Persistent homology is a prominent tool in topological data analysis. This thesis is designed to be an introduction and guide to a beginner in persistent homology. This comprehensive overview discusses the math used behind it, the code needed to apply it, and its current place in the field. We explain and demonstrate the algebraic topology which fuels persistent homology. Homotopies inspire homology groups, which are able to determine how many holes a shape has. By visualizing data as a shape, persistent homology determines what type of holes are present.
We demonstrate this by using the package TDA in the manipulation software R on controlled datasets. A kernel density estimate diagram presents the results. We showcase applying TDA to an external, uncontrolled dataset. The limits on memory allowed us to process no more than four columns of data at a time. To more thoroughly explore the dataset, we analyzed several four-column subsets, but found no special features aside from a base level of closeness.
","abstract_html":"<p>Persistent homology is a prominent tool in topological data analysis. This thesis is designed to be an introduction and guide to a beginner in persistent homology. This comprehensive overview discusses the math used behind it, the code needed to apply it, and its current place in the field. We explain and demonstrate the algebraic topology which fuels persistent homology. Homotopies inspire homology groups, which are able to determine how many holes a shape has. By visualizing data as a shape, persistent homology determines what type of holes are present.</p> <p>We demonstrate this by using the package TDA in the manipulation software R on controlled datasets. A kernel density estimate diagram presents the results. We showcase applying TDA to an external, uncontrolled dataset. The limits on memory allowed us to process no more than four columns of data at a time. To more thoroughly explore the dataset, we analyzed several four-column subsets, but found no special features aside from a base level of closeness.</p>","abstract_has_math":false,"creators":["Flynn, R. Anne"],"institution":null,"degree_name":"Master of Science","degree_level":"Thesis","degree_discipline":"Math and Computer Science","degree_department":null,"school":null,"contributors":["Josh Thompson"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-05-01T07:00:00Z","date_published":"2024-05-01T07:00:00Z","updated_at":"2026-07-24T03:24:36Z","subjects":["TDA","topological data analysis","topology","data science","persistence","persistent homology","algebraic topology","R","Geometry and Topology"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://commons.nmu.edu/theses/844","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Josh Thompson"]},{"key":"dc:creator","label":"Author","values":["Flynn, R. Anne"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2024-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Math and Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["TDA","topological data analysis","topology","data science","persistence","persistent homology","algebraic topology","R","Geometry and Topology"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://commons.nmu.edu/theses/844"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Persistent homology is a prominent tool in topological data analysis. This thesis is designed to be an introduction and guide to a beginner in persistent homology. This comprehensive overview discusses the math used behind it, the code needed to apply it, and its current place in the field. We explain and demonstrate the algebraic topology which fuels persistent homology. Homotopies inspire homology groups, which are able to determine how many holes a shape has. By visualizing data as a shape, persistent homology determines what type of holes are present.
We demonstrate this by using the package TDA in the manipulation software R on controlled datasets. A kernel density estimate diagram presents the results. We showcase applying TDA to an external, uncontrolled dataset. The limits on memory allowed us to process no more than four columns of data at a time. To more thoroughly explore the dataset, we analyzed several four-column subsets, but found no special features aside from a base level of closeness.
"]},{"key":"dc:title","label":"Title","values":["Plumbing the Depths of the Shallow End: Exploring Persistent Homology Using Small Data"]}]}],"canonical_facts":{"dc:contributor":["Josh Thompson"],"dc:creator":["Flynn, R. Anne"],"dc:date.available":["2024-01-01T08:00:00Z"],"dc:description.abstract":["Persistent homology is a prominent tool in topological data analysis. This thesis is designed to be an introduction and guide to a beginner in persistent homology. This comprehensive overview discusses the math used behind it, the code needed to apply it, and its current place in the field. We explain and demonstrate the algebraic topology which fuels persistent homology. Homotopies inspire homology groups, which are able to determine how many holes a shape has. By visualizing data as a shape, persistent homology determines what type of holes are present.
We demonstrate this by using the package TDA in the manipulation software R on controlled datasets. A kernel density estimate diagram presents the results. We showcase applying TDA to an external, uncontrolled dataset. The limits on memory allowed us to process no more than four columns of data at a time. To more thoroughly explore the dataset, we analyzed several four-column subsets, but found no special features aside from a base level of closeness.
"],"dc:identifier":["https://commons.nmu.edu/theses/844"],"dc:subject":["TDA","topological data analysis","topology","data science","persistence","persistent homology","algebraic topology","R","Geometry and Topology"],"dc:title":["Plumbing the Depths of the Shallow End: Exploring Persistent Homology Using Small Data"],"thesis:degree_discipline":["Math and Computer Science"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T03:24:36Z"}