Learning and Analysing Discrete Geometric Structure in Statistical Models

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The quantity and dimensionality of genetic data have been rapidly increasing in recent decades. Phylogenetic trees are a popular tool for summarising the underlying mutations inferred from genetic data, but we lack a rigorous statistical framework with which to study large tree data. Currently, such tools include the Robinson-Foulds metric and BHV tree space. The Robinson-Foulds metric is particularly popular for its computational efficiency but lacks sensitivity; meanwhile BHV space can fully capture the rich geometry of tree space, but calculating distances is computationally intensive.In 2004 Speyer and Sturmfels established an equivalence between tree space and the tropical Grassmannian. This formulation embeds tree space in the tropical projective torus via the tropical Plucker relations. The tropical projective torus is a Banach space whose dimensionality increases quadratically with the number of leaves, providing a computationally tractable ambient space for trees. The geometry of tree space within the tropical projective torus is also well-studied from an algebraic and geometric perspective. More recently, this tropical tree space has been studied for its statistical potential, and initial investigations using Influenza data have shown it to offer more efficient statistical summaries of tree data than BHV space.This project will focus on establishing the probabilistic groundwork on the tropical projective torus and tropical tree space to develop a rich statistical theory for tree datasets. We aim to formalise probabilistic distances for measures on different spaces, allowing us to compare tree datasets with different taxa. We will also study the behaviour of Fréchet means on the tropical projective torus, both in terms of their intersection with tropical tree space and their limiting behaviour for empirical measures.We hope to carry this work through to practical application, establishing a methodology for hypothesis testing for the comparison of different tree datasets. We will be able to use these novel methods to study the clonal evolution of acute myeloid leukaemia using data from the Haematopoietic Stem Cell Laboratory at the Francis Crick Institute. The study of this data will highlight the impact of this research by establishing a rigorous and detailed statistical study of the evolutionary patterns exhibited by cancers.This project falls within the EPSRC Mathematical Biology and Statistics and Applied Probability research areas. The project is supervised by Dr Anthea Monod (Imperial College London) and Prof. Mathias Drton (Technical University of Munich). It is part of the ICL-TUM Joint Academy of Doctoral Studies, which fosters collaboration between our two research groups. We also have the support of Dominique Bonnet at the Francis Crick institute, whose Haematopoietic Stem Cell Laboratory have pledged proprietary data on the clonal evolution of AML.