Tropical Principal Component Analysis and its Application to Phylogenetics

Yoshida, Ruriko; Zhang, Leon; Zhang, Xu

doi:10.48550/arxiv.1710.02682

Cited by 5 publications

(12 citation statements)

References 3 publications

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“…For example, a space of phylogenetic trees with a fixed set of leaves is a union of lower dimensional cones over R e , where e = n 2 with n as the number of leaves [3]. Since the space of phylogenetic trees is a union of lower dimensional cones, we cannot just apply statistical models in data science to a set of phylogenetic trees [19].…”

Section: Introductionmentioning

confidence: 99%

“…In 2004, Speyer and Sturmfels showed a space of phylogenetic trees with a given set of labels on their leaves is a tropical Grassmannian [16], which is a tropicalization of a linear space defined by a set of linear equations [19] with the max-plus algebra. It is important to note that the tree space defined by Speyer and Sturmfels is not isometric to the tree space defined by Billera-Holmes-Vogtman while it is homeomorphic to each other.…”

Section: Introductionmentioning

confidence: 99%

“…, n}. For example, Yoshida et al defined tropical principal component analysis (PCA) with the tropical metric over the space of equidistant trees to reduce dimensionality and to visualize data sets [19]. Also Tang et al developed hard and soft tropical support vector machines (SVMs) and the authors applied them to classifying sets of equidistant trees [18].…”

Section: Introductionmentioning

confidence: 99%

“…For example, the principal geodesic developed by Nye in [14] has a natural interpretation of the geodesic with the BHV metric over the space of phylogenetic trees with n leaves. However, it is not obvious how to interpret a tropical principal polytope developed by Yoshida et al in [19,15]. Interpretation of the output from a statistical learning model is one of the most important processes in data analysis.…”

Section: Introductionmentioning

confidence: 99%

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Tree Topologies along a Tropical Line Segment

Yoshida

2021

Preprint

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Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces have some nice properties. One of the challenges in applications of tropical geometry to tree spaces is the difficulty to interpret outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Especially we show that a probability of a tropical line segment of randomly chosen two trees going thorough the origin (the star tree) is zero and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tree Topologies along a Tropical Line Segment

Yoshida

2021

Preprint

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“…In tropical geometry, one redefines arithmetic over the real numbers so that the sum of two numbers is their maximum and the product is their sum (in the usual sense). There are strong connections between phylogenetics and tropical geometry [3,4,14,15,19,20] so the l ∞ -metric is a natural choice to measure best fit for phylogenetic reconstruction. An algorithm of Chepoi and Fichet computes an ultrametric l ∞ -nearest to a given dissimilarity map in polynomial time [6] but this is generally not the only l ∞ -nearest ultrametric.…”

Section: Introductionmentioning

confidence: 99%

L-Infinity Optimization to Bergman Fans of Matroids with an Application to Phylogenetics

Bernstein¹

2020

SIAM J. Discrete Math.

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Given a dissimilarity map δ on finite set X, the set of ultrametrics (equidistant tree metrics) which are l ∞ -nearest to δ is a tropical polytope. We give an interior description of this tropical polytope. It was shown by Ardila and Klivans [4] that the set of all ultrametrics on a finite set of size n is the Bergman fan associated to the matroid underlying the complete graph on n vertices. Therefore, we derive our results in the more general context of Bergman fans of matroids. This added generality allows our results to be used on dissimilarity maps where only a subset of the entries are known.

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Tropical Data Science over the Space of Phylogenetic Trees

Yoshida

2021

Lecture Notes in Networks and Systems

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When we apply comparative phylogenetic analyses to genome data, it is a well-known problem and challenge that some of given species (or taxa) often have missing genes. In such a case, we have to impute a missing part of a gene tree from a sample of gene trees. In this short paper we propose a novel method to infer a missing part of a phylogenetic tree using an analogue of a classical linear regression in the setting of tropical geometry. In our approach, we consider a tropical polytope, a convex hull with respect to the tropical metric closest to the data points. We show a condition that we can guarantee that an estimated tree from our method has at most two Robinson-Foulds (RF) distance from the ground truth and computational experiments with simulated data show our method works well.

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Tropical Principal Component Analysis and its Application to Phylogenetics

Cited by 5 publications

References 3 publications

Tree Topologies along a Tropical Line Segment

Tree Topologies along a Tropical Line Segment

L-Infinity Optimization to Bergman Fans of Matroids with an Application to Phylogenetics

Tropical Data Science over the Space of Phylogenetic Trees

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