pour saisir des modèles d'interaction riches. Nous proposons un modèle qui combine les marches aléatoires avec une matrice de factorisation sous forme de noyau. Dans la dernière partie de la thèse, nous étudions des modèles permettant un bon compromis entre efficacité et précision. Nous proposons un modèle évolutif qui calcule des représentations binairesGraph Representation Learning aims to embed nodes in a low-dimensional space. In this thesis, we tackle various challenging problems arising in the field. Firstly, we study how to leverage the inherent local community structure of graphs while learning node representations. We learn enhanced community-aware representations by combining the latent information with the embeddings. Moreover, we concentrate on the expressive- ness of node representations. We emphasize exponential family distributions to capture rich interaction patterns. We propose a model that combines random walks with kernelized matrix factorization. In the last part of the thesis, we study models balancing the trade-off between efficiency and accuracy. We propose a scalable embedding model which computes binary node representations
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Franke, Markus. "An update algorithm for restricted random walk clusters." Karlsruhe : Univ.-Verl. Karlsruhe, 2007. http://www.uvka.de/univerlag/volltexte/2007/273/.
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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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Sani, Aliakbar Rahimzadeh. "Martingale convergence in the multitype branching random walk." Thesis, University of Sheffield, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.266032.
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Gonçalves, Antônio Renato Cruz. "Problema de perseguição-evasão baseado em random walk." Universidade Federal de Sergipe, 2016. https://ri.ufs.br/handle/riufs/5023.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPESOne of the greatest reasons to use robotics rather than human beings is to avoid hazardous situations such as activities related to search, surveillance and rescue. The pursuit-evasion problem is a fundamental theoretical base to apply robotics on these cases. This dissertation presents an approach to solve the pursuit-evasion problem with no previous knowledge of the map, which must be simply connected, using multi-robots systems with limited sensing. The approach is based on the random walk, since it is a mathematical formalization probabilistically complete, considering plane and obstacle free environments that shall be treated discretely through a regular occupation grid. This dissertation also presents a variation of this approach, though it considers random walk probabilities, to enhance the previous approach, decreasing the amount of iterations needed to solve the problem. In order to validate what is proposed, a discrete multi-robot simulation environment was developed. Finally, the results obtained on the tests that were performed and possible future works that could improve this approach are discussed.
Uma das principais motivações do uso de sistemas robóticos em detrimento de seres humanos é evitar situações de risco, como as encontradas em atividades de busca, vigilância e resgate. O problema de perseguição-evasão é uma base teórica fundamental para a aplicação da robótica nestes casos. Esta dissertação apresenta uma abordagem para solução do problema de perseguição-evasão sem um conhecimento a priori do mapa, que deverá ser simplesmente conectado, através da coordenação de múltiplos robôs com visão limitada. A abordagem aqui proposta é baseada na random walk, por esta ser uma formalização matemática probabilisticamente completa, sendo contemplados ambientes planos e sem obstáculos, que serão tratados discretamente por meio de uma grade de ocupação regular. Ainda nesta dissertação, foi proposta uma variação dessa abordagem, porém com a ponderação de probabilidades da random walk, com o objetivo de aprimorar a anterior, diminuindo número de iterações necessárias para solução do problema. Para a validação da abordagem proposta, foi desenvolvido um ambiente de simulações para abordagens discretas de múltiplos robôs. Finalmente, são discutidos os resultados obtidos nos testes realizados e propostos trabalhos futuros para melhoria desta abordagem.
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Wu, Tao. "Higher-order Random Walk Methods for Data Analysis." Thesis, Purdue University, 2018. http://pqdtopen.proquest.com/#viewpdf?dispub=10790747.
