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Dissertations / Theses on the topic 'Statistical graphs'
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Ruan, Da. "Statistical methods for comparing labelled graphs." Thesis, Imperial College London, 2014. http://hdl.handle.net/10044/1/24963.
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Due to the availability of the vast amount of graph-structured data generated in various experiment settings (e.g., biological processes, social connections), the need to rapidly identify network structural differences is becoming increasingly prevalent. In many fields, such as bioinformatics, social network analysis and neuroscience, graphs estimated from the same experimental settings are always defined on a fixed set of objects. We formalize such a problem as a labelled graph comparison problem. The main issue in this area, i.e. measuring the distance between graphs, has been extensively studied over the past few decades. Although a large distance value constitutes evidence of difference between graphs, we are more interested in the issue of inferentially justifying whether a distance value as large or larger than the observed distance could have been obtained simply by chance. However, little work has been done to provide the procedures of statistical inference necessary to formally answer this question. Permutation-based inference has been proposed as a theoretically sound approach and a natural way of tackling such a problem. However, the common permutation procedure is computationally expensive, especially for large graphs. This thesis contributes to the labelled graph comparison problem by addressing three different topics. Firstly, we analyse two labelled graphs by inferentially justifying their independence. A permutation-based testing procedure based on Generalized Hamming Distance (GHD) is proposed. We show rigorously that the permutation distribution is approximately normal for a large network, under three graph models with two different types of edge weights. The statistical significance can be evaluated without the need to resort to computationally expensive permutation procedures. Numerical results suggest the validity of this approximation. With the Topological Overlap edge weight, we suggest that the GHD test is a more powerful test to identify network differences. Secondly, we tackle the problem of comparing two large complex networks in which only localized topological differences are assumed. By applying the normal approximation for the GHD test, we propose an algorithm that can effectively detect localised changes in the network structure from two large complex networks. This algorithm is quickly and easily implemented. Simulations and applications suggest that it is a useful tool to detect subtle differences in complex network structures. Finally, we address the problem of comparing multiple graphs. For this topic, we analyse two different problems that can be interpreted as corresponding to two distinct null hypotheses: (i) a set of graphs are mutually independent; (ii) graphs in one set are independent of graphs in another set. Applications for the multiple graphs problem are commonly found in social network analysis (i) or neuroscience (ii). However, little work has been done to inferentially address the problem of comparing multiple networks. We propose two different statistical testing procedures for (i) and (ii), by again using a normality approximation for GHD. We extend the normality of GHD for the two graphs case to multiple cases, for hypotheses (i) and (ii), with two different permutation strategies. We further build a link between the test of group independence to an existing method, namely the Multivariate Exponential Random Graph Permutation model (MERGP). We show that by applying asymptotic normality, the maximum likelihood estimate of MERGP can be analytically derived. Therefore, the original, computationally expensive, inferential procedure of MERGP can be abandoned.
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Cerqueira, Andressa. "Statistical inference on random graphs and networks." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45133/tde-04042018-094802/.
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In this thesis we study two probabilistic models defined on graphs: the Stochastic Block model and the Exponential Random Graph. Therefore, this thesis is divided in two parts. In the first part, we introduce the Krichevsky-Trofimov estimator for the number of communities in the Stochastic Block Model and prove its eventual almost sure convergence to the underlying number of communities, without assuming a known upper bound on that quantity. In the second part of this thesis we address the perfect simulation problem for the Exponential random graph model. We propose an algorithm based on the Coupling From The Past algorithm using a Glauber dynamics. This algorithm is efficient in the case of monotone models. We prove that this is the case for a subset of the parametric space. We also propose an algorithm based on the Backward and Forward algorithm that can be applied for monotone and non monotone models. We prove the existence of an upper bound for the expected running time of both algorithms.Nessa tese estudamos dois modelos probabilísticos definidos em grafos: o modelo estocástico por blocos e o modelo de grafos exponenciais. Dessa forma, essa tese está dividida em duas partes. Na primeira parte nós propomos um estimador penalizado baseado na mistura de Krichevsky-Trofimov para o número de comunidades do modelo estocástico por blocos e provamos sua convergência quase certa sem considerar um limitante conhecido para o número de comunidades. Na segunda parte dessa tese nós abordamos o problema de simulação perfeita para o modelo de grafos aleatórios Exponenciais. Nós propomos um algoritmo de simulação perfeita baseado no algoritmo Coupling From the Past usando a dinâmica de Glauber. Esse algoritmo é eficiente apenas no caso em que o modelo é monotóno e nós provamos que esse é o caso para um subconjunto do espaço paramétrico. Nós também propomos um algoritmo de simulação perfeita baseado no algoritmo Backward and Forward que pode ser aplicado à modelos monótonos e não monótonos. Nós provamos a existência de um limitante superior para o número esperado de passos de ambos os algoritmos.
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Vohra, Neeru Rani. "Three dimensional statistical graphs, visual cues and clustering." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/MQ56213.pdf.
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Chandrasekaran, Venkat. "Convex optimization methods for graphs and statistical modeling." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/66002.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 209-220).
An outstanding challenge in many problems throughout science and engineering is to succinctly characterize the relationships among a large number of interacting entities. Models based on graphs form one major thrust in this thesis, as graphs often provide a concise representation of the interactions among a large set of variables. A second major emphasis of this thesis are classes of structured models that satisfy certain algebraic constraints. The common theme underlying these approaches is the development of computational methods based on convex optimization, which are in turn useful in a broad array of problems in signal processing and machine learning. The specific contributions are as follows: -- We propose a convex optimization method for decomposing the sum of a sparse matrix and a low-rank matrix into the individual components. Based on new rank-sparsity uncertainty principles, we give conditions under which the convex program exactly recovers the underlying components. -- Building on the previous point, we describe a convex optimization approach to latent variable Gaussian graphical model selection. We provide theoretical guarantees of the statistical consistency of this convex program in the high-dimensional scaling regime in which the number of latent/observed variables grows with the number of samples of the observed variables. The algebraic varieties of sparse and low-rank matrices play a prominent role in this analysis. -- We present a general convex optimization formulation for linear inverse problems, in which we have limited measurements in the form of linear functionals of a signal or model of interest. When these underlying models have algebraic structure, the resulting convex programs can be solved exactly or approximately via semidefinite programming. We provide sharp estimates (based on computing certain Gaussian statistics related to the underlying model geometry) of the number of generic linear measurements required for exact and robust recovery in a variety of settings. -- We present convex graph invariants, which are invariants of a graph that are convex functions of the underlying adjacency matrix. Graph invariants characterize structural properties of a graph that do not depend on the labeling of the nodes; convex graph invariants constitute an important subclass, and they provide a systematic and unified computational framework based on convex optimization for solving a number of interesting graph problems. We emphasize a unified view of the underlying convex geometry common to these different frameworks. We describe applications of both these methods to problems in financial modeling and network analysis, and conclude with a discussion of directions for future research.
by Venkat Chandrasekaran.
Ph.D.
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Piotet, Fabien. "Statistical properties of the eigenfunctions on quantum graphs." Thesis, University of Bristol, 2009. http://hdl.handle.net/1983/af3ec57a-5d16-4995-936f-c310696f1093.
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Phadnis, Miti. "Statistical Analysis of Linear Analog Circuits Using Gaussian Message Passing in Factor Graphs." DigitalCommons@USU, 2009. https://digitalcommons.usu.edu/etd/504.
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This thesis introduces a novel application of factor graphs to the domain of analog circuits. It proposes a technique of leveraging factor graphs for performing statistical yield analysis of analog circuits that is much faster than the standard Monte Carlo/Simulation Program With Integrated Circuit Emphasis (SPICE) simulation techniques. We have designed a tool chain to model an analog circuit and its corresponding factor graph and then use a Gaussian message passing approach along the edges of the graph for yield calculation. The tool is also capable of estimating unknown parameters of the circuit given known output statistics through backward message propagation in the factor graph. The tool builds upon the concept of domain-specific modeling leveraged for modeling and interpreting different kinds of analog circuits. Generic Modeling Environment (GME) is used to design modeling environment for analog circuits. It is a configurable tool set that supports creation of domain-specific design environments for different applications. This research has developed a generalized methodology that could be applied towards design automation of different kinds of analog circuits, both linear and nonlinear. The tool has been successfully used to model linear amplifier circuits and a nonlinear Metal Oxide Semiconductor Field Effect Transistor (MOSFET) circuit. The results obtained by Monte Carlo simulations performed on these circuits are used as a reference in the project to compare against the tool's results. The tool is tested for its efficiency in terms of time and accuracy against the standard results.
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Hamdi, Maziyar. "Statistical signal processing on dynamic graphs with applications in social networks." Thesis, University of British Columbia, 2015. http://hdl.handle.net/2429/56256.
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Due to the proliferation of social networks and their significant effects on our day-to-day activities, there has been a growing interest in modeling and analyzing behavior of agents in social networks over the past decade. The unifying theme of this thesis is to develop a set of mathematical theories and algorithmic tools for different estimation and sensing problems over graphs with applications to social networks. The first part of this dissertation is devoted to multi-agent Bayesian estimation and learning problem in social networks. We consider a set of agents that interact over a network to estimate an unknown parameter called state of nature. As a result of the recursive nature of Bayesian models and the correlation introduced by the structure of the underlying communication graph, information collected by one agent can be mistakenly considered independent, that is, mis-information prop- agation, also known as data incest arises. This part presents data incest removal algorithms that ensure complete mitigation of the mis-information associated with the estimates of agents in two different information exchange patterns: First, a scenario where beliefs (posterior distribution of state of nature) are transmitted over the network. Second, a social learning context where agents map their local beliefs into a finite set of actions and broadcast their actions to other agents. We also present a necessary and sufficient condition on the structure of information flow graph to mitigate mis-information propagation. The second part of the thesis considers a Markov-modulated duplication-deletion random graph where at each time instant, one node can either join or leave the network; the probabilities of joining or leaving evolve according to the realization of a finite state Markov chain. This part presents two results. First, motivated by social network applications, the asymptotic behavior of the degree distribution is analyzed. Second, a stochastic approximation algorithm is presented to track empirical degree distribution as it evolves over time. The tracking performance of the algorithm is analyzed in terms of mean square error and a functional central limit theorem is presented for the asymptotic tracking error.Applied Science, Faculty of
Electrical and Computer Engineering, Department of
Graduate
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Alberici, Diego. "Statistical Mechanics of Monomer-Dimer Models on Complete and Erdös-Rényi Graphs." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2012. http://amslaurea.unibo.it/4169/.
