Abstract
Persistent homology allows us to create topological summaries of complex data. In order to analyse these statistically, we need to choose a topological summary and a relevant metric space in which this topological summary exists. While different summaries may contain the same information (as they come from the same persistence module), they can lead to different statistical conclusions since they lie in different metric spaces. The best choice of metric will often be application-specific. In this paper we discuss distance correlation, which is a non-parametric tool for comparing data sets that can lie in completely different metric spaces. In particular we calculate the distance correlation between different choices of topological summaries. We compare some different topological summaries for a variety of random models of underlying data via the distance correlation between the samples. We also give examples of performing distance correlation between topological summaries and other scalar measures of interest, such as a paired random variable or a parameter of the random model used to generate the underlying data. This article is meant to be expository in style, and will include the definitions of standard statistical quantities in order to be accessible to non-statisticians.
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Notes
- 1.
We will consider only persistent homology with intervals with finite death.
- 2.
When analyzing real data, one often cones off the space at some more or less meaningful maximum filtration so as to avoid infinite intervals.
- 3.
The data was provided by the Norwegian Mapping Authority [10] under a CC-BY-4.0 license.
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Acknowledgements
G.S. would like to thank Andreas Prebensen Korsnes of the Norwegian Mapping Authority for going out of his way to facilitate bulk downloads of DEM data before a single region was decided upon for the experiment in Sect. 5.2.
G.S. was supported by Swiss National Science Foundation grant number 200021_172636.
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Turner, K., Spreemann, G. (2020). Same But Different: Distance Correlations Between Topological Summaries. In: Baas, N., Carlsson, G., Quick, G., Szymik, M., Thaule, M. (eds) Topological Data Analysis. Abel Symposia, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-030-43408-3_18
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