Populations of Complex Objects, Visualization and Smoothing

DMS-9971649理解复杂对象（包括曲线、图像和曲面）的总体结构的需求越来越明显，尤其是随着自动数据收集能力在许多领域的迅速发展。 研究功能数据分析方法，提出了解决这一问题的需要。这一努力的一个重要部分将是稳健方法的方向，其中很少有标准方法是适用的，因为非常高的维度加上小样本量。 当对象是图像和曲面时，函数数据分析需要新的可视化方法。 创造性的可视化的想法提出了一个不同的方向，在拟议的工作平滑。 平滑的真实的数据应用的中心问题是平滑中的哪些特征是真实存在的？SiZer就是这样一种方法，并且提出了各种改进。 新的视觉范例将开发的相关数据和二维平滑的挑战性和重要的情况下。 还提出了研究SiZer的数理统计基础和相关方法的工作，所提出的工作的动机是眼科学，计算机科学，软件工程和地质学中的问题。 眼科和计算机科学的工作是寻找统计和视觉方法来深入了解复杂物体群体的结构。可能的物体类型是非常多样的，目前正在研究表示角膜曲率的图像和表示collosa的形状。 开发的方法预计将在许多其他领域是有用的，但一个直接的好处将是提高诊断角结膜炎和精神分裂症。 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，既在软件工程中建立了完善的维护和降低故障率之间的联系，也在冰川作用和火山作用之间提供了第一个统计上显着的联系。 另一种新开发的方法显示了遗留软件系统的年龄，并提供了由于气候变化而导致的软体动物灭绝时期的统计证据。

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.