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Nella tesi sono trattate due famiglie di modelli meccanico statistici su vari grafi: i modelli di spin ferromagnetici (o di Ising) e i modelli di monomero-dimero.
Il primo capitolo è dedicato principalmente allo studio del lavoro di Dembo e Montanari, in cui viene risolto il modello di Ising su grafi aleatori.
Nel secondo capitolo vengono studiati i modelli di monomero-dimero, a partire dal lavoro di Heilemann e Lieb,con l'intento di dare contributi nuovi alla teoria. I principali temi trattati sono disuguaglianze di correlazione, soluzioni esatte su alcuni grafi ad albero e sul grafo completo, la concentrazione dell'energia libera intorno al proprio valor medio sul grafo aleatorio diluito di Erdös-Rényi.
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Díaz-Levicoy, Danilo, Miluska Osorio, Pedro Arteaga, and Francisco Rodríguez-Alveal. "Gráficos estadísticos en libros de texto de matemática de educación primaria en Perú." BOLEMA Departamento de Matematica, 2018. http://hdl.handle.net/10757/624628.
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El presente artículo presenta los resultados del análisis de los gráficos estadísticos según las directrices curriculares y su implementación en dieciocho libros de texto de matemática de Educación Primaria en Perú, los que corresponden a tres series completas y de diferentes editoriales. En ellos se analizan, mediante análisis de contenido, las secciones en las que aparecen estas representaciones, identificado el tipo de actividad que se plantea, los gráficos involucrados, el nivel de lectura y el nivel de complejidad semiótica involucrado. Los libros de texto se adecúan parcialmente a las directrices curriculares en cuanto a la presentación de los gráficos por nivel educativo, el número de actividades propuestas por las tres editoriales es similar. La principal actividad que se solicita en los libros es de calcular y construir. Se observa un predominio del gráfico de barras, un nivel de lectura básico y la representación de una distribución de datos en el gráfico.Revisión por pares
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Leger, Jean-Benoist. "Modelling the topology of ecological bipartite networks with statistical models for heterogeneous random graphs." Paris 7, 2014. http://www.theses.fr/2014PA077185.
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Un réseau écologique constitue une représentation de l'ensemble des interactions entre espèces dans un contexte donné. L'analyse de la structure topologique de ces réseaux permet aux écologues d'identifier et de comprendre les processus sous-jacents. La détection de sous-groupes d'espèces interagissant fortement ensemble, souvent nommés communautés o compartiments, est un des principaux moyens pour interpréter la structure sous-jacente des réseaux. Il existe de nombreuses méthodes de classification non supervisée, qui peuvent être utilisées pour analyser des réseaux écologiques. L'analyse des réseaux est actuellement un sujet de recherche en pleine expansion avec des applications dans des domaines de recherches variés. À notre connaissance, il n'existe pas de comparaison des méthodes de classification non supervisée dans le cas des réseaux écologique. Nous avons effectué une revue des méthodes disponibles de classification non supervisées des noeuds, et nous les avons comparé dans un contexte écologique. Afin d'évaluer la contribution des différents processus expliquant la structure d'un réseau, nous avons introduit de l'information extérieure au réseau. Nous avons analysé deux réseaux d'interaction arbre-champignon et arbre-insecte. Ces résultats sont préliminaires, mais la méthode semble ouvrir des perspectives intéressantes en écologie. Nous avons également étudié un réseau écologique de nature différente, un réseau de reproduction entre arbres. Nous avons utilisé ces résultats pour discuter d'un concept central en écologie, le concept d'espèceAn ecological network is a representation of the whole set of interactions between species in a given context. Ecological scientists analyse the topological structure of such networks, in order to understand the underlying processes. The identification of sub-groups of highly-interacting species (usually called communities, or compartments) is an important stream of research. The most popular method for the search of communities in ecological networks is the modularity optimization method. However this popularity is more due to the first paper published on this topic than to a rational choice based on solid grounds. There are many other clustering methods that could be used to delimit communities in ecological networks. The analysis of complex networks is indeed a rapidly growing topic with many applications in several scientific fields. To our knowledge, no comparison of different clustering methods is available in the case of ecological networks. Here we reviewed the whole set of methods available for clustering networks and we compared them using an ecological benchmark. In order to assess the relative contribution of several processes to the network structure, we integrated exogenous information in the clustering model. We analysed two bipartite antagonistic networks with this method, a tree-fungus and tree-insect network. The results are still preliminary but the method seems to us very promising for future ecological studies. Finally we searched communities in a different kind of network, a mating network between individuals belonging to two hybridizing tree species. We used our results to discuss a concept which is central in ecology, the species concept
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Matulewicz, Gustaw. "Statistical inference of Ornstein-Uhlenbeck processes : generation of stochastic graphs, sparsity, applications in finance." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLX066/document.
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Le sujet de cette thèse est l'inférence statistique de processus d'Ornstein-Uhlenbeck multi-dimensionnels. Dans une première partie, nous introduisons un modèle de graphes stochastiques définis comme observations binaires de trajectoires. Nous montrons alors qu'il est possible de déduire la dynamique de la trajectoire sous-jacente à partir des observations binaires. Pour ceci, nous construisons des statistiques à partir du graphe et montrons de nouvelles propriétés de convergence dans le cadre d'une observation en temps long et en haute fréquence. Nous analysons aussi les propriétés des graphes stochastiques du point de vue des réseaux évolutifs. Dans une deuxième partie, nous travaillons sous l'hypothèse d'information complète et en temps continu et ajoutons une hypothèse de sparsité concernant le paramètre de textit{drift} du processus d'Ornstein-Uhlenbeck. Nous montrons alors des propriétés d'oracle pointues de l'estimateur Lasso, prouvons une borne inférieure sur l'erreur d'estimation au sens minimax et démontrons des propriétés d'optimalité asymptotique de l'estimateur Lasso Adaptatif. Nous appliquons ensuite ces méthodes pour estimer la vitesse de retour à la moyenne des retours journaliers d'actions américaines ainsi que des prix de futures de dividendes pour l'indice EURO STOXX 50The subject if this thesis is the statistical inference of multi-dimensional Ornstein-Uhlenbeck processes. In a first part, we introduce a model of stochastic graphs, defined as binary observations of a trajectory. We show then that it is possible to retrieve the dynamic of the underlying trajectory from the binary observations. For this, we build statistics of the stochastic graph and prove new results on their convergence in the long-time, high-frequency setting. We also analyse the properties of the stochastic graph from the point of view of evolving networks. In a second part, we work in the setting of complete information and continuous time. We add then a sparsity assumption applied to the drift matrix coefficient of the Ornstein-Uhlenbeck process. We prove sharp oracle inequalities for the Lasso estimator, construct a lower bound on the estimation error for sparse estimators and show optimality properties of the Adaptive Lasso estimator. Then, we apply the methods to estimate mean-return properties of real-world financial datasets: daily returns of SP500 components and EURO STOXX 50 Dividend Future prices
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FRAGOSO, LUANE DA COSTA PINTO LINS. "INTEGRATION OF LINGUISTIC AND GRAPHIC INFORMATION IN MULTIMODAL COMPREHENSION OF STATISTICAL GRAPHS: A PSYCHOLINGUISTIC ASSESSMENT." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2015. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=25595@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCOORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
Esta tese possui como objetivo investigar o mapeamento entre o conteúdo de sentenças e aquele apresentado em gráficos no processo de compreensão multimodal. Assume-se uma abordagem experimental, baseada nos aportes teórico-metodológicos da Psicologia Cognitiva e da Psicolinguística, aliada a discussões pertinentes à área de Educação Matemática e aos estudos sobre multimodalidade e letramento. Consideram-se duas propostas acerca da integração entre informação linguística e visual: uma vinculada à hipótese de modularidade representacional de Jackendoff (1996), em que se defende a ideia de módulos de interface, de natureza híbrida, e uma proposta alternativa, assumida no presente trabalho, segundo a qual tanto o processamento linguístico como o visual gerariam representações de natureza abstrata/proposicional, que seriam integradas em uma interface conceitual. Buscou-se verificar (i) se fatores top-down como conhecimento prévio do assunto afetam essa integração e (ii) em que medida informação linguística instaura expectativas acerca da informação expressa no gráfico. Foram conduzidos dois experimentos de comparação sentença-figura com gráficos de coluna e de linha, utilizando o programa psyscope, e um envolvendo gráficos de linha com a técnica de rastreamento ocular. Não foram encontradas evidências de efeitos top-down no experimento com gráfico de colunas. Foram obtidos, contudo, efeitos significativos para tempo de resposta associados a outros fatores, quais sejam correção do gráfico, expressão lexical usada para comparar itens do gráfico (maior vs menor, p. ex.) e número de itens referidos na sentença a serem localizados no gráfico. Nos dois experimentos com gráficos de linha, as variáveis independentes foram (i) congruência (linha congruente/incongruente em relação ao verbo – exemplo: linha inclinada para cima ou para baixo vs. verbo subir) e (ii) correção do gráfico em expressar o conteúdo da frase, manipulada com alterações na linha e na ordenação (ascendente/descendente) de informação temporal no eixo x. No experimento com psyscope, os resultados indicaram não haver dificuldade de julgar a compatibilidade frase/gráfico quando congruência e correção não divergiam. Para tempo de resposta, houve efeito principal de congruência e correção, com menores tempos associados, respectivamente, às condições em que a linha era congruente com o verbo e o gráfico correto. Também houve efeito de interação entre as variáveis. No experimento com rastreador ocular, foram analisados índice de acertos, número e tempo total de duração das fixações e trajetória do olhar nas áreas de interesse demarcadas. Em relação a índice de acerto, assim como no experimento com psycope, maior dificuldade de processamento estava associada à condição incongruente correta, em que há quebra de expectativa em relação à posição da linha (vs. verbo) e ao modo usual de organização dos gráficos no eixo x. Quanto aos movimentos oculares, na área do gráfico, observou-se maior número e tempo total de duração das fixações nas condições corretas; na área da frase, tais condições apresentaram resultados opostos. Quanto à trajetória do olhar, os dados sugerem ser a informação linguística acessada em primeiro lugar, orientando a leitura do gráfico. Considerando os resultados em conjunto, pode-se afirmar que o custo de integração é determinado pela compatibilidade (ou não) entre as proposições geradas pelos módulos linguístico e visual.
