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1	Visual Analysis of High-Dimensional Point Clouds using Topological Abstraction Oesterling, Patrick 17 May 2016 (has links) (PDF) This thesis is about visualizing a kind of data that is trivial to process by computers but difficult to imagine by humans because nature does not allow for intuition with this type of information: high-dimensional data. Such data often result from representing observations of objects under various aspects or with different properties. In many applications, a typical, laborious task is to find related objects or to group those that are similar to each other. One classic solution for this task is to imagine the data as vectors in a Euclidean space with object variables as dimensions. Utilizing Euclidean distance as a measure of similarity, objects with similar properties and values accumulate to groups, so-called clusters, that are exposed by cluster analysis on the high-dimensional point cloud. Because similar vectors can be thought of as objects that are alike in terms of their attributes, the point cloud\'s structure and individual cluster properties, like their size or compactness, summarize data categories and their relative importance. The contribution of this thesis is a novel analysis approach for visual exploration of high-dimensional point clouds without suffering from structural occlusion. The work is based on implementing two key concepts: The first idea is to discard those geometric properties that cannot be preserved and, thus, lead to the typical artifacts. Topological concepts are used instead to shift away the focus from a point-centered view on the data to a more structure-centered perspective. The advantage is that topology-driven clustering information can be extracted in the data\'s original domain and be preserved without loss in low dimensions. The second idea is to split the analysis into a topology-based global overview and a subsequent geometric local refinement. The occlusion-free overview enables the analyst to identify features and to link them to other visualizations that permit analysis of those properties not captured by the topological abstraction, e.g. cluster shape or value distributions in particular dimensions or subspaces. The advantage of separating structure from data point analysis is that restricting local analysis only to data subsets significantly reduces artifacts and the visual complexity of standard techniques. That is, the additional topological layer enables the analyst to identify structure that was hidden before and to focus on particular features by suppressing irrelevant points during local feature analysis. This thesis addresses the topology-based visual analysis of high-dimensional point clouds for both the time-invariant and the time-varying case. Time-invariant means that the points do not change in their number or positions. That is, the analyst explores the clustering of a fixed and constant set of points. The extension to the time-varying case implies the analysis of a varying clustering, where clusters appear as new, merge or split, or vanish. Especially for high-dimensional data, both tracking---which means to relate features over time---but also visualizing changing structure are difficult problems to solve. Hochdimensionale Daten Punktwolken Clustering Topologie Visualisierung Dichtefunktion High-Dimensional Data Point Clouds Clustering Topology Visualization Temporal Clustering Scalar Fields Merge Tree Density Function ddc:500
2	Visual Analysis of High-Dimensional Point Clouds using Topological Abstraction Oesterling, Patrick 14 April 2016 (has links) This thesis is about visualizing a kind of data that is trivial to process by computers but difficult to imagine by humans because nature does not allow for intuition with this type of information: high-dimensional data. Such data often result from representing observations of objects under various aspects or with different properties. In many applications, a typical, laborious task is to find related objects or to group those that are similar to each other. One classic solution for this task is to imagine the data as vectors in a Euclidean space with object variables as dimensions. Utilizing Euclidean distance as a measure of similarity, objects with similar properties and values accumulate to groups, so-called clusters, that are exposed by cluster analysis on the high-dimensional point cloud. Because similar vectors can be thought of as objects that are alike in terms of their attributes, the point cloud\''s structure and individual cluster properties, like their size or compactness, summarize data categories and their relative importance. The contribution of this thesis is a novel analysis approach for visual exploration of high-dimensional point clouds without suffering from structural occlusion. The work is based on implementing two key concepts: The first idea is to discard those geometric properties that cannot be preserved and, thus, lead to the typical artifacts. Topological concepts are used instead to shift away the focus from a point-centered view on the data to a more structure-centered perspective. The advantage is that topology-driven clustering information can be extracted in the data\''s original domain and be preserved without loss in low dimensions. The second idea is to split the analysis into a topology-based global overview and a subsequent geometric local refinement. The occlusion-free overview enables the analyst to identify features and to link them to other visualizations that permit analysis of those properties not captured by the topological abstraction, e.g. cluster shape or value distributions in particular dimensions or subspaces. The advantage of separating structure from data point analysis is that restricting local analysis only to data subsets significantly reduces artifacts and the visual complexity of standard techniques. That is, the additional topological layer enables the analyst to identify structure that was hidden before and to focus on particular features by suppressing irrelevant points during local feature analysis. This thesis addresses the topology-based visual analysis of high-dimensional point clouds for both the time-invariant and the time-varying case. Time-invariant means that the points do not change in their number or positions. That is, the analyst explores the clustering of a fixed and constant set of points. The extension to the time-varying case implies the analysis of a varying clustering, where clusters appear as new, merge or split, or vanish. Especially for high-dimensional data, both tracking---which means to relate features over time---but also visualizing changing structure are difficult problems to solve. info:eu-repo/classification/ddc/500 ddc:500
