Persistent Homology of Geospatial Data: A Case Study with Voting
- Creators
- Feng, Michelle
- Porter, Mason A.
Abstract
A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (which, in our case, is a simplicial complex). Software packages for computing persistent homology typically construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute. We investigate alternative methods of constructing simplicial complexes and the effects of making associated choices during simplicial-complex construction on the output of persistent-homology algorithms. We present two new methods for constructing simplicial complexes from two-dimensional geospatial data (such as maps). We apply these methods to a California precinct-level voting data set, and we thereby demonstrate that our new constructions can capture geometric characteristics that are missed by distance-based constructions. Our new constructions can thus yield more interpretable persistence modules and barcodes for geospatial data. In particular, they are able to distinguish short-persistence features that occur only for a narrow range of distance scales (e.g., voting patterns in densely populated cities) from short-persistence noise by incorporating information about other spatial relationships between regions.
Additional Information
© 2021 SIAM. Received by the editors January 29, 2019; accepted for publication (in revised form) March 25, 2020; published electronically February 4, 2021. We thank Moon Duchin, Joshua Gensler, Mike Hill, Stan Osher, Nina Otter, Bernadette Stolz, BaoWang, and two anonymous referees for helpful comments. We also thank Emilia Alvarez, Eion Blanchard, Austin Eide, Patrick Girardet, Everett Meike, Dmitriy Morozov, Justin Solomon, Courtney Thatcher, Jim Thatcher, and Maia Woluchem for insightful discussions.Attached Files
Published - 19m1241519.pdf
Submitted - 1902.05911.pdf
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Additional details
- Eprint ID
- 108472
- Resolver ID
- CaltechAUTHORS:20210318-084444863
- Created
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2021-03-19Created from EPrint's datestamp field
- Updated
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2021-03-19Created from EPrint's last_modified field