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Markov random walk models are powerful analytical tools for multiple areas in machine learning, numerical optimizations and data mining tasks. The key assumption of a first-order Markov chain is memorylessness, which restricts the dependence of the transition distribution to the current state only. However in many applications, this assumption is not appropriate. We propose a set of higher-order random walk techniques and discuss their applications to tensor co-clustering, user trails modeling, and solving linear systems. First, we develop a new random walk model that we call the super-spacey random surfer, which simultaneously clusters the rows, columns, and slices of a nonnegative three-mode tensor. This algorithm generalizes to tensors with any number of modes. We partition the tensor by minimizing the exit probability between clusters when the super-spacey random walk is at stationary. The second application is user trails modeling, where user trails record sequences of activities when individuals interact with the Internet and the world. We propose the retrospective higher-order Markov process as a two-step process by first choosing a state from the history and then transitioning as a first-order chain conditional on that state. This way the total number of parameters is restricted and thus the model is protected from overfitting. Lastly we propose to use a time-inhomogeneous Markov chain to approximate the solution of a linear system. Multiple simulations of the random walk are conducted to approximate the solution. By allowing the random walk to transition based on multiple matrices, we decrease the variance of the simulations, and thus increase the speed of the solver.
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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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Boutaud, Pierre. "Branching random walk : limit cases and minimal hypothesis." Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASM025.
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La marche aléatoire branchante est un système de particules sur la droite réelle où partant au temps 0 d’une particule initiale en position 0, chaque particule vivante au temps n meurt au temps n + 1 en donnant indépendemment naissance à un nombre aléatoire de particules se dispersant aléatoirement autour de la position de la particule parente. Dans un premier chapitre introductif, nous définissons en détails le modèle de la marche aléatoire branchante ainsi que certains des enjeux de la recherche autour de ce modèle, notamment l’étude de la martingale additive. Cette martingale peut-être étudié au travers de sa convergence vers une limite triviale ou non ainsi que l’étude d’une renormalisation appropriée, dite de Seneta-Heyde, lorsque cette limite est triviale. Elle peut aussi être étudiée au travers d’équations récursives stochastiques menant à des équations de points fixes en loi. Cette dernière question correspond à des travaux non-publiés effectués en première année de thèse en continuité avec ceux effectués en mémoire de master. Le second chapitre est une traduction en anglais de certaines sections du précédent chapitre pour faciliter la compréhension de certains lecteurs sur les points importants de cette thèse.Dans un troisième chapitre nous présentons une nouvelle méthode de preuve développée avec Pascal Maillard pour le théorème d’Aïdékon et Shi sur la renormalisation de Seneta-Heyde de la martingale additive critique dans le cas où la marche de l’épine admet une variance finie. Cette nouvelle preuve se passe du recours à un lemme d’épluchage et à des calculs de seconds moments pour lui préférer une étude de la transformée de Laplace conditionnée. Les propriétés des fonctions de renouvellements permettent une approche plus générale qui ne demande pas de s’attarder en particulier sur la martingale dérivée. Ceci est d’ailleurs illustré dans le quatrième chapitre où dans de nou veaux travaux avec Pascal Maillard, nous trouvons la renormalisation de Seneta-Heyde de la martingale additive critique dans le cas où la marche de l’épine est dans le domaine d’attraction d’une loi stable. On voit alors que les fonctions de renouvellement nous fournissent un candidat mieux adapté à cette étude que la martingale dérivée, qui n’est plus toujours une martingale dans ce nouveau contexte.Enfin, le cinquième chapitre étudie la question de l’optimalité des hypothèses faites dans le chapitre précédent quant à la trivialité ou non de la limite obtenue après renormalisation de Seneta-HeydeThe branching random walk is a particle system on the real line starting at time 0 with an initial particle at position 0, then each particle living at time n proceeds to die at time n+1 and give birth, independently from the other particles of generation n, to a random number of particles at random positions. In a first chapter, we define in details the branching random walk model and some key elements of the scientific research on this model, including the study of the additive martingale. This martingale can be stuided through its convergence towards a limit that may be trivial, raising the question of an appropriate scaling, called Seneta-Heyde sclaing, in the case the limit is trivial. The additive martingale can also be studied with stochastic recursive equations lezading to fixed points equations in law. This latter question is adressed in some unpublished works from the first year of PhD, in continuioty with works from the masters thesis. The second chapter is a translation in english of some sections of the preivous chapter so that every reader can grasp the key elements and goals of this manuscript.In a third chapter, we present a new proof developed with Pascal Maillard for Aîdékon and Shi's theorem on the Seneta-Heyde scaling of the critical additive martingale in the finite variance case. This new proof no longer need a peeling lemma and the use of second moment arguments and prefers studying the conditional Laplace transform. the properties of some renewal functions allow a much more general approach without the need to foucs to much on the derivative martingale. This is also illustrated in a fourth chapter where in new works with Pascal Maillard, we find the Seneta-Heyde scaling for the critical additive martingale in the case where the spinal random walk is in the attraction domain of a stable law. We then observe that the renewal functions provide us with a better suited candidate for this study than the derivative artingale, which is no longer always a martingale in this context.Finally, the fifth chapter focus on the question of the optimality of the assumptions made in the preivous chapter concerning the non-triviality of the limit obtained with the Seneta-Heyde scaling
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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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Mulatier, Clélia de. "A random walk approach to stochastic neutron transport." Thesis, Université Paris-Saclay (ComUE), 2015. http://www.theses.fr/2015SACLS029/document.
