Learning and Analysing Discrete Geometric Structure in Statistical Models

近几十年来，遗传数据的数量和维度都在迅速增加。系统发育树是一种流行的工具，用于总结从基因数据推断的潜在突变，但我们缺乏一个严格的统计框架来研究大型树数据。目前，这类工具包括Robinson-Foulds度量和BHV树空间。Robinson-Foulds度量因其计算效率而特别受欢迎，但缺乏敏感性；同时BHV空间能够充分捕捉树空间的丰富几何特征，但计算距离的计算非常密集。2004年，Speyer和Sturmfels建立了树空间与热带Grassman空间的等价关系。该公式通过热带Plucker关系将树空间嵌入热带射影环面。热带射影环面是一种Banach空间，它的维度随着树叶数的增加而二次增加，为树木提供了一个计算上容易处理的环境空间。从代数和几何的角度，也很好地研究了热带射影环面内树空间的几何。最近，人们研究了这种热带树木空间的统计潜力，使用流感数据的初步调查表明，它比BHV空间提供了更有效的树木数据统计汇总。本项目将专注于建立热带投影环面和热带树木空间的概率基础，以发展丰富的树木数据集的统计理论。我们的目标是将不同空间上测量的概率距离正式化，使我们能够比较具有不同分类群的树木数据集。我们还将研究Fréchet均值在热带投影环面上的行为，包括它们与热带树空间的交集以及它们对经验度量的极限行为。我们希望将这项工作进行到实际应用中，建立一种假设检验的方法，用于不同树木数据集的比较。我们将能够利用弗朗西斯·克里克研究所的造血干细胞实验室的数据，使用这些新方法来研究急性髓系白血病的克隆进化。对这些数据的研究将突出这项研究的影响，通过建立对癌症所表现出的进化模式的严格和详细的统计研究。该项目属于EPSRC数学生物学和统计学以及应用概率研究领域。该项目由伦敦帝国理工学院的Anthea Monod博士和慕尼黑工业大学的Mathias Drton教授监督。它是ICL-TUM联合博士生学院的一部分，该学院促进了我们两个研究小组之间的合作。我们还得到了弗朗西斯·克里克研究所的多米尼克·博内特的支持，该研究所的造血干细胞实验室承诺提供关于急性髓细胞白血病克隆进化的专有数据。

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.