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，用于总结从遗传数据推断出的潜在突变，但我们缺乏一个严格的统计框架来研究大型树数据。目前，此类工具包括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，但缺乏灵敏度；同时BHV空间可以充分捕捉树空间丰富的几何特征，但计算距离的计算量较大。2004年，Speyer和Sturmfels建立了树木空间和热带格拉斯曼年之间的等价关系。该公式通过热带采摘关系将树空间嵌入到热带投影环面中。热带射影环面是一个巴拿赫空间，其维数随树叶数量的增加呈二次增长，为树木提供了一个计算上可处理的环境空间。从代数和几何的角度来看，热带射影环面内树木空间的几何形状也得到了很好的研究。最近，对热带树木空间的统计潜力进行了研究，使用流感数据进行的初步调查表明，它比BHV空间提供更有效的树木数据统计摘要。本项目将重点建立热带射影环面和热带树木空间的概率基础，为树木数据集发展丰富的统计理论。我们的目标是形式化不同空间测量的概率距离，使我们能够比较不同分类群的树数据集。我们还将研究热带射影环面上的fr<s:1>均值的行为，包括它们与热带树空间的相交以及它们在经验测度上的极限行为。我们希望将这项工作贯彻到实际应用中，为不同树数据集的比较建立一种假设检验的方法。我们将能够利用弗朗西斯克里克研究所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通过对癌症表现出的进化模式进行严格而详细的统计研究来突出这项研究的影响。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Anthea Monod博士（伦敦帝国理工学院）和Mathias Drton教授（慕尼黑工业大学）监督。它是ICL-TUM博士研究联合学院的一部分，促进了我们两个研究小组之间的合作。我们还得到了Francis Crick研究所Dominique Bonnet的支持，他的造血干细胞实验室承诺提供AML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.