This thesis aims at investigating the mapping between the sentential content and the content presented in graphs in a multimodal comprehension process. We assume an experimental approach, based on Cognitive Pyschology and Psycholinguistics methodological and theoretical contributions as well as literacy and multimodality studies. Two proposals concerning the integration between linguistic and visual information are considered: one linked to Jackendoff s (1996) representational modularity hypothesis, in which, the idea of interface modules, of hybrid nature, is defended; and an alternative one according to which linguistic and visual processing could generate propositional/abstract representations which could be integrated into a conceptual interface. We tried to check (i) if top-down aspects such as prior knowledge can affect this integration and (ii) in what extent linguistic information may bring expectations about the information expressed in the graph. Sentence-picture comparison experiments were conducted with line and columns graphs using the pyscope software, and another one concerning line graphs with eye tracking technique. Top-down effects were not found in columns graphs experiment. However, significant effects related to response time associated with other aspects such as graph accuracy, lexical expression used in order to compare graph elements (larger x smaller, for example) and the number of elements in the sentence that must be found in the graph. In both experiments with line graphs, the independent variables were (i) congruency (congruent/incongruent line in relation to the verb - line up or down vs verb increase) and (ii) accuracy of the graph in order to express the content of the sentence, manipulated with changes in the line and time information order (ascendant/descendent) in x axis. In psyscope experiment, there was no difficulty in judging the sentence-picture compatibility when congruency and correction were not different. Concerning the response time, there was effect of congruency and correction, with shorter times associated, respectively, to the conditions in which line was congruent to the verb and correct graph. There was also effect of interaction. In eye tracking experiment, accuracy rates, number of fixations, total fixation duration and the scanpath in areas of interest were analysed. In relation to accuracy rates, similar to psyscope experiment, more difficulty in processing was associated to incongruent/incorrect condition, in which there is a break in the expectation related to the line position (vs.verb) and the common organization of the elements displayed in x axis. Concerning eye movements, in the graph area, number of fixations and total fixation duration were higher in correct conditions; in the sentence area, these results were opposite. Analyzing the scanpath, data suggest that linguistic information is accessed first, guiding the graph reading. To conclude, it s possible to state that the cost of integration is determined by compatibility (or not) between the propositions from both linguistic and visual modules.
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van, Woerden Irene. "A statistical investigation of the risk factors for tuberculosis." Thesis, University of Canterbury. Mathematics and Statistics, 2013. http://hdl.handle.net/10092/8662.
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Tuberculosis (TB) is called a disease of poverty and is the main cause of death from infectious diseases among adults. In 1993 the World Health Organisation (WHO) declared TB to be a global emergency; however there were still approximately 1.4 million deaths due to TB in 2011. This thesis contains a detailed study of the existing literature regarding the global risk factors of TB. The risk factors identified from the literature review search which were also available from the NFHS-3 survey were then analysed to determine how well we could identify respondents who are at high risk of TB. We looked at the stigma and misconceptions people have regarding TB and include detailed reports from the existing literature of how a persons wealth, health, education, nutrition, and HIV status affect how likely the person is to have TB. The difference in the risk factor distribution for the TB and non-TB populations were examined and classification trees, nearest neighbours, and logistic regression models were trialled to determine if it was possible for respondents who were at high risk of TB to be identified. Finally gender-specific statistically likely directed acyclic graphs were created to visualise the most likely associations between the variables.
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Decelle, Aurélien. "Statistical physics of disordered networks - Spin Glasses on hierarchical lattices and community inference on random graphs." Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00653375.
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Cette thèse aborde des aspects fondamentales et appliquées de la théorie des verres de spin etplus généralement des systèmes complexes. Les premiers modèles théoriques décrivant la transitionvitreuse sont apparues dans les années 1970. Ceux-ci décrivaient les verres à l'aide d'interactionsaléatoires. Il a fallu alors plusieurs années avant qu'une théorie de champs moyen pour ces systèmessoient comprises. De nos jours il existe un grand nombre de modèles tombant dans la classe de" champs moyen " et qui sont bien compris à la fois analytiquement, mais également numériquementgrâce à des outils tels que le monte-carlo ou la méthode de la cavité. Par ailleurs il est bien connu quele groupe de renormalisation a échoué jusque ici à pouvoir prédire le comportement des observablescritiques dans les verres hors champs moyen. Nous avons donc choisi d'étudier des systèmes eninteraction à longue portée dont on ignore encore si la physique est identique à celle du champmoyen. Nous avons montré dans une première partie, la facilité avec laquelle on peut décrire unetransformation du groupe de renormalisation dans les systèmes ferromagnétiques en interaction àlongue portée dé finies sur le réseau hiérarchique de Dyson. Dans un second temps, nous avons portéenotre attention sur des modèles de verre de spin sur ce même réseau. Un début d'analyse sur cestransformations dans l'espace réel est présenté ainsi qu'une comparaison de la mesure de l'exposantcritique nu par différentes méthodes. Si la transformation décrite semble prometteuse il faut cependantnoter que celle-ci doit encore être améliorée afin d'être considérée comme une méthode valide pournotre système. Nous avons continué dans cette même direction en analysant un modèle d'énergiesaléatoires toujours en utilisant la topologie du réseau hiérarchique. Nous avons étudié numériquementce système dans lequel nous avons pu observer l'existence d'une transition de phase de type " criseentropique " tout à fait similaire à celle du REM de Derrida. Toutefois, notre modèle présente desdifférences importantes avec ce dernier telles que le comportement non-analytique de l'entropie à latransition, ainsi que l'émergence de " criticalité " dont la présence serait à confirmer par d'autres études.Nous montrons également à l'aide de notre méthode numérique comment la température critique dece système peut-être estimée de trois façon différentes.Dans une dernière partie nous avons abordé des problèmes liés aux systèmes complexes. Il aété remarqué récemment que les modèles étudiés dans divers domaines, par exemple la physique, labiologie ou l'informatique, étaient très proches les uns des autres. Ceci est particulièrement vrai dansl'optimisation combinatoire qui a en partie été étudiée par des méthodes de physique statistique. Cesméthodes issues de la théories des verres de spin et des verres structuraux ont été très utilisées pourétudier les transitions de phase qui ont lieux dans ces systèmes ainsi que pour inventer de nouveauxalgorithmes pour ces modèles. Nous avons étudié le problème de l'inférence de modules dans lesréseaux à l'aide de ces même méthodes. Nous présentons une analyse sur la détection des modules topologiques dans des réseaux aléatoires et démontrons la présence d'une transition de phase entre une région où ces modules sont indétectables et une région où ils sont détectables. Par ailleurs, nous avons implémenté pour ces problèmes un algorithme utilisant Belief Propagation afin d'inférer les modules ainsi que d'apprendre leurs propriétés en ayant pour unique information la structure du réseau. Finalementnous avons appliqué cet algorithme sur des réseaux construits à partir de données réelles et discutonsles développements à apporter à notre méthode.
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Schmidt, Hinnerk Christian. "Statistical Physics of Sparse and Dense Models in Optimization and Inference." Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLS366.
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Une donnée peut avoir diverses formes et peut provenir d'un large panel d'applications. Habituellement, une donnée possède beaucoup de bruit et peut être soumise aux effets du hasard. Les récents progrès en apprentissage automatique ont relancé les recherches théoriques sur les limites des différentes méthodes probabilistes de traitement du signal. Dans cette thèse, nous nous intéressons aux questions suivantes : quelle est la meilleure performance possible atteignable ? Et comment peut-elle être atteinte, i.e., quelle est la stratégie algorithmique optimale ?La réponse dépend de la forme des données. Les sujets traités dans cette thèse peuvent tous être représentés par des modèles graphiques. Les propriétés des données déterminent la structure intrinsèque du modèle graphique correspondant. Les structures considérées ici sont soit éparses, soit denses. Les questions précédentes peuvent être étudiées dans un cadre probabiliste, qui permet d'apporter des réponses typiques. Un tel cadre est naturel en physique statistique et crée une analogie formelle avec la physique des systèmes désordonnés. En retour, cela permet l'utilisation d'outils spécifiques à ce domaine et de résoudre des problèmes de satisfaction de contraintes et d'inférence statistique. La problématique de performance optimale est directement reliée à la structure des extrema de la fonction d'énergie libre macroscopique, tandis que les aspects algorithmiques proviennent eux de la minimisation de la fonction d'énergie libre microscopique (c'est-à-dire, dans la forme de Bethe).Cette thèse est divisée en quatre parties. Premièrement, nous aborderons par une approche de physique statistique le problème de la coloration de graphes aléatoires et mettrons en évidence un certain nombre de caractéristiques. Dans un second temps, nous calculerons une nouvelle limite supérieure de la taille de l'ensemble contagieux. Troisièmement, nous calculerons le diagramme de phase du modèle de Dawid et Skene dans la région dense en modélisant le problème par une factorisation matricielle de petit rang. Enfin, nous calculerons l'erreur optimale de Bayes pour une classe restreinte de l'estimation matricielle de rang élevéDatasets come in a variety of forms and from a broad range of different applications. Typically, the observed data is noisy or in some other way subject to randomness. The recent developments in machine learning have revived the need for exact theoretical limits of probabilistic methods that recover information from noisy data. In this thesis we are concerned with the following two questions: what is the asymptotically best achievable performance? And how can this performance be achieved, i.e., what is the optimal algorithmic strategy? The answer depends on the properties of the data. The problems in this thesis can all be represented as probabilistic graphical models. The generative process of the data determines the structure of the underlying graphical model. The structures considered here are either sparse random graphs or dense (fully connected) models. The above questions can be studied in a probabilistic framework, which leads to an average (or typical) case answer. Such a probabilistic formulation is natural to statistical physics and leads to a formal analogy with problems in disordered systems. In turn, this permits to harvest the methods developed in the study of disordered systems, to attack constraint satisfaction and statistical inference problems. The formal analogy can be exploited as follows. The optimal performance analysis is directly related to the structure of the extrema of the macroscopic free energy. The algorithmic aspects follow from the minimization of the microscopic free energy (that is, the Bethe free energy in this work) which is closely related to message passing algorithms. This thesis is divided into four contributions. First, a statistical physics investigation of the circular coloring problem is carried out that reveals several distinct features. Second, new rigorous upper bounds on the size of minimal contagious sets in random graphs, with bounded maximum degree, are obtained. Third, the phase diagram of the dense Dawid-Skene model is derived by mapping the problem onto low-rank matrix factorization. The associated approximate message passing algorithm is evaluated on real-world data. Finally, the Bayes optimal denoising mean square error is derived for a restricted class of extensive rank matrix estimation problems
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Araujo, Marcus Vinicius. "Gráficos estatísticos: uma postura crítica." Universidade Federal de Juiz de Fora (UFJF), 2016. https://repositorio.ufjf.br/jspui/handle/ufjf/3849.