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L’un des principaux objectifs de la physique des réacteurs nucléaires est de caractériser la répartition aléatoire de la population de neutrons au sein d’un réacteur. Les fluctuations de cette population sont liées à la nature stochastique des interactions des neutrons avec les noyaux fissiles du milieu : diffusion, capture stérile, ou encore émission de plusieurs neutrons lors de la fission d’un noyau. L’ensemble de ces mécanismes physiques confère une structure aléatoire branchante à la trajectoire des neutrons, alors modélisée par des marches aléatoires. Avec environs 10⁸ neutrons par centimètre cube dans un réacteur de type REP à pleine puissance en conditions stationnaires, les grandeurs physiques du système (flux, taux de réaction, énergie déposée) sont, en première approximation, bien représentées par leurs valeurs moyennes respectives. Ces observables physiques moyennes obéissent alors à l’équation de transport linéaire de Boltzmann. Au cours de ma thèse, je me suis penchée sur deux aspects du transport qui ne sont pas décrits par cette équation, et pour lesquels je me suis appuyée sur des outils issus de la théorie des marches aléatoires. Tout d’abord, grâce au formalisme de Feynman-Kac, j’ai étudié les fluctuations statistiques de la population de neutrons, et plus particulièrement le phénomène de « clustering neutronique », qui a été mis en évidence numériquement pour de faibles densités de neutrons (typiquement un réacteur au démarrage). Je me suis ensuite intéressée à différentes propriétés de la statistique d’occupation des neutrons effectuant un transport anormal (càd non-exponentiel). Ce type de transport permet de modéliser le transport dans des matériaux fortement hétérogènes et désordonnés, tel que les réacteurs à lit de boulets. L’un des aspects novateurs de ce travail est la prise en compte de la présence de bords. En effet, bien que les systèmes réels soient de taille finie, la plupart des résultats théoriques pré-existants sur ces thématiques ont été obtenus sur des systèmes de taille infinieOne of the key goals of nuclear reactor physics is to determine the distribution of the neutron population within a reactor core. This population indeed fluctuates due to the stochastic nature of the interactions of the neutrons with the nuclei of the surrounding medium: scattering, emission of neutrons from fission events and capture by nuclear absorption. Due to these physical mechanisms, the stochastic process performed by neutrons is a branching random walk. For most applications, the neutron population considered is very large, and all physical observables related to its behaviour, such as the heat production due to fissions, are well characterised by their average values. Generally, these mean quantities are governed by the classical neutron transport equation, called linear Boltzmann equation. During my PhD, using tools from branching random walks and anomalous diffusion, I have tackled two aspects of neutron transport that cannot be approached by the linear Boltzmann equation. First, thanks to the Feynman-Kac backward formalism, I have characterised the phenomenon of “neutron clustering” that has been highlighted for low-density configuration of neutrons and results from strong fluctuations in space and time of the neutron population. Then, I focused on several properties of anomalous (non-exponential) transport, that can model neutron transport in strongly heterogeneous and disordered media, such as pebble-bed reactors. One of the novel aspects of this work is that problems are treated in the presence of boundaries. Indeed, even though real systems are finite (confined geometries), most of previously existing results were obtained for infinite systems
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Gabucci, Ilenia. "Random walks classici e quantistici." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/17759/.
