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A recent cohort of research aims to apply topological and geometric theory to data analysis. However, more effort is needed to incorporate statistical ideas and structure to these analysis methods. To this end, we present persistent homology clustering techniques through the perspective of data analysis. These techniques provide insight into the structure of the underlying dynamic and are able to recognize important shape properties such as periodicity, chaos, and multi-stability. Moreover, introducing quantitative structure on the topological data space allows for rigorous understanding of the data's geometry, a powerful tool for scrutinizing the morphology of the inherent dynamic. Additionally, we illustrate the advantages of these techniques and results through examples derived from dynamical systems applications.
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