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O presente trabalho tem por objetivo instigar nos leitores que tenham acesso a ele,um olhar mais crítico e questionador em relação aos gráficos estatísticos que surgem diariamente na mídia, e também servir como material de apoio em uma aula de Estatística, esclarecendo aos discentes a importância da compreensão de tal conteúdo para um pleno exercer da cidadania. Para atingir a tais fins, o mesmo foi dividido em duas partes. A primeira, um referencial teórico, traz uma revisão sobre os conceitos iniciais da Estatística, mais especificamente sobre as estruturas necessárias para compreensão e construção de gráficos. A segunda, um estudo de cinco casos (cinco gráficos estatísticos divulgados pela mídia) e as respectivas análises sobre cada um deles, pretende assim levar o leitor a uma postura mais amadurecida e cautelosa perante os mesmos.
This study aims to entice in readers who have access to it, a look more critical and questioning in relation to the statistical graphs that appear daily in the media, and also serveassupportmaterialinaStatisticsclass,explainingtostudentstheimportanceofsuch content understanding to a full exercise of citizenship. To achieve such purposes, it was divided into two parts. The first, a theoretical referential, provides a review of the initial concepts of Statistics, more specifically on the structures necessary for understanding and building graphics. The second one, a study of five cases (five statistical charts released by the media) and the analysis of each of them, plans to take the reader to become aware of such material to a more mature and cautious stance towards them.
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Grönlund, Andreas. "Complex patterns : from physical to social interactions." Doctoral thesis, Umeå universitet, Institutionen för fysik, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-801.
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Interactions are what gives us the knowledge of the world around us. Interactions on all levels may fundamentally be seen as an exchange of information and a possible response of the same. Whether it is an electron in an electrical field or a handsome dude in a bar responding to a flirtation---interactions make things happen. In this sense we can see that objects without the capability of interacting with each other also are invisible to each other. Chains of pairwise interacting entities can serve as mediators of indirect interactions between objects. Nonetheless, in the limit of no interactions, we get into a philosophical debate whether we actually may consider anything to exist since it can not be detected in any way. Interactions between matter tend to be organized and show a hierarchical structure in which smaller sub-systems can be seen as parts of a bigger system, which in turn might be a smaller part of an even bigger system. This is reflected by the fact that we have sciences that successfully study specific interactions between objects or matter---physics, chemistry, biology, ecology, sociology,... What happens in a situation where all length scales are important? How does the structure of the underlying network of interactions affect the dynamical properties of a system? What network structures do we find and how are they created? This thesis is a physicist's view of collective dynamics, from superconductors to social systems and navigation in city street networks.
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Rombach, Michaela Puck. "Colouring, centrality and core-periphery structure in graphs." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:7326ecc6-a447-474f-a03b-6ec244831ad4.
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Krivelevich and Patkós conjectured in 2009 that χ(G(n, p)) ∼ χ=(G(n, p)) ∼ χ∗=(G(n, p)) for C/n < p < 1 − ε, where ε > 0. We prove this conjecture for n−1+ε1 < p < 1 − ε2 where ε1, ε2 > 0. We investigate several measures that have been proposed to indicate centrality of nodes in networks, and find examples of networks where they fail to distinguish any of the vertices nodes from one another. We develop a new method to investigate core-periphery structure, which entails identifying densely-connected core nodes and sparsely-connected periphery nodes. Finally, we present an experiment and an analysis of empirical networks, functional human brain networks. We found that reconfiguration patterns of dynamic communities can be used to classify nodes into a stiff core, a flexible periphery, and a bulk. The separation between this stiff core and flexible periphery changes as a person learns a simple motor skill and, importantly, it is a good predictor of how successful the person is at learning the skill. This temporally defined core-periphery organisation corresponds well with the core- periphery detected by the method that we proposed earlier the static networks created by averaging over the subjects dynamic functional brain networks.
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Miolane, Léo. "Fundamental limits of inference : a statistical physics approach." Thesis, Paris Sciences et Lettres (ComUE), 2019. http://www.theses.fr/2019PSLEE043.
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Nous étudions des problèmes statistiques classiques, tels que la détection de communautés dans un graphe, l’analyse en composantes principales, les modèles de mélanges Gaussiens, les modèles linéaires (généralisés ou non), dans un cadre Bayésien. Nous calculons pour ces problèmes le “risque de Bayes” qui est la plus petite erreur atteignable par une méthode statistique, dans la limite de grande dimension. Nous observons alors un phénomène surprenant : dans de nombreux cas il existe une valeur critique de l’intensité du bruit au-delà de laquelle il n’est plus possible d’extraire de l’information des données. En dessous de ce seuil, nous comparons la performance d’algorithmes polynomiaux à celle optimale. Dans de nombreuses situations nous observons un écart : bien qu’il soit possible – en théorie – d’estimer le signal, aucune méthode algorithmiquement efficace ne parvient à être optimale. Dans ce manuscrit, nous adoptons une approche issue de la physique statistique qui explique ces phénomènes en termes de transitions de phase. Les méthodes et outils que nous utilisons proviennent donc de la physique, plus précisément de l’étude mathématique des verres de spinsWe study classical statistical problems such as community detection on graphs, Principal Component Analysis (PCA), sparse PCA, Gaussian mixture clustering, linear and generalized linear models, in a Bayesian framework. We compute the best estimation performance (often denoted as “Bayes Risk”) achievable by any statistical method in the high dimensional regime. This allows to observe surprising phenomena: for many problems, there exists a critical noise level above which it is impossible to estimate better than random guessing. Below this threshold, we compare the performance of existing polynomial-time algorithms to the optimal one and observe a gap in many situations: even if non-trivial estimation is theoretically possible, computationally efficient methods do not manage to achieve optimality. From a statistical physics point of view that we adopt throughout this manuscript, these phenomena can be explained by phase transitions. The tools and methods of this thesis are therefore mainly issued from statistical physics, more precisely from the mathematical study of spin glasses
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Cami, Aurel. "ANALYZING THE COMMUNITY STRUCTURE OF WEB-LIKE NETWORKS: MODELS AND ALGORITHMS." Doctoral diss., University of Central Florida, 2005. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/2590.
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This dissertation investigates the community structure of web-like networks (i.e., large, random, real-life networks such as the World Wide Web and the Internet). Recently, it has been shown that many such networks have a locally dense and globally sparse structure with certain small, dense subgraphs occurring much more frequently than they do in the classical Erdös-Rényi random graphs. This peculiarity--which is commonly referred to as community structure--has been observed in seemingly unrelated networks such as the Web, email networks, citation networks, biological networks, etc. The pervasiveness of this phenomenon has led many researchers to believe that such cohesive groups of nodes might represent meaningful entities. For example, in the Web such tightly-knit groups of nodes might represent pages with a common topic, geographical location, etc., while in the neural networks they might represent evolved computational units. The notion of community has emerged in an effort to formalize the empirical observation of the locally dense globally sparse structure of web-like networks. In the broadest sense, a community in a web-like network is defined as a group of nodes that induces a dense subgraph which is sparsely linked with the rest of the network. Due to a wide array of envisioned applications, ranging from crawlers and search engines to network security and network compression, there has recently been a widespread interest in finding efficient community-mining algorithms. In this dissertation, the community structure of web-like networks is investigated by a combination of analytical and computational techniques: First, we consider the problem of modeling the web-like networks. In the recent years, many new random graph models have been proposed to account for some recently discovered properties of web-like networks that distinguish them from the classical random graphs. The vast majority of these random graph models take into account only the addition of new nodes and edges. Yet, several empirical observations indicate that deletion of nodes and edges occurs frequently in web-like networks. Inspired by such observations, we propose and analyze two dynamic random graph models that combine node and edge addition with a uniform and a preferential deletion of nodes, respectively. In both cases, we find that the random graphs generated by such models follow power-law degree distributions (in agreement with the degree distribution of many web-like networks). Second, we analyze the expected density of certain small subgraphs--such as defensive alliances on three and four nodes--in various random graphs models. Our findings show that while in the binomial random graph the expected density of such subgraphs is very close to zero, in some dynamic random graph models it is much larger. These findings converge with our results obtained by computing the number of communities in some Web crawls. Next, we investigate the computational complexity of the community-mining problem under various definitions of community. Assuming the definition of community as a global defensive alliance, or a global offensive alliance we prove--using transformations from the dominating set problem--that finding optimal communities is an NP-complete problem. These and other similar complexity results coupled with the fact that many web-like networks are huge, indicate that it is unlikely that fast, exact sequential algorithms for mining communities may be found. To handle this difficulty we adopt an algorithmic definition of community and a simpler version of the community-mining problem, namely: find the largest community to which a given set of seed nodes belong. We propose several greedy algorithms for this problem: The first proposed algorithm starts out with a set of seed nodes--the initial community--and then repeatedly selects some nodes from community's neighborhood and pulls them in the community. In each step, the algorithm uses clustering coefficient--a parameter that measures the fraction of the neighbors of a node that are neighbors themselves--to decide which nodes from the neighborhood should be pulled in the community. This algorithm has time complexity of order , where denotes the number of nodes visited by the algorithm and is the maximum degree encountered. Thus, assuming a power-law degree distribution this algorithm is expected to run in near-linear time. The proposed algorithm achieved good accuracy when tested on some real and computer-generated networks: The fraction of community nodes classified correctly is generally above 80% and often above 90% . A second algorithm based on a generalized clustering coefficient, where not only the first neighborhood is taken into account but also the second, the third, etc., is also proposed. This algorithm achieves a better accuracy than the first one but also runs slower. Finally, a randomized version of the second algorithm which improves the time complexity without affecting the accuracy significantly, is proposed. The main target application of the proposed algorithms is focused crawling--the selective search for web pages that are relevant to a pre-defined topic.Ph.D.
