Tropical principal component analysis on the space of phylogenetic trees

Abstract

Motivation Due to new technology for efficiently generating genome data, machine learning methods are urgently needed to analyze large sets of gene trees over the space of phylogenetic trees. However, the space of phylogenetic trees is not Euclidean, so ordinary machine learning methods cannot be directly applied. In 2019, Yoshida et al. introduced the notion of tropical principal component analysis (PCA), a statistical method for visualization and dimensionality reduction using a tropical polytope with a fixed number of vertices that minimizes the sum of tropical distances between each data point and its tropical projection. However, their work focused on the tropical projective space rather than the space of phylogenetic trees. We focus here on tropical PCA for dimension reduction and visualization over the space of phylogenetic trees. Results Our main results are twofold: (1) theoretical interpretations of the tropical principal components over the space of phylogenetic trees, namely, the existence of a tropical cell decomposition into regions of fixed tree topology; and (2) the development of a stochastic optimization method to estimate tropical PCs over the space of phylogenetic trees using a Markov Chain Monte Carlo (MCMC) approach. This method performs well with simulation studies, and it is applied to three empirical datasets: Apicomplexa and African coelacanth genomes as well as sequences of hemagglutinin for influenza from New York. Availability Dataset: http://polytopes.net/Data.tar.gz, Code: http://polytopes.net/tropica_MCMC_codes.tar.gz Supplementary information Supplementary data are available at http://polytopes.net/supplement.pdf.

Tropical optimal transport and Wasserstein distances

“…An application of tropical geometry that has gained much interest is the tropical geometric representation of the space of phylogenetic trees. In particular, there has very recently been active work in using tropical geometry as a data analytic tool for sets of phylogenetic trees [33,37,45,50]. In this paper, we study the tropical projective torus, which is the ambient space of phylogenetic trees, and build upon it to provide a set of tools for statistical, probabilistic, and geometric studies using optimal transport theory.…”

Projections of Tropical Fermat-Weber Points

“…Recently, R. Yoshida, L. Zhang and X. Zhang proposed the tropical principal component analysis (tropical PCA) [8], which is of great use in the analysis of phylogenetic trees in Phylogenetics (also see [9]). Phylogenetics is a subject that is very powerful for explaining genome evolution, processes of speciation and relationships among species.…”

“…Analysing data sets of phylogenetic trees with a fixed number of leaves is difficult because the space of phylogenetic trees is high dimensional and not Euclidean; it is a union of lower dimensional polyhedra cones in R ( n 2 ) , where n is the number of leaves [9]. Many multivariate statistical procedures have been applied to such data sets [10][11][12][13][14][15].…”

“…Nye [16] used a two-point convex hull under the CAT(0)-metric as the first order principal component over the Billera-Holmes-Vogtman (BHV) tree space introduced in [17]. However, Lin et al [18] showed that the three-point convex hull in the BHV tree space can have arbitrarily high dimension, which means that the idea in [16] cannot be generalized to higher order principal components (e.g., see [9]). In addition, Nye et al [19] used the locus of the weighted Fréchet mean when the weights vary over the k-simplex as the k-th principal component in the BHV tree space, and this approach performed well in simulation studies.…”

Tropical Gaussians: a brief survey

“…Tropical mathematics has found many applications in both pure and applied areas, as documented by a growing number of monographs on its interactions with various other areas of mathematics: algebraic geometry [Baker and Payne 2016;Gross 2011;Huh 2018;Maclagan and Sturmfels 2015], discrete event systems [Baccelli et al 1992;Butkovič 2010], large deviations and calculus of variations [Kolokoltsov and Maslov 1997;Puhalskii 2001], and combinatorial optimization [Joswig ≥ 2020]. At the same time, new applications are emerging in phylogenetics [Monod et al 2018;Yoshida et al 2019;Page et al 2020], statistics [Hook 2017], economics [Baldwin and Klemperer 2019;Crowell and Tran 2016;Elsner and van den Driessche 2004;Gursoy et al 2013;Joswig 2017;Shiozawa 2015;Tran 2013;Tran and Yu 2019], game theory, and complexity theory [Allamigeon et al 2018;Akian et al 2012]. There is a growing need for a systematic study of probability distributions in tropical settings.…”