Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective

Monod, Anthea; Lin, Benyao; Yoshida, Ruriko; Kang, Qiwen

doi:10.48550/arxiv.1805.12400

Cited by 9 publications

(27 citation statements)

References 72 publications

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“…This equivalence relation generates a quotient space known as the tropical projective torus, denoted by R n /R1, which is the ambient space of tree space, U N ⊂ T N ⊂ R n /R1. The tropical projective torus R n /R1 may be embedded into R n−1 by considering representatives of the equivalence classes with first coordinate equal to zero, ( Maclagan and Sturmfels (2015); Monod et al (2018); Lee et al (2019) for more detail.…”

Section: The Tropical Projective Torusmentioning

confidence: 99%

“…Owen and Provan (2011) give the fastest available algorithm to date to find the geodesic path between any two trees in BHV space, which runs in quartic time in the number of leaves N . Monod et al (2018) present an alternative geometric construction of phylogenetic tree space based on tropical geometry and study its analytic and topological properties with the aim of statistical inference and data analysis in mind. Trees are represented by cophenetic vectors (2); external edge lengths are thus included.…”

Section: Comparing Trees Via Metricsmentioning

confidence: 99%

“…It has also been used in previous work studying phylogenetic trees (Lin et al, 2017;Lin and Yoshida, 2018;Yoshida et al, 2019). The tropical metric is combinatorial in nature and is given by the difference between the maximum and minimum of the differences between the tree coordinates; it is a proper metric (Monod et al, 2018). Under this metric, properties that are desirable for statistical inference (such as hypothesis testing) and exact probabilistic studies (such as concentration inequalities and convergence studies) are satisfied and well-defined.…”

Section: Palm Tree Spacementioning

confidence: 99%

“…Proposition 7 (Proposition 12, Monod et al (2018)). A tree metric w for a phylogenetic tree T is a tree ultrametric if and oly if T is equidistant.…”

Section: Introductionmentioning

confidence: 99%

“…The above technical condition characterizes phylogenetic trees; in particular, the space of phylogenetic trees with N leaves, T N , is the collection of all n-tuples {w(i, j) } 1≤i<j≤N that satisfy the four-point condition (4), or equivalently, the maximum among ( 3) is achieved at least twice. For examples and counterexamples of the four-point condition with illustrations, see Monod et al (2018).…”

Section: Introductionmentioning

confidence: 99%

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Tropical Geometric Variation of Phylogenetic Tree Shapes

Lin¹,

Monod

Yoshida³

2020

Preprint

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We study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is given by the tree's branching configuration and leaf labeling. We use the tropical line segment as a framework to define notions of variance as well as invariance of tree topologies: we provide a combinatorial search theorem that gives all tree topologies that occur along a tropical line segment, as well as a setting under which tree topologies do not change along a tropical line segment. Our study is motivated by comparison to the moduli space endowed with a geodesic metric proposed by Billera, Holmes, and Vogtmann (referred to as BHV space); we consider the tropical geometric setting as an alternative framework to BHV space for sets of phylogenetic trees. We give an algorithm to compute tropical line segments which is lower in computational complexity than the fastest method currently available for BHV geodesics and show that its trajectory behaves more subtly: while the BHV geodesic traverses the origin for vastly different tree topologies, the tropical line segment bypasses it.

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Section: The Tropical Projective Torusmentioning

confidence: 99%

Section: Comparing Trees Via Metricsmentioning

confidence: 99%

Section: Palm Tree Spacementioning

confidence: 99%

“…Proposition 7 (Proposition 12, Monod et al (2018)). A tree metric w for a phylogenetic tree T is a tree ultrametric if and oly if T is equidistant.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tropical Geometric Variation of Phylogenetic Tree Shapes

Lin¹,

Monod

Yoshida³

2020

Preprint

Self Cite

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Weighted lens depth: Some applications to supervised classification

et al. 2022

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Starting with Tukey's pioneering work in the 1970s, the notion of depth in statistics has been widely extended, especially in the last decade. Such extensions include those to high‐dimensional data, functional data, and manifold‐valued data. In particular, in the learning paradigm, the depth‐depth method has become a useful technique. In this article, we extend the lens depth to the case of data in metric spaces and study its main properties. We also introduce, for Riemannian manifolds, the weighted lens depth. The weighted lens depth is nothing more than a lens depth for a weighted version of the Riemannian distance. To build it, we replace the geodesic distance on the manifold with the Fermat distance, which has the important property of taking into account the density of the data together with the geodesic distance. Next, we illustrate our results with some simulations and also in some interesting real datasets, including pattern recognition in phylogenetic trees, using the depth‐depth approach.

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Wald Space for Phylogenetic Trees

Lueg

Garba

Nye

et al. 2021

Lecture Notes in Computer Science

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Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order to properly quantify this uncertainty during the statistical analysis of collections of possible evolutionary trees inferred from biological data. Recently, the wald space has been introduced: a length space for trees which is a certain subset of the manifold of symmetric positive definite matrices. In this work, the wald space is introduced formally and its topology and structure is studied in detail. In particular, we show that wald space has the topology of a disjoint union of open cubes, it is contractible, and by careful characterization of cube boundaries, we demonstrate that wald space is a Whitney stratified space of type (A). Imposing the metric induced by the affine invariant metric on symmetric positive definite matrices, we prove that wald space is a geodesic Riemann stratified space. A new numerical method is proposed and investigated for construction of geodesics, computation of Fréchet means and calculation of curvature in wald space. This work is intended to serve as a mathematical foundation for further geometric and statistical research on this space.

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Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective

Cited by 9 publications

References 72 publications

Tropical Geometric Variation of Phylogenetic Tree Shapes

Tropical Geometric Variation of Phylogenetic Tree Shapes

Weighted lens depth: Some applications to supervised classification

Wald Space for Phylogenetic Trees

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