School of Computer Science
Engineering and Computer Science
Computer Science
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Bell, M. L., M. H. Fiero, H. M. Dhillon, V. J. Bray, and J. L. Vardy. "Statistical controversies in cancer research: using standardized effect size graphs to enhance interpretability of cancer-related clinical trials with patient-reported outcomes." Oxford University Press, 2017. http://hdl.handle.net/10150/626025.
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Patient reported outcomes (PROs) are becoming increasingly important in cancer studies, particularly with the emphasis on patient centered outcome research. However, multiple PROs, using different scales, with different directions of favorability are often used within a trial, making interpretation difficult. To enhance interpretability, we propose the use of a standardized effect size graph, which shows all PROs from a study on the same figure, on the same scale. Plotting standardized effects with their 95% confidence intervals (CIs) on a single graph clearly showing the null value conveys a comprehensive picture of trial results. We demonstrate how to create such a graph using data from a randomized controlled trial that measured 12 PROs at two time points. The 24 effect sizes and CIs are shown on one graph and clearly indicate that the intervention is effective and sustained.
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Krompaß, Denis [Verfasser], and Volker [Akademischer Betreuer] Tresp. "Exploiting prior knowledge and latent variable representations for the statistical modeling and probabilistic querying of large knowledge graphs / Denis Krompaß. Betreuer: Volker Tresp." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2015. http://d-nb.info/1081628847/34.
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Mullins, Sherry Lynn. "Statistics: Raising the Bar for the Seventh Grade Classroom." Digital Commons @ East Tennessee State University, 2006. https://dc.etsu.edu/etd/2221.
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After recognizing the need for a more thorough concentration of statistics at the seventh grade level, the author concluded that it would be a good idea to include statistics that cover both seventh and eighth grade Virginia Standards of Learning. Many years of administering the SOL mathematics test at the eighth grade level led the author to the understanding that some of the more advanced seventh graders would be missing some key concepts taught in eighth grade because those advanced students would be taking algebra in the eighth grade. In this thesis, the author has developed four units that she feels are appropriate for this level and will fill the gap.
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Panthulu, Pradeep. "Intelligent Memory Management Heuristics." Thesis, University of North Texas, 2003. https://digital.library.unt.edu/ark:/67531/metadc4399/.
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Automatic memory management is crucial in implementation of runtime systems even though it induces a significant computational overhead. In this thesis I explore the use of statistical properties of the directed graph describing the set of live data to decide between garbage collection and heap expansion in a memory management algorithm combining the dynamic array represented heaps with a mark and sweep garbage collector to enhance its performance. The sampling method predicting the density and the distribution of useful data is implemented as a partial marking algorithm. The algorithm randomly marks the nodes of the directed graph representing the live data at different depths with a variable probability factor p. Using the information gathered by the partial marking algorithm in the current step and the knowledge gathered in the previous iterations, the proposed empirical formula predicts with reasonable accuracy the density of live nodes on the heap, to decide between garbage collection and heap expansion. The resulting heuristics are tested empirically and shown to improve overall execution performance significantly in the context of the Jinni Prolog compiler's runtime system.
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Warnke, Lutz. "Random graph processes with dependencies." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:71b48e5f-a192-4684-a864-ea9059a25d74.
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Random graph processes are basic mathematical models for large-scale networks evolving over time. Their systematic study was pioneered by Erdös and Rényi around 1960, and one key feature of many 'classical' models is that the edges appear independently. While this makes them amenable to a rigorous analysis, it is desirable, both mathematically and in terms of applications, to understand more complicated situations. In this thesis the main goal is to improve our rigorous understanding of evolving random graphs with significant dependencies. The first model we consider is known as an Achlioptas process: in each step two random edges are chosen, and using a given rule only one of them is selected and added to the evolving graph. Since 2000 a large class of 'complex' rules has eluded a rigorous analysis, and it was widely believed that these could give rise to a striking and unusual phenomenon. Making this explicit, Achlioptas, D'Souza and Spencer conjectured in Science that one such rule yields a very abrupt (discontinuous) percolation phase transition. We disprove this, showing that the transition is in fact continuous for all Achlioptas process. In addition, we give the first rigorous analysis of the more 'complex' rules, proving that certain key statistics are tightly concentrated (i) in the subcritical evolution, and (ii) also later on if an associated system of differential equations has a unique solution. The second model we study is the H-free process, where random edges are added subject to the constraint that they do not complete a copy of some fixed graph H. The most important open question for such 'constrained' processes is due to Erdös, Suen and Winkler: in 1995 they asked what the typical final number of edges is. While Osthus and Taraz answered this in 2000 up to logarithmic factors for a large class of graphs H, more precise bounds are only known for a few special graphs. We close this gap for the cases where a cycle of fixed length is forbidden, determining the final number of edges up to constants. Our result not only establishes several conjectures, it is also the first which answers the more than 15-year old question of Erdös et. al. for a class of forbidden graphs H.
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Pace, Bruno. "O modelo de Axelrod com tensão superficial." Universidade de São Paulo, 2011. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-26042012-123155/.
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Nesta dissertação foram estudados alguns modelos vetoriais que pretendem modelar e descrever alguns aspectos de sistemas sociais e de sua organização cultural. Partimos do modelo de Axelrod, um processo estocástico definido em uma rede, e introduzimos uma pequena alteração no modelo que desencadeou mudanças qualitativas interessantes, especialmente o surgimento de uma tensão superficial, que leva ao aparecimento de estados metaestáveis e de regiões culturais mais fixamente localizadas no espaço. Através da ótica da mecânica estatística e de extensas simulações computacionais, exploramos alguns dos aspectos que julgamos mais importantes na caracterização desse rico modelo.Axelrod\'s model for cultural dissemination is a discrete vector representation for modeling social and cultural systems. In this work we have studied it and other related models, and a subtle change in the model\'s rule was proposed. Our slight alterations to the model yielded significant qualitative changes, specifically the emergence of surface tension, driving the system to metastable states. Using concepts from statistical mechanics and extensive numerical simulations, we explored some of the aspects that better describe the rich model devised, such as its transient and stationary behaviour.
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Kim, Jae Ill S. M. Massachusetts Institute of Technology. "Graph polynomials and statistical physics." Thesis, Massachusetts Institute of Technology, 2007. http://hdl.handle.net/1721.1/39000.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.Includes bibliographical references (p. 53-54).
We present several graph polynomials, of which the most important one is the Tutte polynomial. These various polynomials have important applications in combinatorics and statistical physics. We generalize the Tutte polynomial and establish its correlations to the other graph polynomials. Finally, our result about the decomposition of planar graphs and its application to the ice-type model is presented.
by Jae Ill Kim.
S.M.
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Deel, Troy A. "A statistical study of graph algorithms." Virtual Press, 1985. http://liblink.bsu.edu/uhtbin/catkey/424871.
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The object of this paper is to investigate the behavior of some important graph properties and to statistically analyze the execution times of certain graph are the average degree of a vertex, connectivity of a graph, the existence of Hamilton cycles, Euler tours, and bipartitions in graphs. This study is unique in that it is based on statistical rather than deterministic methods.
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Goldman, Nathan. "Quantum transport and phase transitions in lattices subjected to external gauge fields." Doctoral thesis, Universite Libre de Bruxelles, 2009. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210333.
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The first and main part of this thesis concerns the quantization of the transverse transport in diverse periodic quantum systems. From a theoretical point of view, the Hall conductivity's quantization may be understood at the single-particle level in terms of topological invariants. In periodic media such as crystals, the single-particle energy spectrum depicts a specific band structure. A modern approach, based on topology and differential geometry, consists in assigning an abstract mathematical object, a fibre bundle, to each energy band. The fibre bundle's topology is measured by a topological invariant, called the Chern number, which only takes integral values. Surprisingly, the transverse conductivity can be expressed as a sum of Chern numbers. In this work, one provides a rigorous derivation of this fact and one presents several methods which allow the numerical and analytical computation of the Chern numbers for diverse systems. The first original study concerns the physics of ultracold atoms trapped in optical lattices. These very popular experimental setups, which are currently designed in several laboratories worldwide, allow for the exploration of fundamental problems encountered in modern physics. In particular atoms trapped in optical lattices reproduce with a very high accuracy the physics of the Hubbard-type models which describe a huge variety of condensed
matter phenomena, such as high-Tc superconductivity and the Mott quantum phase transition. Particularly interesting is the possibility to create artificial magnetic fields in optical lattices. Generated by complex laser configurations or by rotation of the trap, these artificial fields allow the simulation of electronic systems subjected to intense magnetic fields. In this thesis, one explores the possibility of a quantum Hall-like effect for neutral particles in such arrangements. In particular one focuses on the exotic situation in which non-Abelian gauge potentials are generated in the system. In these interesting arrangements, the atomic hoppings are assisted by external lasers and are described by non-commutating translation operators. The non-Abelian fields which are generated in these systems are well known in high-energy physics, where they play a key role in modern theories of fundamental interactions.
Thereafter, our study of the IQHE in periodic systems concerns quantum graphs. These models which describe the propagation of a quantum wave within an arbitrary complex object are extremely versatile and hence allow the study of various interesting quantum phenomena. Quantum graphs appear in diverse fields such as solid state physics, quantum chemistry, quantum chaology and wave physics. On the other hand, in the context of quantum chaology, graphs have been the vehicle to confirm important conjectures about chaos signatures. In this thesis, one studies the spectral and chaological properties of infinite rectangular quantum graphs in the presence of a magnetic field. One then establishes the quantization of the Hall transverse conductivity for these systems.