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Questa tesi si propone di fornire uno studio sul random walk. Partendo da un approfondimento sulla teoria della probabilita` alla base di tale processo ed in particolare le distribuzioni Binomiale e Gaussiana, si `e potuto studiare il caso del random walk classico, sia nel caso del reticolo monodimensionale che per reticoli a piu` dimensioni. Sempre nell’ambito classico si sono analizzati anche i processi stocastici dipendenti dal tempo, detti Processi di Markov, e il moto browniano, per cui si sono ricavate le equazioni del moto della particella browniana, ovvero le equazioni di Langevin. Si sono in seguito definiti i quattro postulati della meccanica quantistica, introducendo il concetto di quantum bit, per cui si `e fornito un confronto con il bit classico. Si `e inoltre trattato il calcolo quantistico, definendo il concetto di porta logica e affrontando l’esempio importante del gate di Hadamard, utile nel caso del random walk quantistico. Quest’ultimo `e stato introdotto separando il caso nel discreto e nel continuo, cercando di sottolineare le differenze che presenta con il caso classico.
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Zou, Shanshan. "Empirical analysis on random walk behavior of foreign exchange rates." Thesis, Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/33836.
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This thesis conducts a comprehensive examination on the random walk behavior of 29 foreign exchange rates over the period of floating exchange regime, using variance-ratio tests. The cross-country and time-series test show that random walk model cannot be rejected on majority, and the random walk behavior is quite volatile across the whole floating exchange regime period. It then goes further to explore possible factors that can explain the probability of rejection/ non-rejections on random walk model using linear as well as nonlinear probability models, and find that the factors such as capital openness and investment-to-trade ratio significantly increases the chance of its exchange rate exhibiting random walk behavior.
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Chilukuri, Raghu N. "Spin Diffusion Associated with a Quantum Random Walk on a One-Dimensional Lattice." University of Cincinnati / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1406820159.
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fr, kaimanov@univ-rennes1. "Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1085.ps.
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Uguccioni, Marco. "Introduzione alla meccanica statistica dei random walk su network." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21027/.
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Lo scopo della tesi è quello di studiare la dinamica stocastica di particelle che realizzano un random walk su network, focalizzandosi in particolare sulle proprietà dello stato stazionario del processo.
Dopo aver introdotto i concetti fondamentali di network e random walk per singola particella, si generalizza la trattazione a N particelle non interagenti su network a capacità di trasporto e storage infinita (ISTC) e si ricava il relativo stato stazionario con una trattazione meccano-statistica, applicando il principio di massima entropia.
Successivamente, per avvicinarsi a modelli di trasporto reali, si introducono nel sistema dei vincoli sulla capacità di trasporto e di storage (1-FTC+q-FSC) e si derivano le relazioni reciproche di Onsager, mostrando ancora una volta di poter trattare il sistema in termini termodinamici.
Durante la trattazione si realizzano numerose simulazioni numeriche utilizzando codici sviluppati in C++ e ROOT, volte a confermare l'andamento analitico atteso dei sistemi studiati.
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Kipp, Andreas Michael. "Finanzprognosen versus Random-Walk-Theorie : Perspektiven eines nichtlinearen Kapitalmarktmodells /." Aachen : Shaker, 2002. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=010169925&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.
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Böhm, Walter. "The Correlated Random Walk with Boundaries. A Combinatorial Solution." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 1999. http://epub.wu.ac.at/1058/1/document.pdf.
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The transition fundions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's Theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes. (author's abstract)Series: Forschungsberichte / Institut für Statistik
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Yu, Wei, and 余韡. "Reverse Top-k search using random walk with restart." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2013. http://hdl.handle.net/10722/197515.
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With the increasing popularity of social networking applications, large volumes of graph data are becoming available. Large graphs are also derived by structure extraction from relational, text, or scientific data (e.g., relational tuple networks, citation graphs, ontology networks, protein-protein interaction graphs). Nodeto-node proximity is the key building block for many graph based applications that search or analyze the data. Among various proximity measures, random walk with restart (RWR) is widely adapted because of its ability to consider the global structure of the whole network.
