Estimating Tropical Principal Components Using Metropolis Hasting Algorithm

  • Qiwen Kang
  • Ruriko YoshidaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10931)


Principal component analysis is one of the most popular unsupervised learning methods for reducing the dimension of a given data set in a high-dimensional Euclidean space. However, computing principal components on a space of phylogenetic trees with fixed labels of leaves is a challenging task since a space of phylogenetic tree is not Euclidean. In 2017, Yoshida et al. defined a notion of tropical principal component analysis and they have applied it to a space of phylogenetic trees. The challenge, however, they encountered was a computational times.

In this paper we estimate tropical principal components in a space of phylogenetic trees using the Metropolis-Hasting algorithm. We have implemented an R software package to efficiently estimate tropical principal components and then we have applied it to African coelacanth genomes data set.


Phylogenetic trees Polytopes Tropical geometry 



R. Y. is supported by NSF Division of Mathematical Sciences: CDS&E-MSS program. Proposal number:1622369.


  1. 1.
    Akian, M., Gaubert, S., Viorel, N., Singer, I.: Best approximation in max-plus semimodules. Linear Algebra Appl. 435, 3261–3296 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ardila, F., Klivans, C.: The Bergman complex of a matroid and phylogenetic trees. J. Comb. Theory Ser. B 96, 38–49 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Billera, L., Holmes, S., Vogtman, K.: Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27, 733–767 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cohen, G., Gaubert, S., Quadrat, J.P.: Duality and separation theorems in idempotent semimodules. Linear Algebra Appl. 379, 395–422 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hedges, S.B.: Vertebrates (Vertebrata). In: Hedges, S.B., Kumar, S. (eds.) The Timetree of Life, pp. 309–314. Oxford University Press (2009)Google Scholar
  6. 6.
    Joswig, M., Essentials of Tropical Combinatorics (2017).
  7. 7.
    Levine, N.D.: Progress in taxonomy of the Apicomplexan protozoa. J. Eukaryot Microbiol. 35, 518–520 (1988)Google Scholar
  8. 8.
    Liang, D., Shen, X.X., Zhang, P.: One thousand two hundred ninety nuclear genes from a genome-wide survey support lungfishes as the sister group of tetrapods. Mol. Biol. Evol. 8, 1803–1807 (2013)CrossRefGoogle Scholar
  9. 9.
    Lin, B., Sturmfels, B., Tang, X., Yoshida, R.: Convexity in tree spaces. SIAM Discrete Math. 3, 2015–2038 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015)CrossRefGoogle Scholar
  11. 11.
    Nye, T.: Principal components analysis in the space of phylogenetic trees. Ann. Stat. 39, 2716–2739 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pachter, L., Sturmfels, B.: Algebraic Statistics for Computational Biology. Cambridge University Press, New York (2005)CrossRefGoogle Scholar
  13. 13.
    Speyer, D., Sturmfels, B.: Tropical mathematics. Math. Mag. 82, 163–173 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yoshida, R., Zhang, L., Zhang, X.: Tropical principal component analysis and its application to phylogenetics (2017).

Copyright information

© This is a U.S. government work and its text is not subject to copyright protection in the United States; however, its text may be subject to foreign copyright protection 2018

Authors and Affiliations

  1. 1.University of KentuckyLexingtonUSA
  2. 2.Naval Postgraduate SchoolMontereyUSA

Personalised recommendations