Geometry, Shape and Objects

细丝、脊线、图形、轮廓和路径是现代数据分析挑战中几何对象的例子。开发和分析统计方法来解决与这些对象相关的挑战是这个项目的总体目标。特别是，首席研究员和他的合作者将(i)开发统计方法和算法，用于估计脊线和/或丝状结构，涉及路径平滑方法。在这里，路径是梯度的积分曲线或适当的Hessians特征向量的估计。他们还将为随后的高维数据聚类制订降维方法，其中将考虑向非线性流形的投影。进一步的目标是（iii）适应高维图形模型的模型选择方法，以纳入空间和时间的平滑性，以及（iv）调查统计密度/回归水平集方法和深度轮廓方法之间的共性和差异。除了所考虑的问题的固有几何性质外，几何还通过选择适当的距离度量来发挥作用，例如。该项目将产生以相关理论为基础的新的统计方法，并在几何统计领域产生相应的数值算法。在这个统计学领域，感兴趣的对象本身是真正的几何动机。一个相关科学问题的一个重要的指导性实例是对宇宙网的分析，它由空间中星系的位置组成。这些位置的网状结构需要使用非标准的方法。几何驱动的物体在医学成像、指纹识别和遥感等其他几个科学领域也发挥着重要作用。该项目的研究结果将推动统计领域的发展，并将直接影响相关的应用领域。培养现代统计学领域的研究生和本科生是该项目的另一个重要目标。

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.