The second part of the thesis is devoted to the physics of interacting atoms trapped in optical lattices and subjected to artificial gauge potentials. One explores the Mott quantum phase transition in both bosonic and fermionic optical lattices subjected to such fields. The optical lattices are described through the Hubbard model in which the dynamics is ruled by two competing parameters: the interaction strength U and the tunneling amplitude t. The Mott phase is characterized by a commensurate filling of the lattice and is reached by increasing the ration U/t, which can be easily achieved experimentally by varying the depth of the optical potential. In this thesis one studies how this quantum phase transition is modified when the optical lattice is subjected to diverse artificial gauge potentials.
Moreover, one shows that vortices are created in bosonic optical lattices in the vicinity of the Mott regime. The vortices are topological defects in the macroscopic wave function that describes the superfluid. One comments on the vortex patterns that are observed for several configurations of the gauge potential.
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La physique statistique quantique prédit l’émergence de propriétés remarquables lorsque la matière est soumise à des conditions extrêmes de basses températures. Aujourd’hui ces nouvelles phases de la matière jouent un rôle fondamental pour les technologies actuelles et ainsi méritent d’être étudiées sur le plan théorique.
Dans le cadre de ma thèse, j’ai étudié l’effet Hall quantique qui se manifeste dans des systèmes bidimensionnels ultra froids et soumis à des champs magnétiques intenses. Cet effet remarquable se manifeste par la quantification parfaite d’un coefficient de transport appelé conductivité de Hall. Cette grandeur physique évolue alors sur divers plateaux qui correspondent à des valeurs entières d’une constante fondamentale de la nature. D’un point de vue théorique, cette quantification peut être approchée par la théorie des espaces fibrés qui permet d’exprimer la conductivité de Hall en termes d’invariants topologiques.
Nous explorons l'effet Hall quantique pour différents systèmes en nous appuyant sur l’interprétation topologique de la quantification de la conductivité de Hall. Nous démontrons ainsi que l’effet Hall quantique se manifeste aussi bien dans les métaux que dans les graphes quantiques et les réseaux optiques. Les graphes quantiques sont des modèles permettant l’étude du transport dans des circuits fins, alors que les réseaux optiques sont des dispositifs actuellement réalisés en laboratoire qui piègent des atomes froids de façon périodique. Considérant différents champs magnétiques externes et variant la géométrie des systèmes, nous montrons que cet effet subit des modifications remarquables. Notamment, l’effet Hall quantique est représenté par des diagrammes des phases impressionnants :les multiples phases correspondant à la valeur entière de la conductivité de Hall se répartissent alors dans des structures fractales. De plus, ces diagrammes des phases se révèlent caractéristiques des différents systèmes étudiés.
D’autre part, nous étudions la transition quantique de Mott dans les réseaux optiques. En augmentant l’interaction entre les particules, le système devient isolant et se caractérise par le remplissage homogène du réseau. Nous étudions également l’apparition de tourbillons quantiques lorsque le système est soumis à un champ magnétique au voisinage de la phase isolante.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished
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Penman, David Binnie. "Random graphs with correlation structure." Thesis, University of Sheffield, 1998. http://etheses.whiterose.ac.uk/14768/.
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In this thesis we consider models of random graphs where, unlike in the classical models G (n, p) the probability of an edge arising can be correlated with that of other edges arising. Attention focuses on graphs whose vertices are each assigned a colour (type) at random and where edges between differently coloured vertices subsequently arise with different probabilities (so-called RRC graphs), especially the special case with two colours. Various properties of these graphs are considered, often by comparing and contrasting them with the classical model with the same probability of each particular edge existing. Topics examined include the probabilities of trees and cycles, how the joint probability of two subgraphs compares with the product of their probabilities, the number of edges in the graph (including large deviations results), connectedness, connectivity, the number and order of complete graphs and cliques, and tournaments with correlation structure.
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Obando, Forero Catalina. "Statistical graph models of temporal brain networks." Electronic Thesis or Diss., Sorbonne université, 2018. https://accesdistant.sorbonne-universite.fr/login?url=https://theses-intra.sorbonne-universite.fr/2018SORUS454.pdf.
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La discipline encore naissante des réseaux complexes est vecteur d’un changement de paradigme dans la neuroscience. Les connectomes estimés à partir de mesures de neuroimagerie comme l’électroencéphalographie, la magnétoencéphalographie ou encore l’imagerie par résonance magnétique fonctionnelle fournissent une représentation abstraite du cerveau sous la forme d’un graphe, ce qui a permis des percées décisives dans la compréhension compacte et objective des propriétés topologiques et physiologiques des cerveaux sains. Cependant, les approches de pointe ignorent souvent l'incertitude et la nature temporelle de données de connectivité fonctionnelles. La plupart des méthodes disponibles dans la littérature ont en effet été développées pour caractériser les réseaux cérébraux fonctionnels comme des graphes statiques composés de nœuds (des régions cérébrales) et des liens (intensité de connectivité fonctionnelle) par métrique de réseau. En conséquence, la théorie des réseaux complexes a été principalement appliquée à des études transversales avec une unique mesure par sujet, produisant au final une caractérisation consistant en une moyenne de phénomènes neuronaux spatiotemporels. Nous avons implémenté des méthodes statistiques pour modéliser et simuler des réseaux cérébraux temporels. Nous avons utilisé des modèles de graphe qui permettent d'étudier simultanément à quel point les différentes propriétés des réseaux influencent la topologie observée dans les réseaux de connectivité cérébrale fonctionnelle. Nous avons identifié avec succès les mécanismes de connectivité locale fondamentaux qui gouvernent les propriétés des réseaux cérébraux. Nous avons proposé l'adaptation temporelle de ces mécanismes fondamentaux pour modéliser et simuler les changements physiologiques dynamiques d'un réseau cérébral. Plus spécifiquement, nous avons exploité des métriques temporelles pour construire des modèles temporels informatifs du rétablissement de patients ayant subit un accident vasculaire cérébralThe emerging area of complex networks has led to a paradigm shift in neuroscience. Connectomes estimated from neuroimaging techniques such as electroencephalography (EEG), magnetoencephalography (MEG) and functional magnetic resonance imaging (fMRI) results in an abstract representation of the brain as a graph, which has allowed a major breakthrough in the understanding of topological and physiological properties of healthy brains in a compact and objective way. However, state of the art approaches often ignore the uncertainty and temporal nature of functional connectivity data. Most of the available methods in the literature have been developed to characterize functional brain networks as static graphs composed of nodes (brain regions) and links (FC intensity) by network metrics. As a consequence, complex networks theory has been mainly applied to cross-sectional studies referring to a single point in time and the resulting characterization ultimately represents an average across spatiotemporal neural phenomena. Here, we implemented statistical methods to model and simulate temporal brain networks. We used graph models that allow to simultaneously study how different network properties influence the emergent topology observed in functional connectivity brain networks. We successfully identified fundamental local connectivity mechanisms that govern properties of brain networks. We proposed a temporal adaptation of such fundamental connectivity mechanisms to model and simulate physiological brain network dynamic changes. Specifically, we exploited the temporal metrics to build informative temporal models of recovery of patients after stroke
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Sbihi, Amine M. (Amine Mohammed). "Covering times for random walks on graphs." Thesis, McGill University, 1990. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74538.
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This thesis is a contribution to the covering times problems for random walks on graphs. By considering uniform random walks on finite connected graphs, the covering time is defined as the time (number of steps) taken by the random walk to visit every vertex. The motivating problem of this thesis is to find bounds for the expected covering times. We provide explicit bounds that are uniformly valid over all starting points and over large classes of graphs. In some cases the asymptotic distribution of the suitably normalized covering time is given as well.
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Dibble, Emily. "The interpretation of graphs and tables /." Thesis, Connect to this title online; UW restricted, 1997. http://hdl.handle.net/1773/9101.
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Chaudhuri, Sanjay. "Using the structure of d-connecting paths as a qualitative measure of the strength of dependence /." Thesis, Connect to this title online; UW restricted, 2005. http://hdl.handle.net/1773/8948.
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Wang, Yi Ming. "Graph-based data selection for statistical machine translation." Thesis, University of Macau, 2017. http://umaclib3.umac.mo/record=b3691542.
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Fairbanks, James Paul. "Graph analysis combining numerical, statistical, and streaming techniques." Diss., Georgia Institute of Technology, 2016. http://hdl.handle.net/1853/54972.
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Graph analysis uses graph data collected on a physical, biological, or social
phenomena to shed light on the underlying dynamics and behavior of the agents in that system. Many fields contribute to this topic including graph theory, algorithms, statistics, machine learning, and linear algebra. This dissertation advances a novel framework for dynamic graph analysis that combines numerical, statistical, and streaming algorithms to provide deep
understanding into evolving networks. For example, one can be interested in the changing influence structure over time. These disparate techniques each
contribute a fragment to understanding the graph; however, their combination
allows us to understand dynamic behavior and graph structure. Spectral partitioning methods rely on eigenvectors for solving data analysis
problems such as clustering. Eigenvectors of large sparse systems must be approximated with iterative methods. This dissertation analyzes how data analysis accuracy depends on the numerical accuracy of the eigensolver. This leads to new bounds on the residual tolerance necessary to guarantee correct partitioning. We present a novel stopping criterion for spectral partitioning guaranteed to satisfy the Cheeger inequality along with an empirical study of the performance on real world networks such as web, social, and e-commerce networks. This work bridges the gap between numerical analysis and computational data analysis.
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Hofbauer, Pamela S. Mooney Edward S. "Characterizing high school students' understanding of the purpose of graphical representations." Normal, Ill. : Illinois State University, 2007. http://proquest.umi.com/pqdweb?index=0&did=1414114601&SrchMode=1&sid=6&Fmt=2&VInst=PROD&VType=PQD&RQT=309&VName=PQD&TS=1207664408&clientId=43838.
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Thesis (Ph. D.)--Illinois State University, 2007.Title from title page screen, viewed on April 8, 2008. Dissertation Committee: Edward S. Mooney (chair), Cynthia W. Langrall, Sherry L. Meier, Norma C. Presmeg. Includes bibliographical references (leaves 112-121) and abstract. Also available in print.
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Martinet, Lucie. "Réseaux dynamiques de terrain : caractérisation et propriétés de diffusion en milieu hospitalier." Thesis, Lyon, École normale supérieure, 2015. http://www.theses.fr/2015ENSL1010/document.
