Tropical geometry for phylogenetic statistics
File(s)
Author(s)
Talbut, Roan
Type
Thesis
Abstract
The identification of the tropical Grassmannian and the space of phylogenetic trees has inspired a range of research on the use of tropical geometry for phylogenetic statistics. This thesis presents novel probabilistic and computational results on phylogenetic statistical methods which utilise this connection.
In Part I of this thesis, we study Fréchet means - natural generalisations of the classical mean - which enable us to calculate an average phylogenetic tree from a set of phylogenetic trees. We present original results on the uniqueness of Fréchet means, their role as maximum likelihood estimates, and their limiting distribution for large samples. We also consider the problem of comparing sets of phylogenetic trees with different leaf sets, and define a Wasserstein distance between the two datasets. We prove that this distance is symmetric, whether we map the first dataset to the state space of the second, or vice versa.
In Part II of this thesis, we study the optimisation problems involved in computing tropical statistics. We introduce the novel method of tropical gradient descent, which tailors gradient methods to optimisation problems in tropical settings. Tropical gradient descent no longer requires classical convexity to find a global solution; instead, tropical convexity is sufficient. Using simulated datasets, we show these convergence results computationally and demonstrate how tropical gradient descent can be seamlessly integrated into more sophisticated gradient methods. We also show how tropical gradient descent can be combined with discrete optimisation methods to compute Wasserstein distances. This thesis concludes with the application of our work to real evolutionary datasets of phonetic spellings and influenza virus mutations, as well as simulated auction data.
In Part I of this thesis, we study Fréchet means - natural generalisations of the classical mean - which enable us to calculate an average phylogenetic tree from a set of phylogenetic trees. We present original results on the uniqueness of Fréchet means, their role as maximum likelihood estimates, and their limiting distribution for large samples. We also consider the problem of comparing sets of phylogenetic trees with different leaf sets, and define a Wasserstein distance between the two datasets. We prove that this distance is symmetric, whether we map the first dataset to the state space of the second, or vice versa.
In Part II of this thesis, we study the optimisation problems involved in computing tropical statistics. We introduce the novel method of tropical gradient descent, which tailors gradient methods to optimisation problems in tropical settings. Tropical gradient descent no longer requires classical convexity to find a global solution; instead, tropical convexity is sufficient. Using simulated datasets, we show these convergence results computationally and demonstrate how tropical gradient descent can be seamlessly integrated into more sophisticated gradient methods. We also show how tropical gradient descent can be combined with discrete optimisation methods to compute Wasserstein distances. This thesis concludes with the application of our work to real evolutionary datasets of phonetic spellings and influenza virus mutations, as well as simulated auction data.
Version
Open Access
Date Issued
2025-09-30
Date Awarded
2026-03-01
Copyright Statement
Attribution-ShareAlike 4.0 International (CC BY-SA)
Advisor
Monod, Anthea
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)