Although RWR-based similarity search has been well studied before, there is no prior work on reverse top-k proximity search in graphs based on RWR. We discuss the applicability of this query and show that the direct application of existing methods on RWR-based similarity search to solve reverse top-k queries has very high computational and storage demands. To address this issue, we propose an indexing technique, paired with an on-line reverse top-k search algorithm.
In the indexing step, we compute from the graph G a graph index, which is based on a K X |V| matrix, containing in each column v the K largest approximate proximity values from v to any other node in G. K is application-dependent and represents the highest value of k in a practical reverse top-k query. At each column v of the index, the approximate values are lower bounds of the K largest proximity values from v to all other nodes.
Given the graph index and a reverse top-k query q (k _ K), we prove that the exact proximities from any node v to query q can be efficiently computed by applying the power method. By comparing these with the corresponding lower bounds taken from the k-th row of the graph index, we are able to determine which nodes are certainly not in the reverse top-k result of q. For some of the remaining nodes, we may also be able to determine that they are certainly in the reverse top-k result of q, based on derived upper bounds for the k-th largest proximity value from them. Finally, for any candidate that remains, we progressively refine its approximate proximities, until based on its lower or upper bound it can be determined not to be or to be in the result. The proximities refined during a reverse top-k are used to update the graph index, making its values progressively more accurate for future queries.
Our experimental evaluation shows that our technique is efficient and has manageable storage requirements even when applied on very large graphs. We also show the effectiveness of the reverse top-k search in the scenarios of spam detection and determining the popularity of authors.published_or_final_version
Computer Science
Master
Master of Philosophy
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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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Hashimoto, Tatsunori B. (Tatsunori Benjamin). "Continuous representations and models from random walk diffusion limits." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/105670.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2016.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 193-202).
Structured data such as sequences and networks pose substantial difficulty for traditional statistical theory which has focused on data drawn independently from a vector space. A popular and empirically effective technique for dealing with such data is to map elements of the data to a vector space and to operate over the embedding as a summary statistic. Such a vector representation of discrete objects is known as a 'continuous representation'. Continuous space models of words, objects, and signals have become ubiquitous tools for learning rich representations of data, from natural language processing to computer vision. Even in cases that the embedding is not explicit, many algorithms operate over similarity measures which implicitly embed the original dataset. In this thesis, we attempt to understand the intuition behind continuous representations. Can we construct a general theory of continuous representations? Are there general principles for semantically meaninguful representations? In order to answer these questions, we develop a framework for analyzing continuous representations through diffusion limits of random walks. We show that measureable quantities of discrete random walks with a latent metric structure have closed form diffusion limits. These diffusion limits allow us to approximate attributes of the discrete random walk such as the stationary distribution, hitting time, or co-occurrence using closed-form expressions from diffusions. We establish limits which guarantee asymptotic consistency of such estimators, and show they work well in practice. Using this new approach, we solve three classes of problems: first, we derive principled network algorithms which connect statistical estimation tasks such as density estimation to network algorithms such as PageRank. Next, we demonstrate that continuous representations of words are a type of random walk metric estimator with close connections to manifold learning. Finally, we apply our theory to single-cell RNA seq data, and derive a way to learn time-series models without trajectories by using stochastic recurrent neural networks.
by Tatsunori B. Hashimoto.
Ph. D.
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Sidenko, Sergiy. "Kac's random walk and coupon collector's process on posets." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/43781.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.Includes bibliographical references (p. 100-104).
In the first part of this work, we study a long standing open problem on the mixing time of Kac's random walk on SO(n, R) by random rotations. We obtain an upper bound mix = O (n2.5 log n) for the weak convergence which is close to the trivial lower bound [Omega] (n2). This improves the upper bound O (n4 log n) by Diaconis and SaloffCoste 1131. The proof is a variation on the coupling technique we develop to bound the mixing time for compact Markov chains, which is of independent interest. In the second part, we consider a generalization of the coupon collector's problem in which coupons are allowed to be collected according to a partial order. Along with the discrete process, we also study the Poisson version which admits a tractable parametrization. This allows us to prove convexity of the expected completion time E T with respect to sample probabilities, which has been an open question for the standard coupon collector's problem. Since the exact computation of E - is formidable, we use convexity to establish the upper and the lower bound (these bounds differ by a log factor). We refine these bounds for special classes of posets. For instance, we show the cut-off phenomenon for shallow posets that are closely connected to the classical Dixie Cup problem. We also prove the linear growth of the expectation for posets whose number of chains grows at most exponentially with respect to the maximal length of a chain. Examples of these posets are d-dimensional grids, for which the Poisson process is usually referred as the last passage percolation problem. In addition, the coupon collector's process on a poset can be used to generate a random linear extension.
