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Tropical Principal Component Analysis and Its Application to Phylogenetics

Bulletin of Mathematical Biology volume 81, pages 568–597 (2019)Cite this article

Abstract

Principal component analysis is a widely used method for the dimensionality reduction of a given data set in a high-dimensional Euclidean space. Here we define and analyze two analogues of principal component analysis in the setting of tropical geometry. In one approach, we study the Stiefel tropical linear space of fixed dimension closest to the data points in the tropical projective torus; in the other approach, we consider the tropical polytope with a fixed number of vertices closest to the data points. We then give approximative algorithms for both approaches and apply them to phylogenetics, testing the methods on simulated phylogenetic data and on an empirical dataset of Apicomplexa genomes.

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Notes

  1. Our software for all computations can be downloaded at http://polytopes.net/computations/tropicalPCA/.

  2. A more detailed tutorial for how to use Mesquite to generate data can be accessed at https://www.youtube.com/watch?v=94tXmA4ods4&t=238s. We followed the same simulation steps and parameter settings as in this video.

  3. In keeping with the format of Mesquite Maddison and Maddison (2017), leaves of the species tree are rendered as capital letters. Tree topologies of all projected points can be found in the supplement at http://polytopes.net/computations/tropicalPCA/.

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Acknowledgements

R. Y. was supported by Research Initiation Proposals from the Naval Postgraduate School and NSF Division of Mathematical Sciences 1622369. L. Z. was supported by an NSF Graduate Research Fellowship. X. Z. was supported by travel funding from the Department of Statistics at the University of Kentucky. The authors thank Bernd Sturmfels (UC Berkeley and MPI Leipzig) for many helpful conversations. The authors also thank Daniel Howe (University of Kentucky) for his input on Apicomplexa tree topologies.

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Authors and Affiliations

  1. Naval Postgraduate School, 1411 Cunningham Road, Monterey, CA, 93943-5219, USA

    Ruriko Yoshida

  2. University of California, Berkeley, 1042 Evans Hall, Berkeley, CA, 94720-3840, USA

    Leon Zhang

  3. University of Kentucky, 725 Rose Street, Multidisciplinary Science Building 0082, Room 311, Lexington, KY, 40536-0082, USA

    Xu Zhang

Authors
  1. Ruriko Yoshida
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  2. Leon Zhang
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  3. Xu Zhang
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Correspondence to Ruriko Yoshida.

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Yoshida, R., Zhang, L. & Zhang, X. Tropical Principal Component Analysis and Its Application to Phylogenetics. Bull Math Biol 81, 568–597 (2019). https://doi.org/10.1007/s11538-018-0493-4

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  • DOI: https://doi.org/10.1007/s11538-018-0493-4

Keywords

  • Dimensionality reduction
  • Phylogenomics
  • Tropical geometry