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Empirical geodesic graphs and CAT(k) metrics for data analysis

Kobayashi, Kei and Wynn, Henry P. ORCID: 0000-0002-6448-1080 (2020) Empirical geodesic graphs and CAT(k) metrics for data analysis. Statistics and Computing, 30 (1). 1 - 18. ISSN 0960-3174

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Identification Number: 10.1007/s11222-019-09855-3


A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. For a probability distribution, the length along a path between two points can be defined as the amount of probability mass accumulated along the path. The geodesic, then, is the shortest such path and defines a geodesic metric. Such metrics are transformed in a number of ways to produce parametrised families of geodesic metric spaces, empirical versions of which allow computation of intrinsic means and associated measures of dispersion. These reveal properties of the data, based on geometry, such as those that are difficult to see from the raw Euclidean distances. Examples of application include clustering and classification. For certain parameter ranges, the spaces become CAT(0) spaces and the intrinsic means are unique. In one case, a minimal spanning tree of a graph based on the data becomes CAT(0). In another, a so-called “metric cone” construction allows extension to CAT(k) spaces. It is shown how to empirically tune the parameters of the metrics, making it possible to apply them to a number of real cases.

Item Type: Article
Official URL:
Additional Information: © 2019 Springer Science+Business Media, LLC, part of Springer Nature
Divisions: Centre for Analysis of Time Series
Subjects: H Social Sciences > HA Statistics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited: 27 Mar 2020 15:21
Last Modified: 20 Oct 2021 02:49

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