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Durant cette thèse, nous nous sommes intéressés aux outils permettant d'extraire les propriétés structurelles et temporelles de réseaux dynamiques ainsi que les caractéristiques de certains scénarios de diffusion pouvant s'opérer sur ces réseaux. Nous avons travaillé sur un jeu de données spécifiques, issu du projet MOSAR, qui comporte entre autre le réseau de proximité des personnes au cours du temps durant 6 mois à l'hôpital de Berk-sur-mer. Ce réseau est particulier dans le sens où il est constitué de trois dimensions: temporelle, structurelle par la répartition des personnes en services et fonctionnelle car chaque personne appartient à une catégorie socio-professionnelle. Pour chacune des dimensions, nous avons utilisé des outils existants en physique statistique ainsi qu'en théorie des graphes pour extraire des informations permettant de décrire certaines propriétés du réseau. Cela nous a permis de souligner le caractère très structuré de la répartition des contacts qui suit la répartition en services et mis en évidence les accointances entre certaines catégories professionnelles. Concernant la partie temporelle, nous avons mis en avant l'évolution périodique circadienne et hebdomadaire ainsi que les différences fondamentales entre l'évolution des interactions des patients et celle des personnels. Nous avons aussi présenté des outils permettant de comparer l'activité entre deux périodes données et de quantifier la similarité de ces périodes. Nous avons ensuite utilisé la technique de simulation pour extraire des propriétés de diffusion de ce réseau afin de donner quelques indices pour établir une politique de préventionIn this thesis, we focus on tools whose aim is to extract structural and temporal properties of dynamic networks as well as diffusion characteristics which can occur on these networks. We work on specific data, from the European MOSAR project, including the network of individuals proximity from time to time during 6 months at the Brek-sur-Mer Hospital. The studied network is notable because of its three dimensions constitution : the structural one induced by the distribution of individuals into distinct services, the functional dimension due to the partition of individual into groups of socio-professional categories and the temporal dimension.For each dimension, we used tools well known from the areas of statistical physics as well as graphs theory in order to extract information which enable to describe the network properties. These methods underline the specific structure of the contacts distribution which follows the individuals distribution into services. We also highlight strong links within specific socio-professional categories. Regarding the temporal part, we extract circadian and weekly patterns and quantify the similarities of these activities. We also notice distinct behaviour within patients and staff evolution. In addition, we present tools to compare the network activity within two given periods. To finish, we use simulations techniques to extract diffusion properties of the network to find some clues in order to establish a prevention policy
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Williams, Trevor K. "Combinatorial Games on Graphs." DigitalCommons@USU, 2017. https://digitalcommons.usu.edu/etd/6502.
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Combinatorial games are intriguing and have a tendency to engross students and lead them into a serious study of mathematics. The engaging nature of games is the basis for this thesis. Two combinatorial games along with some educational tools were developed in the pursuit of the solution of these games.
The game of Nim is at least centuries old, possibly originating in China, but noted in the 16th century in European countries. It consists of several stacks of tokens, and two players alternate taking one or more tokens from one of the stacks, and the player who cannot make a move loses. The formal and intense study of Nim culminated in the celebrated Sprague-Grundy Theorem, which is now one of the centerpieces in the theory of impartial combinatorial games. We study a variation on Nim, played on a graph. Graph Nim, for which the theory of Sprague-Grundy does not provide a clear strategy, was originally developed at the University of Colorado Denver. Graph Nim was first played on graphs of three vertices.
The winning strategy, and losing position, of three vertex Graph Nim has been discovered, but we will expand the game to four vertices and develop the winning strategies for four vertex Graph Nim.
Graph Theory is a markedly visual field of mathematics. It is extremely useful for graph theorists and students to visualize the graphs they are studying. There exists software to visualize and analyze graphs, such as SAGE, but it is often extremely difficult to learn how use such programs. The tools in GeoGebra make pretty graphs, but there is no automated way to make a graph or analyze a graph that has been built. Fortunately GeoGebra allows the use of JavaScript in the creation of buttons which allow us to build useful Graph Theory tools in GeoGebra. We will discuss two applets we have created that can be used to help students learn some of the basics of Graph Theory.
The game of thrones is a two-player impartial combinatorial game played on an oriented complete graph (or tournament) named after the popular fantasy book and TV series. The game of thrones relies on a special type of vertex called a king. A king is a vertex, k, in a tournament, T, which for all x in T either k beats x or there exists a vertex y such that k beats y and y beats x. Players take turns removing vertices from a given tournament until there is only one king left in the resulting tournament. The winning player is the one which makes the final move. We develop a winning position and classify those tournaments that are optimal for the first or second-moving player.
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Colin, Igor. "Adaptation des méthodes d’apprentissage aux U-statistiques." Thesis, Paris, ENST, 2016. http://www.theses.fr/2016ENST0070/document.
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L’explosion récente des volumes de données disponibles a fait de la complexité algorithmique un élément central des méthodes d’apprentissage automatique. Les algorithmes d’optimisation stochastique ainsi que les méthodes distribuées et décentralisées ont été largement développés durant les dix dernières années. Ces méthodes ont permis de faciliter le passage à l’échelle pour optimiser des risques empiriques dont la formulation est séparable en les observations associées. Pourtant, dans de nombreux problèmes d’apprentissage statistique, l’estimation précise du risque s’effectue à l’aide de U-statistiques, des fonctions des données prenant la forme de moyennes sur des d-uplets. Nous nous intéressons tout d’abord au problème de l’échantillonnage pour la minimisation du risque empirique. Nous montrons que le risque peut être remplacé par un estimateur de Monte-Carlo, intitulé U-statistique incomplète, basé sur seulement O(n) termes et permettant de conserver un taux d’apprentissage du même ordre. Nous établissons des bornes sur l’erreur d’approximation du U-processus et les simulations numériques mettent en évidence l’avantage d’une telle technique d’échantillonnage. Nous portons par la suite notre attention sur l’estimation décentralisée, où les observations sont désormais distribuées sur un réseau connexe. Nous élaborons des algorithmes dits gossip, dans des cadres synchrones et asynchrones, qui diffusent les observations tout en maintenant des estimateurs locaux de la U-statistique à estimer. Nous démontrons la convergence de ces algorithmes avec des dépendances explicites en les données et la topologie du réseau. Enfin, nous traitons de l’optimisation décentralisée de fonctions dépendant de paires d’observations. De même que pour l’estimation, nos méthodes sont basées sur la concomitance de la propagation des observations et l’optimisation local du risque. Notre analyse théorique souligne que ces méthodes conservent une vitesse de convergence du même ordre que dans le cas centralisé. Les expériences numériques confirment l’intérêt pratique de notre approcheWith the increasing availability of large amounts of data, computational complexity has become a keystone of many machine learning algorithms. Stochastic optimization algorithms and distributed/decentralized methods have been widely studied over the last decade and provide increased scalability for optimizing an empirical risk that is separable in the data sample. Yet, in a wide range of statistical learning problems, the risk is accurately estimated by U-statistics, i.e., functionals of the training data with low variance that take the form of averages over d-tuples. We first tackle the problem of sampling for the empirical risk minimization problem. We show that empirical risks can be replaced by drastically computationally simpler Monte-Carlo estimates based on O(n) terms only, usually referred to as incomplete U-statistics, without damaging the learning rate. We establish uniform deviation results and numerical examples show that such approach surpasses more naive subsampling techniques. We then focus on the decentralized estimation topic, where the data sample is distributed over a connected network. We introduce new synchronous and asynchronous randomized gossip algorithms which simultaneously propagate data across the network and maintain local estimates of the U-statistic of interest. We establish convergence rate bounds with explicit data and network dependent terms. Finally, we deal with the decentralized optimization of functions that depend on pairs of observations. Similarly to the estimation case, we introduce a method based on concurrent local updates and data propagation. Our theoretical analysis reveals that the proposed algorithms preserve the convergence rate of centralized dual averaging up to an additive bias term. Our simulations illustrate the practical interest of our approach
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馮榮錦 and Wing-kam Tony Fung. "Analysis of outliers using graphical and quasi-Bayesian methods." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1987. http://hub.hku.hk/bib/B31230842.
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Pickle, Maria Consuelo (suzie) Capiral. "Statistical Content in Middle Grades Mathematics Textbooks." Scholar Commons, 2012. http://scholarcommons.usf.edu/etd/4203.
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Statistical Content in Middle Grades Mathematics Textbooks
Maria Consuelo (Suzie) Capiral Pickle
Abstract
This study analyzed the treatment and scope of
statistical concepts in four, widely-used, contemporary, middle
grades mathematics textbook series: Glencoe Math Connects,
Prentice Hall Mathematics, Connected Mathematics Project, and
University of Chicago School Mathematics Project. There were
three phases for the data analysis. Phase 1 addressed the location
and sequence of the statistical concepts. Phase 2 focused upon an
examination of the lesson narrative, its components and scope.
Phase 3 analyzed the level of cognitive demand required of the
students to complete the exercises, and the total number of
exercises per statistical concept. These three phases taken
together provided insight into students' potential opportunity to learn
statistical topics found in middle grades mathematics textbooks.
Results showed that concepts, such as measures of central
tendency, were repeated in several grades while other topics such
as circle graphs were presented earlier than the recommendations
in documents such as the National Council of Teachers of
Mathematics Principles and Standards (2000) and the Common
Core State Standards (2010). Further results showed that most of
the statistical content was found in a chapter near the end of the
book that would likely not be covered should time run short. Also,
each textbook had a particular lesson narrative style. Moreover,
most of the statistical exercises required low level cognitive
demand of the students to complete the exercises, potentially
hindering the development of deep understanding of the concepts.
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Nyberg, Brodda Carl-Fredrik. "Deterministic and Random Pebbling of Graphs." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-325833.
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Ježek, Jan. "Statistický modul k pracovní databázi TIVOLE." Master's thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2008. http://www.nusl.cz/ntk/nusl-217191.
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Presented thesis thesis is written for IBM Company, specifically IBM IDC Czech Republic, s.r.o. branch. The thesis describes development of statistic module for work database TIVOLE utilizing Lotus Notes program. The basic aim of the thesis is to ease creation of reports and graphical data outputs in relation to speed of operator´s work while narrowing down the required time of human resources for solving problems.
