Geometry, Shape and Objects

几何、形状和物体

• 批准号: 1107206
• 负责人: Wolfgang Polonik
• 金额: $ 17万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1107206&HistoricalAwards=false
• 关键词: Geometry; Shape; Objects

Filaments, ridges, graphs, contours, and paths are examples of geometric objects in modern data analytic challenges. The development and the analysis of statistical methodology to tackle challenges related to such objects is the overarching goal of this project. In particular, the principal investigator and his collaborators will (i) develop statistical methodology and algorithms for the estimation of ridge lines and/or filamentary structure involving path smoothing methodologies. Here, paths are estimates of integral curves of gradients or of appropriate eigenvectors of Hessians. (ii) They will also develop dimension reduction methodology for subsequent clustering of high-dimensional data, where projection onto non-linear manifolds will be considered. Further goals are (iii) the adaptation of model selection methodologies for high-dimensional graphical models to incorporate spatial and temporal smoothness, and (iv) the investigation of commonalities and differences between statistical density/regression level set methodology and depth contour approaches. Besides the intrinsic geometric nature of the problems under consideration, geometry also comes into play via the choice of an appropriate distance measure, for instance.This project will lead to new statistical methodologies underpinned by relevant theory, and to corresponding numerical algorithms in the area of geometric statistics. In this field of statistics, the objects of interest itself are genuinely geometrically motivated. One important guiding instance of a relevant scientific problem is the analysis of the cosmic web, which consists of locations of galaxies in the space. The web-like structure of these locations requires the use of nonstandard methodologies. Geometrically motivated objects also play a major role in several other scientific areas, such as medical imaging, fingerprint identification and remote sensing. The findings of this project will advance the field of statistics, and they will directly impact the relevant fields of application. The education of graduate and undergraduate students in a modern field of statistics is another important goal of this project.

细丝、脊线、图形、轮廓和路径是现代数据分析挑战中的几何对象的例子。制定和分析统计方法以应对与这些目标有关的挑战是本项目的首要目标。特别是，首席研究人员和他的合作者将(I)开发统计方法和算法，用于估计脊线和/或丝状结构，包括路径平滑方法。这里，路径是梯度的积分曲线或适当的Hessians特征向量的估计。(2)他们还将为随后的高维数据聚类制定降维方法，其中将考虑对非线性流形的投影。进一步的目标是(3)调整高维图形模型的模型选择方法，以纳入空间和时间的平稳性，以及(4)调查统计密度/回归水平集方法和深度等值线方法之间的共性和差异。除了所讨论问题的内在几何性质外，几何还通过选择适当的距离度量发挥作用。这个项目将导致以相关理论为基础的新统计方法，并在几何统计领域产生相应的数值算法。在这个统计领域，感兴趣的对象本身就是真正的几何动机。相关科学问题的一个重要指导实例是对宇宙网络的分析，宇宙网络由星系在空间中的位置组成。这些地点的网络式结构需要使用非标准方法。几何激励的物体在其他几个科学领域也发挥着重要作用，如医学成像、指纹识别和遥感。该项目的研究结果将推动统计领域的发展，并将直接影响到相关的应用领域。对研究生和本科生进行现代统计领域的教育是该项目的另一个重要目标。