Topological Hierarchies and Decomposition: From Clustering to Persistence

Hierarchical clustering is a class of algorithms commonly used in exploratory data analysis (EDA) and supervised learning. However, they suffer from some drawbacks, including the difficulty of interpreting the resulting dendrogram, arbitrariness in the choice of cut to obtain a flat clustering, and the lack of an obvious way of comparing individual clusters. In this dissertation, we develop the notion of a topological hierarchy on recursively-defined subsets of a metric space. We look to the field of topological data analysis (TDA) for the mathematical background to associate topological structures such as simplicial complexes and maps of covers to clusters in a hierarchy. Our main results include the definition of a novel hierarchical algorithm for constructing a topological hierarchy, and an implementation of the MAPPER algorithm and our topological hierarchies in pure Python code as well as a web app dashboard for exploratory data analysis. We show that the algorithm scales well to high-dimensional data due to the use of dimensionality reduction in most TDA methods, and analyze the worst-case time complexity of MAPPER and our hierarchical decomposition algorithm. Finally, we give a use case for exploratory data analysis with our techniques

Women and Stability: A Topological View of the Relationship between Women and Armed Conflict in West Africa

The relationship between women and stability, if any, is a topic of much debate and research. Several large and influential organizations have all researched women\u27s effect on stability. Furthermore, several of these world organizations, the United Nations, in particular, have declared gender equality to be a driving force in promoting stability and conflict prevention. Due to the United States active involvement in conflict prevention in such regions as West Africa, research concerning the relationship between women and stability is of particular interest to the United States Africa Command. As such, this research applied Topological Data Analysis, combined with other machine learning algorithms, to Demographic and Health Survey Program data combined with Armed Conflict Location and Event Data so as to observe the relationship between women\u27s status and armed conflicts in the West African region. While this team did not observe any direct correlation between women\u27s well-being and stability - defined as a lack of armed conflict events - the chosen methodologies and data usage have potential implications for future research concerning stability and conflict