CAREER: Statistical and Geometric Analysis for Tree-Shaped Data

该项目将建立一个几何数学框架，用于分析疾病、癌症和医学成像中的树形数据。病毒的进化历史，肿瘤突变随时间的关系，以及肺气道和动脉的图像都是树形的。在分析过程中必须考虑树木的几何形状，以避免误差和偏差，但传统的统计方法的作用有限。此外，基因测序等技术进步正在产生越来越大、越来越复杂的此类树木数据集。该项目将开发新的数学工具，以理解并从树形数据中获得见解。教育部分将为本科生提供培训项目和课外机会，让他们参与研究并学习有价值的统计和技术技能。预计许多学生将来自低收入和代表性不足的群体，从而扩大这些群体在STEM行业和研究中的参与度。这项研究将开发分析树木数据的方法，这些方法使用树木的形状和边缘的长度，以一种数学集成的方式。现有的大多数树分析方法只关注树的形状，而所提出的方法将基于数据所在的底层非欧几里德几何空间。这项研究将有助于几何统计的不断发展的领域与数学公式和计算方法的偏差在树形数据，并在生物现实数据验证的双样本测试。在扩展这些统计方法以应对大数据挑战的过程中，本研究将继续将计算几何领域扩展到欧几里得、非正弯曲空间。最后，这项研究将导致表征系统发育网络的第一个连续几何空间。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will build a geometric, mathematical framework for analyzing tree-shaped data, which occurs in disease, cancer, and medical imaging. The evolutionary histories of viruses, relationships of tumor mutations over time, and images of lung airways and arteries are all tree-shaped. The geometry of trees must be considered during analysis to avoid errors and bias, but traditional statistical methods are of limited use. Furthermore, technological improvements such as genetic sequencing are producing increasingly large and complex datasets of such trees. This project will develop new mathematical tools to understand and derive insights from tree-shaped data. An educational component will create training programs and extra-curricular opportunities for undergraduates to participate in research and learn valuable statistical and technical skills. Many of the students are expected to be from low-income and underrepresented groups, thus broadening the participation of these groups in STEM industry and research.This research will develop methods for analyzing tree data that use both the tree shape and the length of its edges in a mathematically integrated way. The majority of existing tree analysis methods only focus on the tree shape, but the proposed methods will be based on the underlying non-Euclidean geometric space in which the data lie. This research will contribute to the growing field of geometric statistics with a mathematical formulation and method of computation for bias in tree-shaped data, and two-sample tests validated on biologically realistic data. In scaling these statistical methods to meet the Big Data challenge, this research will continue to expand the field of computational geometry to piece-wise Euclidean, non-positively curved spaces. Finally, this research will lead to characterizing the first continuous geometric spaces for phylogenetic networks.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.