3	Beiträge zur expliziten Fehlerabschätzung im zentralen Grenzwertsatz Paditz, Ludwig 04 June 2013 (has links) (PDF) In der Arbeit wird das asymptotische Verhalten von geeignet normierten und zentrierten Summen von Zufallsgrößen untersucht, die entweder unabhängig sind oder im Falle der Abhängigkeit als Martingaldifferenzfolge oder stark multiplikatives System auftreten. Neben der klassischen Summationstheorie werden die Limitierungsverfahren mit einer unendlichen Summationsmatrix oder einer angepaßten Folge von Gewichtsfunktionen betrachtet. Es werden die Methode der charakteristischen Funktionen und besonders die direkte Methode der konjugierten Verteilungsfunktionen weiterentwickelt, um quantitative Aussagen über gleichmäßige und ungleichmäßige Restgliedabschätzungen in zentralen Grenzwertsatz zu beweisen. Die Untersuchungen werden dabei in der Lp-Metrik, 1<p<oo oder p=1 bzw. p=oo, durchgeführt, wobei der Fall p=oo der üblichen sup-Norm entspricht. Darüber hinaus wird im Fall unabhängiger Zufallsgrößen der lokale Grenzwertsatz für Dichten betrachtet. Mittels der elektronischen Datenverarbeitung neue numerische Resultate zu erhalten. Die Arbeit wird abgerundet durch verschiedene Hinweise auf praktische Anwendungen. / In the work the asymptotic behavior of suitably centered and normalized sums of random variables is investigated, which are either independent or occur in the case of dependence as a sequence of martingale differences or a strongly multiplicative system. In addition to the classical theory of summation limiting processes are considered with an infinite summation matrix or an adapted sequence of weighting functions. It will be further developed the method of characteristic functions, and especially the direct method of the conjugate distribution functions to prove quantitative statements about uniform and non-uniform error estimates of the remainder term in central limit theorem. The investigations are realized in the Lp metric, 1 <p <oo or p = 1 or p = oo, where in the case p = oo it is the usual sup-norm. In addition, in the case of independent random variables the local limit theorem for densities is considered. By means of electronic data processing new numerical results are obtained. The work is finished by various references to practical applications. Zentraler Grenzwertsatz Summen unabhängiger Zufallsgrößen Doppelfolge von Zufallssgrößen Martingaldifferenzfolgen stark multiplikative Systeme Zufallsvektoren Zufallselemente Verteilungsfunktion Dichtefunktion charakteristische Funktionen konjugierte Verteilungen zufälliger Index Konvergenzgeschwindigkeit ungleichmäßige Fehlerabschätzung einseitige Momente Pseudomomente abgeschnittene Momente Limitierungsverfahren Lp-Metrik globaler zentraler Grenzwertsatz Ordnung der Konvergenzgeschwindigkeit Fehlerabschätzung mittlere Abweichungen central limit theorem sums of independent random variables triangular arrays of random variables series of independent random variables sequences of martingale differences strongly multiplicative systems random vectors random elements distribution function density function characteristic functions conjugate distributions random index speed of convergence nonuniform error estimate one-sided moments pseudo moments truncated moments limiting methods Lp-metric global central limit theorem nonuniform central limit theorem order of the speed of convergence error estimation moderate deviations ddc:510 rvk:SK 800
4	Beiträge zur expliziten Fehlerabschätzung im zentralen Grenzwertsatz Paditz, Ludwig 27 April 1989 (has links) In der Arbeit wird das asymptotische Verhalten von geeignet normierten und zentrierten Summen von Zufallsgrößen untersucht, die entweder unabhängig sind oder im Falle der Abhängigkeit als Martingaldifferenzfolge oder stark multiplikatives System auftreten. Neben der klassischen Summationstheorie werden die Limitierungsverfahren mit einer unendlichen Summationsmatrix oder einer angepaßten Folge von Gewichtsfunktionen betrachtet. Es werden die Methode der charakteristischen Funktionen und besonders die direkte Methode der konjugierten Verteilungsfunktionen weiterentwickelt, um quantitative Aussagen über gleichmäßige und ungleichmäßige Restgliedabschätzungen in zentralen Grenzwertsatz zu beweisen. Die Untersuchungen werden dabei in der Lp-Metrik, 1<p<oo oder p=1 bzw. p=oo, durchgeführt, wobei der Fall p=oo der üblichen sup-Norm entspricht. Darüber hinaus wird im Fall unabhängiger Zufallsgrößen der lokale Grenzwertsatz für Dichten betrachtet. Mittels der elektronischen Datenverarbeitung neue numerische Resultate zu erhalten. Die Arbeit wird abgerundet durch verschiedene Hinweise auf praktische Anwendungen. / In the work the asymptotic behavior of suitably centered and normalized sums of random variables is investigated, which are either independent or occur in the case of dependence as a sequence of martingale differences or a strongly multiplicative system. In addition to the classical theory of summation limiting processes are considered with an infinite summation matrix or an adapted sequence of weighting functions. It will be further developed the method of characteristic functions, and especially the direct method of the conjugate distribution functions to prove quantitative statements about uniform and non-uniform error estimates of the remainder term in central limit theorem. The investigations are realized in the Lp metric, 1 <p <oo or p = 1 or p = oo, where in the case p = oo it is the usual sup-norm. In addition, in the case of independent random variables the local limit theorem for densities is considered. By means of electronic data processing new numerical results are obtained. The work is finished by various references to practical applications. info:eu-repo/classification/ddc/510 ddc:510

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