Geometric and topological methods in statistics

统计学面临着海量多维数据的挑战。挑战在于表示、可视化、解释和分析，需要开发新的统计方法，而这些进展有赖于越来越复杂的技术。这项拟议的研究计划的主要目标是研究两个这样的新兴领域：几何统计学；以及统计拓扑学。 几何统计学涉及黎曼流形上的参数估计，其中产生数据的基本几何参数的恢复具有定量的统计意义。统计拓扑学涉及恢复定性的全局特征的拓扑参数，如连通性、孔洞数量或某些结构是否存在障碍物。用定量的方法可以获得定性的拓扑信息，其中它们之间的路径可以利用Morse理论的元素来建立。该框架为通过几何参数考察拓扑参数的统计特性提供了足够丰富的统计结构。 我们看到海量多维对象数据爆炸性增长的领域之一是医学成像。由于医疗扫描仪的技术进步，体素数据的分辨率现在非常详细。使用几何统计的技术出现了巨大的扩散，显然需要更高的技术复杂性。这项研究计划打算在此基础上，通过定量方法和定性方法提供贡献。后者是真正的前沿，现在才开始受到重视，特别是因为它在临床上区分亚群体的能力。其他将受到关注的应用领域包括生物信息学、生物力学、微波工程、正畸数据和量子信息处理。

Statistics is challenged by massive multi-dimensional data. The challenges are in representation, visualization, interpretation and analysis, requiring the development of new statistical methodologies, and these advances are dependent on ever more increasing technical sophistication. The main objective of this proposed research program is to examine two such emerging areas: geometric statistics; and, statistical topology. Geometric statistics is concerned with parameter estimation over Riemannian manifolds, where the recovery of the underlying geometric parameter that generates the data is of quantitative statistical interest. Statistical topology, is concerned with recovering topological parameters which are qualitative global features, such as connectedness, or the number of holes, or the existence of obstructions to certain constructions. With the quantitative approach one can obtain qualitative topological information where the pathway between them can be established by using elements of Morse theory. This framework thus provides an abundant enough statistical structure for examining the statistical properties of topological parameters through geometric parameters. One of the areas where we have seen explosive growth of massive multi-dimensional object data is in medical imaging. Due to technological advances in medical scanners, the resolution of voxel data is now extremely detailed. Techniques using geometric statistics have witnessed tremendous proliferation with the need for greater technical sophistication being evident. This research program intends to build upon this and provide contributions both through quantitative approach, as well the qualitative approach. The latter is really frontier, and is only now starting to be appreciated, particularly so for it's ability to clinically discriminate between sub-populations. Other application areas where attention will be given are in bioinformatics, biomechanics, microwave engineering, orthodontal data, and quantum information processing.