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Persistent Homology of Geospatial Data: A Case Study with Voting

Feng, Michelle and Porter, Mason A. (2021) Persistent Homology of Geospatial Data: A Case Study with Voting. SIAM Review, 63 (1). pp. 67-99. ISSN 0036-1445. doi:10.1137/19m1241519.

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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.

Item Type:Article
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URLURL TypeDescription Paper
Porter, Mason A.0000-0002-5166-0717
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.
Subject Keywords:persistent homology, topological data analysis, voting data, geospatial data
Issue or Number:1
Classification Code:AMS subject classifications: Primary, 55N31; Secondary, 55-04, 55U10, 62R40, 91D20
Record Number:CaltechAUTHORS:20210318-084444863
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Official Citation:Persistent Homology of Geospatial Data: A Case Study with Voting. Michelle Feng and Mason A. Porter. SIAM Review 2021 63:1, 67-99; DOI: 10.1137/19m1241519
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108472
Deposited By: Tony Diaz
Deposited On:19 Mar 2021 18:29
Last Modified:19 Mar 2021 18:29

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