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Persistent Homology in Multivariate Data Visualization

Rieck, Bastian Alexander

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Technological advances of recent years have changed the way research is done. When describing complex phenomena, it is now possible to measure and model a myriad of different aspects pertaining to them. This increasing number of variables, however, poses significant challenges for the visual analysis and interpretation of such multivariate data. Yet, the effective visualization of structures in multivariate data is of paramount importance for building models, forming hypotheses, and understanding intrinsic properties of the underlying phenomena. This thesis provides novel visualization techniques that advance the field of multivariate visual data analysis by helping represent and comprehend the structure of high-dimensional data. In contrast to approaches that focus on visualizing multivariate data directly or by means of their geometrical features, the methods developed in this thesis focus on their topological properties. More precisely, these methods provide structural descriptions that are driven by persistent homology, a technique from the emerging field of computational topology.

Such descriptions are developed in two separate parts of this thesis. The first part deals with the qualitative visualization of topological features in multivariate data. It presents novel visualization methods that directly depict topological information, thus permitting the comparison of structural features in a qualitative manner. The techniques described in this part serve as low-dimensional representations that make the otherwise high-dimensional topological features accessible. We show how to integrate them into data analysis workflows based on clustering in order to obtain more information about the underlying data. The efficacy of such combined workflows is demonstrated by analysing complex multivariate data sets from cultural heritage and political science, for example, whose structures are hidden to common visualization techniques.

The second part of this thesis is concerned with the quantitative visualization of topological features. It describes novel methods that measure different aspects of multivariate data in order to provide quantifiable information about them. Here, the topological characteristics serve as a feature descriptor. Using these descriptors, the visualization techniques in this part focus on augmenting and improving existing data analysis processes. Among others, they deal with the visualization of high-dimensional regression models, the visualization of errors in embeddings of multivariate data, as well as the assessment and visualization of the results of different clustering algorithms.

All the methods presented in this thesis are evaluated and analysed on different data sets in order to show their robustness. This thesis demonstrates that the combination of geometrical and topological methods may support, complement, and surpass existing approaches for multivariate visual data analysis.

Item Type: Dissertation
Supervisor: Leitte, Prof. Dr. Heike
Date of thesis defense: 28 April 2017
Date Deposited: 18 May 2017 07:57
Date: 2017
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Computer Science
Subjects: 004 Data processing Computer science
500 Natural sciences and mathematics
510 Mathematics
Controlled Keywords: Visualization, Visual Analysis, Persistent Homology, Multivariate Data Analysis, Algebraic Topology
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