The idea of `probabilistic distances' (also called divergences), which in some sense assess how `close' two probability distributions are from one another, has been widely employed in probability, statistics, information theory, and related fields. Of particular importance due to their generality and applicability are the Renyi divergence measures. While the closely related concept of Renyi entropy of a probability distribution has been studied extensively, and closed-form expressions for the most common univariate and multivariate continuous distributions have been obtained and compiled, the literature currently lacks the corresponding compilation for continuous Renyi divergences. The present thesis addresses this issue for analytically tractable cases. Closed-form expressions for Kullback-Leibler divergences are also derived and compiled, as they can be seen as an extension by continuity of the Renyi divergences. Additionally, we establish a connection between Renyi divergence and the variance of the log-likelihood ratio of two distributions, which extends the work of Song (2001) on the relation between Renyi entropy and the log-likelihood function, and which becomes practically useful in light of the Renyi divergence expressions we have derived. Lastly, we consider the Renyi divergence rate between two zero-mean stationary Gaussian processes. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-08-30 13:37:41.792

This thesis solves possibility of using entropy for anomaly detection in data communication and especially for security attacks. The main advantage of using entropy is ability to identify unknown attacks because entropy detects changes in network traffic but not the content as existing methods. In this work was tested the suitability of different models entropy (Shannon, Renyi, Tsallis). Also been tested the effect of Renyi and Tsallis parameter on resulting entropy. From the resulting values, I found that all tested entropy achieve good result in the identification of anomalies in network traffic.

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This thesis is twofold. In the first part, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given and a new curvature condition on abstract metric measure spaces is defined. In the second part of the thesis a proof of the identification of the q-heat equation with the gradient flow of the Renyi (3-p)-Renyi entropy functional in the p-Wasserstein space is given. For that, a further study of the q-heat flow is presented including a condition for its mass preservation.

This thesis is twofold. In the first part, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given and a new curvature condition on abstract metric measure spaces is defined. In the second part of the thesis a proof of the identification of the q-heat equation with the gradient flow of the Renyi (3-p)-Renyi entropy functional in the p-Wasserstein space is given. For that, a further study of the q-heat flow is presented including a condition for its mass preservation.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Socially-based networks are of particular interest amongst the variety of communication networks arising in reality. They are distinguished by having small average path length and high clustering coefficient, and so are examples of small-world networks. This thesis studies both real examples and theoretical models of small-world networks, with particular attention to average path length. Existing models of small-world networks, due to Watts and Strogatz (1998) and Newman and Watts (1999a), impose boundary conditions on a one dimensional lattice, and rewire links locally and probabilistically in the former or probabilistically adding extra links in the latter. These models are investigated and compared with real-world networks. We consider a model in which randomness is provided by the Erdos-Rényi random network models superposed on a deterministic one dimensional structured network. We reason about this model using tools and results from random graph theory. Given a disordered network C(n, p) formed by adding links randomly with probability p to a one dimensional network C(n). We improve the analytical result regarding the average path length by showing that the onset of smallworld behaviour occurs if pn is bounded away from zero. Furthermore, we show that when pn tends to zero, C(n, p) is no longer small-world. We display that the average path length in this case approaches infinity with the network order. We deduce that at least εn (where ε is a constant bigger than zero) random links should be added to a one dimensional lattice to ensure average path length of order log n. / AFRIKAANSE OPSOMMING: Sosiaal-baseerde netwerke is van besondere belang onder die verskeidenheid kommunikasie netwerke. Hulle word onderskei deur ’n klein gemiddelde skeidingsafstand en hoë samedrommingskoëffisiënt, en is voorbeelde van kleinwêreld netwerke. Hierdie verhandeling bestudeer beide werklike voorbeelde en teoretiese modelle van klein-wêreld netwerke, met besondere aandag op die gemiddelde padlengte. Bestaande modelle van klein-wêreld netwerke, te danke aan Watts en Strogatz (1998) en Newman en Watts (1999a), voeg randvoorwaardes by tot eendimensionele roosters, en herbedraad nedwerkskakels gebaseer op lokale kennis in die eerste geval en voeg willekeurig ekstra netwerkskakels in die tweede. Hierdie modelle word ondersoek en vergelyk met werklike-wêreld netwerke. Ons oorweeg ’n prosedure waarin willekeurigheid verskaf word deur die Erdös- Renyi toevalsnetwerk modelle wat op ’n een-dimensionele deterministiese gestruktureerde netwerk geimposeer word. Ons redeneer oor hierdie modelle deur gebruik te maak van gereedskap en resultate toevalsgrafieke teorie. Gegewe ’n wanordelike netwerk wat gevorm word deur skakels willekeurig met waarskynlikheid p tot ‘n een-dimensionele netwerk C(n) toe te voeg, verbeter ons die analitiese resultaat ten opsigte van die gemiddelde padlengte deur te wys dat die aanvang van klein-wêreld gedrag voorkom wanneer pn weg van nul begrens is. Verder toon ons dat, wanneer pn neig na nul, C(n, p) nie meer klein-wêreld is nie. Ons toon dat die gemiddelde padlengte in hierdie geval na oneindigheid streef saam met die netwerk groote. Ons lei af dat ten minste εn (waar εn n konstante groter as nul is) ewekansige skakels bygevoeg moet word by ’n een-dimensionele rooster om ‘n gemiddelde padlengte van orde log n te verseker.

Nous appelons accroissements généralisés du processus empirique l'estimateur à noyau de la densité centré sur R^d pour lequel le noyau varie dans une classe de fonctions G. Ceci définit des processus stochastiques indéxés par G. Nous étudions le comportement limite de ces trajectoires aléatoires en considérant une suite de taille de fenêtre h_n qui tend vers 0. Nous donnons des résultats limites fonctionnels lorsque h_n vérifie les conditions de Csörgö-Révész-Stute, puis lorsque h_n vérifie les conditions d'Erdös-Renyi. Nous étudions également quelques comportements au second ordre dans les lois limites fonctionnelles standard du logarithme itéré pour le processus empirique uniforme local.

We consider the problem of extracting a specific feature from a noisy signal generated by a multi-channels Remote Field Eddy Current Sensor. The sensor is installed on a mobile robot whose mission is the detection of anomalous regions in metal pipelines. Given the presence of noise that characterizes the data series, anomaly signals could be masked by noise and therefore difficult to identify in some instances. In order to enhance signal peaks that potentially identify anomalies we consider an entropy filter built on a-posteriori probability density functions associated with data series. Thresholds based on the Neyman-Pearson criterion for hypothesis testing are derived. The algorithmic tool is applied to the analysis of data from a portion of pipeline with a set of anomalies introduced at predetermined locations. Critical areas identifying anomalies capture the set of damaged locations, demonstrating the effectiveness of the filter in detection with Remote Field Eddy Current Sensor.

We consider the problem of extracting a specific feature from a noisy signal generated by a multi-channels Remote Field Eddy Current Sensor. The sensor is installed on a mobile robot whose mission is the detection of anomalous regions in metal pipelines. Given the presence of noise that characterizes the data series, anomaly signals could be masked by noise and therefore difficult to identify in some instances. In order to enhance signal peaks that potentially identify anomalies we consider an entropy filter built on a-posteriori probability density functions associated with data series. Thresholds based on the Neyman-Pearson criterion for hypothesis testing are derived. The algorithmic tool is applied to the analysis of data from a portion of pipeline with a set of anomalies introduced at predetermined locations. Critical areas identifying anomalies capture the set of damaged locations, demonstrating the effectiveness of the filter in detection with Remote Field Eddy Current Sensor.

No description available.