Populations of Complex Objects, Visualization and Smoothing

理解复杂物体（包括曲线、图像和曲面）群体结构的需求正变得越来越清晰，特别是随着自动数据收集能力在许多领域迅速发展。研究功能数据分析方法是为了满足这一需求。这一推动力的一个重要部分将朝着稳健方法的方向发展，由于非常高的维数加上小样本量，因此很少有标准方法适用。当对象是图像和表面时，功能数据分析需要新的可视化方法。在提出的平滑工作中，提出了不同方向的创造性可视化想法。平滑的实际数据应用的中心问题是平滑中的哪些特征确实存在？SiZer就是这样一种方法，并提出了各种改进方法。新的视觉范例将为具有挑战性和重要的依赖数据和二维平滑的情况下开发。本文还提出了研究分级的数理统计基础和相关方法的工作。提出的工作是由眼科，计算机科学，软件工程和地质学问题的动机。眼科学和计算机科学的工作是寻找统计和视觉方法，以深入了解复杂物体的群体结构。可能的物体类型非常多样，目前正在研究代表角膜曲率的图像和代表胶体体的形状。所开发的方法预计将在许多其他领域有用，但一个直接的好处将是改善角锥和精神分裂症的诊断。软件工程和地质学的工作是可视化和平滑方法的结合，以发现和验证数据中的意外结构。例如，一种新开发的方法，既建立了软件工程中完美维护和降低故障率之间的联系，又提供了冰川作用和火山作用之间的第一个统计上有意义的联系。另一种新开发的方法显示了遗留软件系统是如何老化的，并且还提供了软体动物由于气候变化而灭绝的统计证据。

DMS-9971649The need for understanding the structure of populations of complex objects, including curves, images and surfaces, is becoming increasingly clear, especially as automatic data gathering capabilities are burgeoning in many fields. Research into functional data analytic methods is proposed to address this need. An important part of this thrust will be in the direction of robust methods, where few of the standard methods are applicable because of the very high dimensionality coupled with small sample sizes. New visualization methods are needed for functional data analysis, when the objects are images and surfaces. Creative visualization ideas are proposed in a different direction in the proposed work on smoothing. The central issue for real data applications of smoothing is which features in the smooth are really there? SiZer was one such methodology, and a variety of improvements are proposed. New visual paradigms will be developed for the challenging and important cases of dependent data and two dimensional smoothing. Work is also proposed on studying the mathematical statistical underpinnings of SiZer, and related methods.The proposed work is motivated by problems in opthalmology, computer science, software engineering, and geology. The work in ophthalmology and computer science is on finding statistical and visual methods for gaining insight into the structure of populations of complex objects. Possible types of objects are very diverse, with work under way on images representing corneal curvature and on shape representations of corpus collosa. The methods developed are expected to be useful in many other fields, but an immediate benefit will be improved diagnosis of kerataconus and schizophrenia. The work in software engineering and geology is a combination of visualization and smoothing methods to find, and verify, unexpected structure in data. For example one newly developed method, both establishes a link between perfective maintenance and decreased fault rate in software engineering, and also provides the first statistically significant connection between glaciation and volcanism. Another newly developed method shows how legacy software systems age, and also gives statistical proof of periods of extinction of mollusks, due to climatic changes.