New Statistical Modeling Procedures for Object Oriented Data Analysis (OODA)

这项研究的目标是开发新的统计模型、理论、算法和应用程序，用于分析复杂的面向对象的数据，包括树形结构的对象和随机图。这项研究不仅引入了许多创新技术，还为统计基础提供了各种新的和深入的见解，例如复杂数据对象的建模过程。这项研究极大地增强了可用于面向对象数据分析的工具包。特别是，提出了三个相互关联的主题进行调查。首先，研究人员为理解树形结构的对象开发了一种仔细的公理结构，从而避免了定义线性运算的需要。此外，研究人员还研究了如何在树空间中进行基于度量诱导概率度量的统计推理。在树和图的空间中还提出了非参数和半参数的建模方法。其次，研究了利用Hellinger距离进行模型选择的方法。讨论了估计的Hellinger偏差的渐近性质，并检验了逼近的充分性。通过对微阵列基因表达数据的应用，检验了所提出的模型选择过程的性能。第三，研究人员开发了分析在流形上收集的数据的新技术。各种各样的数据，例如在生态学研究中沿着河流收集的数据，以及在地表收集的数据，在许多科学领域变得流行起来。需要新的统计方法来从各种数据中提取有用的特征。在这里，正在研究一种测地线低阶薄板样条方法，该研究项目为面向对象的数据分析提供了一个基础良好和全面的框架。它通过发展包括生物信息学、计算机科学、神经科学、数学和统计学在内的跨学科研究，极大地加强了面向对象的数据分析的研究。树状结构物体的研究可以通过发展图像分析的新技术和提高医学诊断水平来显著造福社会。研究人员通过与来自不同领域的本科生和研究生，特别是代表人数较少的群体密切合作，将研究和教育结合起来。此外，还将通过专题介绍、教程和会议以及通过互联网传播成果。

The objectives of this research are to develop novel statistical models, theory, algorithms and applications geared towards the analysis of complex object oriented data, including tree-structured objects and random graphs. The research not only introduces a number of innovative techniques, but also provides various new and deep insights into statistical foundations, e.g., modeling procedure for complex data objects. This research significantly enhances the toolkit available for the analysis of object oriented data. In particular, three inter-related topics are proposed for investigation. First, the investigator develops a careful axiomatic structure for understanding tree-structured objects, which circumvents the need to define linear operations. Moreover, the investigator studies how to carry out statistical inference, based on the metric induced probability measures, in tree space. Nonparametric and semiparametric modeling procedures are also proposed in the space of trees and graphs. Second, a model selection procedure is studied using Hellinger distance. The asymptotic behavior of the estimated Hellinger discrepancy, and testing the adequacy of the approximation are considered. The performance of the proposed model selection procedure is examined through its application to the microarray gene expression data. Third, the investigator develops new techniques for analyzing data collected on manifolds. Manifold data, such as data collected along a river in an ecological study, and data gathered over a surface, have become popular in many scientific fields. New statistical methodology to extract useful features from manifold data is needed. Here, a geodesic low-rank thin plate splines method is under investigation.The research project lays out a well-grounded and comprehensive framework for analysis of object oriented data. It greatly enhances the research on object oriented data analysis by developing interdisciplinary research including bioinformatics, computer science, neuroscience, mathematics and statistics. The research on tree-structured objects can significantly benefit society by developing new techniques in image analysis and improving medical diagnoses. The investigator integrates research and education by working closely with both undergraduate and graduate students, especially underrepresented groups, from various fields. In addition, the results are to be disseminated through presentations, tutorials and conferences and via internet.