@article{feng_persistent_2021, 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.}, author = {Feng, Michelle and Porter, Mason A.}, date = {2021-01-01}, doi = {10.1137/19M1241519}, issn = {0036-1445}, journaltitle = {{SIAM} Review}, keywords = {1 - Anthropology, 1 - Spatial relations analysis, 1 - Voting, 2 - Persistent homology, 3 - Maps, 3 - Point cloud, 3 - Votes}, note = {Publisher: Society for Industrial and Applied Mathematics}, number = {1}, pages = {67--99}, shortjournal = {{SIAM} Rev.}, shorttitle = {Persistent Homology of Geospatial Data}, title = {Persistent Homology of Geospatial Data: A Case Study with Voting}, url = {https://epubs.siam.org/doi/abs/10.1137/19M1241519}, urldate = {2021-04-17}, volume = {63} }