The persistence transformation; a new methodology of topological data analysis

Abstract

The field of Topological Data Analysis (TDA) continues to evolve as a powerful tool for the analysis of complex data. The motivation behind this research lies in the need to extend existing TDA tools to provide more accurate, efficient, and comprehensive analyses of intricate datasets. The primary research problem addressed herein pertains to the limitations of the Persistence Diagram, a fundamental TDA tool that does not inherently incorporate positional information of topological features. The absence of this crucial spatial context can lead to inaccurate results, especially when analyzing low-dimensional data. To tackle this issue, this dissertation introduces the Persistence Transformation, an innovative extension of the Persistence Diagram. It is designed to capture the positional information of topological peaks, enhancing the robustness and depth of TDA analyses. Key findings of this research include a comprehensive analysis of the properties and stability of the Persistence Tansformation. Furthermore, a real-world application of this method demonstrates its effectiveness in the classification of MALDI data, highlighting the practical utility of the extension. The originality and contribution of this work are underscored by the extension of the traditional Persistence Diagram. The introduction of the Persistence Transformation empowers mathematicians and data analysts to tackle a broader spectrum of complex problems, fostering more accurate results across diverse application domains. However, it is important to note that the Persistence Transformation generates results of a higher dimensionality when compared to the Persistence Diagram. While this enables a richer analysis of complex data, it may necessitate additional computational resources. Nevertheless, this research not only advances the domain of TDA but also opens the door to a wider array of analytical possibilities for complex datasets, offering valuable insights across various fields.