(cont.) We show that for forests of rooted directed trees it is possible to assign sample probabilities such that the induced distribution over all linear extensions will be uniform. Finally, we show the connection of the process with some combinatorial properties of posets.
by Sergiy Sidenko.
Ph.D.
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Alves, Gonçalo Filipe Rodrigues. "Testing the random walk hypothesis with technical trading rules." Master's thesis, Instituto Superior de Economia e Gestão, 2015. http://hdl.handle.net/10400.5/10939.
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Mestrado em FinançasNeste trabalho são testadas as hipóteses de passeio aleatório ao mercado acionista português, examinando as dezoito ações e o índice PSI-20. Considerando cotações diárias e mensais durante o período de 1999-2015. Foram utilizados os testes Augmented Dickey-Fuller (ADF), os testes de rácio de variância automático assim como os rácios de variâncias individuais e múltiplos propostos por Lo e Mackinlay, e Chow e Denning, respetivamente. Os vários testes utilizados para confirmar a hipótese de passeio aleatório das dezoito ações assim como do índice PSI-20, obtiveram resultados mistos contra a hipótese testada. Enquanto o teste Augmented Dickey-Fuller (ADF) rejeitou a hipótese de raiz unitária para todas as ações e também para o índice PSI-20 confirmando assim um passeio aleatório. Por outro lado, os testes de rácios de variâncias, rejeitam a hipótese testada para algumas das ações consideradas assim como para o índice PSI-20, contudo tende esse número de ações tende a diminuir quando se utiliza as cotações mensais.
This paper investigates the efficiency of the eighteen stocks that constitute the main Portuguese stock index, the PSI-20 of the Lisbon Stock Exchange. Tools used for the investigation were daily and monthly data from January 1999 to May of 2015, using the Augmented Dickey-Fuller (ADF) test, the automatic variance ratio by Choi and the individual and multiple variance ratios, by Lo and Mackinlay, and, Chow and Denning, which test the efficiency of the eighteen stocks and PSI-20 index. The Augmented Dickey-Fuller (ADF) tests the null hypothesis that the series has a unit root, while the variance ratio tests the random walk hypothesis. Based on these tests, the results provide mixed evidence against the random walk hypothesis. The results for the unit root tests do not reject the efficient market hypothesis for the entire sample, while the results from the variance ratio tests do, but tend to decrease in monthly data.
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Lains, João Luís da Silva. "Testing the random walk hypothesis with variance ratio statistics." Master's thesis, Instituto Superior de Economia e Gestão, 2015. http://hdl.handle.net/10400.5/11801.
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Mestrado em FinançasEsta dissertação tem como objetivo testar a hipótese de passeio aleatório na curva das yields relativa ás obrigações do tesouro dos Estados Unidos da América para o period entre 1980 e 2014. Para alcançar este objetivo e após revisão da literatura foram efectuados testes de variância e de raiz unitária considerados os mais indicados e poderosos. Os dados necessários para a realização deste estudo foram recolhidos tendo por base um estudo da Reserva Federal dos Estados Unidos da América, que efectua cálculo das yields desde 1961 até ao presente. O método escolhido para obter os resultados referentes à raiz unitária foi o Augmented Dickey-Fuller Unit Root Test e para os testes de variância foram usados: Chow Denning (1993) multiple variance test, Joint wright multiple version of Wrights rank and sign tests e Choi (1999) Automatic Variance ratio. A amostra inclui mais de 8000 observações para cada uma das yields estudadas(1,5,10 e 20 anos Zero-Coupon e Par Yields) durante um período de 34 anos. Os resultados permitiram a detecção de diversos periodos em que o passeio aleatório nas yields das obrigações do tesouro Norte-Americano é real mas também outros em que isso não se verificou. Para isso efectuámos uma análise comparativa entre os resultados dos testes de variância e eventos marcantes na economia americana entres os quais decidimos destacar 3 períodos: a década de 80, a expansao económica dos anos 90 até inicio do século XXI e o pós-crise de 2008 onde é implementado o quantitative Easing.