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Gibert, Domingo Jaume. "Vector Space Embedding of Graphs via Statistics of Labelling Information." Doctoral thesis, Universitat Autònoma de Barcelona, 2012. http://hdl.handle.net/10803/96240.
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El reconeixement de patrons és la tasca que pretén distingir objectes entre diferents classes. Quan aquesta tasca es vol solucionar de forma automàtica un pas crucial és el com representar formalment els patrons a l'ordinador. En funció d'aquests formalismes, podem distingir entre el reconeixement estadístic i l'estructural. El primer descriu objectes com un conjunt de mesures col·locats en forma del que s'anomena un vector de característiques. El segon assumeix que hi ha relacions entre parts dels objectes que han de quedar explícitament representades i per tant fa servir estructures relacionals com els grafs per codificar la seva informació inherent. Els espais vectorials són una estructura matemàtica molt flexible que ha permès definir diverses maneres eficients d'analitzar patrons sota la forma de vectors de característiques. De totes maneres, la representació vectorial no és capaç d'expressar explícitament relacions binàries entre parts dels objectes i està restrigida a mesurar sempre, independentment de la complexitat dels patrons, el mateix nombre de característiques per cadascun d'ells. Les representacions en forma de graf presenten la situació contrària. Poden adaptar-se fàcilment a la complexitat inherent dels patrons però introdueixen un problema d'alta complexitat computational, dificultant el disseny d'eines eficients per al procés i l'anàlisis de patrons.
Resoldre aquesta paradoxa és el principal objectiu d'aquesta tesi. La situació ideal per resoldre problemes de reconeixement de patrons seria el representar-los fent servir estructures relacionals com els grafs, i a l'hora, poder fer ús del ric repositori d'eines pel processament de dades del reconeixement estadístic. Una solució elegant a aquest problema és la de transformar el domini dels grafs en el domini dels vectors, on podem aplicar qualsevol algorisme de processament de dades. En altres paraules, assignant a cada graf un punt en un espai vectorial, automàticament tenim accés al conjunt d'algorismes del món estadístic per aplicar-los al domini dels grafs. Aquesta metodologia s'anomena graph embedding.
En aquesta tesi proposem de fer una associació de grafs a vectors de característiques de forma simple i eficient fixant l'atenció en la informació d'etiquetatge dels grafs. En particular, comptem les freqüències de les etiquetes dels nodes així com de les aretes entre etiquetes determinades. Tot i la seva localitat, aquestes característiques donen una representació prou robusta de les propietats globals dels grafs. Primer tractem el cas de grafs amb etiquetes discretes, on les característiques són sencilles de calcular. El cas continu és abordat com una generalització del cas discret, on enlloc de comptar freqüències d'etiquetes, ho fem de representants d'aquestes. Ens trobem que les representacions vectorials que proposem pateixen d'alta dimensionalitat i correlació entre components, i tractem aquests problems mitjançant algorismes de selecció de característiques. També estudiem com la diversitat de diferents representacions pot ser explotada per tal de millorar el rendiment de classificadors base en el marc d'un sistema de múltiples classificadors. Finalment, amb una extensa evaluació experimental mostrem com la metodologia proposada pot ser calculada de forma eficient i com aquesta pot competir amb altres metodologies per a la comparació de grafs.Pattern recognition is the task that aims at distinguishing objects among different classes. When such a task wants to be solved in an automatic way a crucial step is how to formally represent such patterns to the computer. Based on the different representational formalisms, we may distinguish between statistical and structural pattern recognition. The former describes objects as a set of measurements arranged in the form of what is called a feature vector. The latter assumes that relations between parts of the underlying objects need to be explicitly represented and thus it uses relational structures such as graphs for encoding their inherent information. Vector spaces are a very flexible mathematical structure that has allowed to come up with several efficient ways for the analysis of patterns under the form of feature vectors. Nevertheless, such a representation cannot explicitly cope with binary relations between parts of the objects and it is restricted to measure the exact same number of features for each pattern under study regardless of their complexity. Graph-based representations present the contrary situation. They can easily adapt to the inherent complexity of the patterns but introduce a problem of high computational complexity, hindering the design of efficient tools to process and analyze patterns. Solving this paradox is the main goal of this thesis. The ideal situation for solving pattern recognition problems would be to represent the patterns using relational structures such as graphs, and to be able to use the wealthy repository of data processing tools from the statistical pattern recognition domain. An elegant solution to this problem is to transform the graph domain into a vector domain where any processing algorithm can be applied. In other words, by mapping each graph to a point in a vector space we automatically get access to the rich set of algorithms from the statistical domain to be applied in the graph domain. Such methodology is called graph embedding. In this thesis we propose to associate feature vectors to graphs in a simple and very efficient way by just putting attention on the labelling information that graphs store. In particular, we count frequencies of node labels and of edges between labels. Although their locality, these features are able to robustly represent structurally global properties of graphs, when considered together in the form of a vector. We initially deal with the case of discrete attributed graphs, where features are easy to compute. The continuous case is tackled as a natural generalization of the discrete one, where rather than counting node and edge labelling instances, we count statistics of some representatives of them. We encounter how the proposed vectorial representations of graphs suffer from high dimensionality and correlation among components and we face these problems by feature selection algorithms. We also explore how the diversity of different embedding representations can be exploited in order to boost the performance of base classifiers in a multiple classifier systems framework. An extensive experimental evaluation finally shows how the methodology we propose can be efficiently computed and compete with other graph matching and embedding methodologies.
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Yang, Mengta. "Depth Functions, Multidimensional Medians and Tests of Uniformity on Proximity Graphs." Thesis, The George Washington University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3615104.
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We represent the d-dimensional random vectors as vertices of a complete weighted graph and propose depth functions that are applicable to distributions in d-dimensional spaces and data on graphs. We explore the proximity graphs, examine their connection to existing depth functions, define a family of depth functions on the β-skeleton graph, suggest four types of depth functions on the minimal spanning tree (MST) and define depth functions including path depth, path depth of path length at most δ, all paths probability depth, eccentricity depth, peeling depth and RUNT depth. We study their properties, including affine invariance, maximality at the center, monotonicity and vanishing at infinity. We show that the β-skeleton depth is a family of statistical depth functions and define the sample β-skeleton depth function. We show that it has desirable asymptotic properties, including uniform consistency and asymptotic normality. We consider the corresponding multidimensional medians, investigate their robustness, computational complexity, compare them in a simulation study to find the multivariate medians under different distributions and sample sizes and explore the asymptotic properties of β-skeleton median. We generalize the univariate Greenwood statistic and Hegazy-Green statistic using depth induced order statistics and propose two new test statistics based on normal copula and interpoint distances for testing multivariate uniformity. We generalize the path depth under two-sample setting and propose a new multivariate equality of DF test. We study their empirical power against several copula and multivariate Beta alternatives. The topic is complemented with a discussion on the distribution and moments of the interpoint distances (ID) between bivariate uniform random vectors and the IDs between FGM copula random vectors.
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Behjat, Hamid. "Statistical Parametric Mapping of fMRI data using Spectral Graph Wavelets." Thesis, Linköpings universitet, Medicinsk informatik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-81143.
Full textAbstract:
In typical statistical parametric mapping (SPM) of fMRI data, the functional data are pre-smoothed using a Gaussian kernel to reduce noise at the cost of losing spatial specificity. Wavelet approaches have been incorporated in such analysis by enabling an efficient representation of the underlying brain activity through spatial transformation of the original, un-smoothed data; a successful framework is the wavelet-based statistical parametric mapping (WSPM) which enables integrated wavelet processing and spatial statistical testing. However, in using the conventional wavelets, the functional data are considered to lie on a regular Euclidean space, which is far from reality, since the underlying signal lies within the complex, non rectangular domain of the cerebral cortex. Thus, using wavelets that function on more complex domains such as a graph holds promise. The aim of the current project has been to integrate a recently developed spectral graph wavelet transform as an advanced transformation for fMRI brain data into the WSPM framework. We introduce the design of suitable weighted and un-weighted graphs which are defined based on the convoluted structure of the cerebral cortex. An optimal design of spatially localized spectral graph wavelet frames suitable for the designed large scale graphs is introduced. We have evaluated the proposed graph approach for fMRI analysis on both simulated as well as real data. The results show a superior performance in detecting fine structured, spatially localized activation maps compared to the use of conventional wavelets, as well as normal SPM. The approach is implemented in an SPM compatible manner, and is included as an extension to the WSPM toolbox for SPM.
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Lindström, Alfred Minh. "Cutting and Destroying Graphs using k-cuts." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-395663.
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Cortina, Borja Mario Jose Francisco. "Graph-theoretic multivariate nonparametric procedures." Thesis, University of Bath, 1992. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.302730.
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Sylvester, John A. "Random walks, effective resistance and neighbourhood statistics in binomial random graphs." Thesis, University of Warwick, 2017. http://wrap.warwick.ac.uk/106467/.
Full textAbstract:
The binomial random graph model G(n; p), along with its near-twin sibling G(n; m), were the starting point for the entire study of random graphs and even probabilistic combinatorics as a whole. The key properties of these models are woven into the fabric of the field and their behaviour serves as a benchmark to compare any other model of random structure. In this thesis we contribute to the already rich literature on G(n; p) in a number of directions. Firstly, vertex to vertex hitting times of random walks in G(n; p) are considered via their interpretation as potential differences in an electrical network. In particular we show that in a graph satisfying certain connectivity properties the effective resistance between two vertices is typically determined, up to lower order terms, by the degrees of these vertices. We apply this to obtain the expected values of hitting times and several related indices in G(n; p), and to prove that these values are concentrated around their mean. We then study the statistics of the size of the r-neighbourhood of a vertex in G(n; p). We show that the sizes of these neighbourhoods satisfy a central limit theorem. We also bound the probability a vertex in G(n; p) has an r-neighbourhood of size k from above and below by functions of n; p and k which match up to constants. Finally, in the last chapter the extreme values of the r-degree sequence are studied. We prove a novel neighbourhood growth estimate which states that with high probability the size of a vertex's r neighbourhood is determined, up to lower order terms, by the size of its first neighbourhood. We use this growth estimate to bound the number of vertices attaining a smallest r-neighbourhood.