The random-walk hypothesis in the U.S. treasury yield curve was not previous studied and is surprising that researchers do not filled that void by testing it. However, the U.S treasury securities market is a benchmark, as the U.S treasury is considered to be risk-free. This benchmark is used to forecast economic development, to analyse securities in other markets, to price other fixed-income securities and to hedge positions taken in other markets. This study applies Chow Denning (1993) multiple variance test, Joint wright multiple version of Wright?s rank and sign tests, Choi (1999) Automatic Variance ratio Test and we also use the well-known Augmented Dickey-Fuller unit roots test to enable us to define the methodology to be used in the study. The database used permits the estimation of relative daily variation on U.S. treasury yield curve from January 1980 to December 2014. We hope that this analysis can provide useful information to traders and investors and will make a contribution in assisting to understand the pattern and behaviour of yields movement.
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Chwu, Thomas Kai Yuan. "Random walk modelling of turbulent dispersion within the atmosphere." Thesis, University of Cambridge, 1997. https://www.repository.cam.ac.uk/handle/1810/272825.
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Rüdrich, Stefan [Verfasser]. "Random Walk Approaches to Clustering Directed Networks / Stefan Rüdrich." Berlin : Freie Universität Berlin, 2020. http://d-nb.info/1215572298/34.
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Falconer, Steven. "Subdiffusive transport in non-homogeneous media and nonlinear fractional equations." Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/subdiffusive-transport-in-nonhomogeneous-media-and-nonlinear-fractional-equations(a695fe6e-02d2-4fa1-b90b-6a57505973fc).html.
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Jamshidpey, Arash. "Population Dynamics in Random Environment, Random Walks on Symmetric Group, and Phylogeny Reconstruction." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/34623.
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This thesis concerns applications of some probabilistic tools to phylogeny reconstruction and population genetics. Modelling the evolution of species by continuous-time random walks on the signed permutation groups, we study the asymptotic medians of a set of random permutations sampled from simple random walks at time 0.25cn, for c> 0. Running k independent random walks all starting at identity, we prove that the medians approximate the ancestor (identity permutation) up to time 0.25n, while there exists a constant c>1 after which the medians loose credibility as an estimator. We study the median of a set of random permutations on the symmetric group endowed with different metrics. In particular, for a special metric of dissimilarity, called breakpoint, where the space is not geodesic, we find a large group of medians of random permutations using the concept of partial geodesics (or geodesic patches). Also, we study the Fleming-Viot process in random environment (FVRE) via martingale and duality methods. We develop the duality method to the case of time-dependent and quenched martingale problems. Using a family of dual processes we prove the convergence of the Moran processes in random environments to FVRE in Skorokhod topology. We also study the long-time behaviour of FVRE and prove the existence of equilibrium for the joint annealed-environment process and prove an ergodic theorem for the latter.
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edu, rdlyons@indiana. "Markov Chain Intersections and the Loop--Erased Walk." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1058.ps.
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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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Sundequist, Blomdahl Kristofer. "An evaluation of random-walk based clustering of multiplex networks." Thesis, Uppsala universitet, Institutionen för informationsteknologi, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-333027.
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A network, or a graph, is a mathematical construct used for modeling relationships between different entities. An extension of an ordinary network is a multiplex network. A multiplex network enables one to model different kinds of relationships between the same entities, or even to model how relationships between entities change over time. A common network analysis task is to find groups of nodes that are unusually tightly connected. This is called community detection, and is a form of clustering. The multiplex extension complicates both the notion of what a community is, and the process of finding them.This project focuses on a random-walk based local method that can be used to find communities centered around supplied seed nodes. An implementation of the methodis made which is used to evaluate its ability to detect communities in different kinds of multiplex networks.