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

本研究的目标是开发新的统计模型，理论，算法和应用面向复杂的面向对象的数据，包括树结构的对象和随机图的分析。该研究不仅引入了许多创新技术，还提供了各种新的和深入的见解统计基础，例如，复杂数据对象的建模过程。这项研究显着提高了工具包可用于面向对象的数据分析。特别是，提出了三个相互关联的主题进行调查。首先，研究人员开发了一个仔细的公理结构来理解树结构的对象，这就避免了定义线性运算的需要。此外，研究如何进行统计推断，基于度量诱导的概率测度，在树空间。非参数和半参数建模程序也提出了在树和图的空间。其次，模型选择过程中使用海林格距离进行了研究。估计的Hellinger差异的渐近行为，和测试的适当性的近似被认为是。建议的模型选择程序的性能进行检查，通过其应用程序的微阵列基因表达数据。第三，研究人员开发新的技术来分析收集到的数据集管。多样的数据，例如在生态研究中沿着沿着收集的数据，以及在表面上收集的数据，在许多科学领域中已经变得流行。需要新的统计方法来从多方面的数据中提取有用的特征。本文研究了一种测地线低秩薄板样条方法，为面向对象的数据分析提供了一个坚实而全面的框架。它通过发展生物信息学、计算机科学、神经科学、数学和统计学等跨学科的研究，极大地促进了面向对象数据分析的研究。对树结构对象的研究可以通过开发图像分析的新技术和改进医学诊断来显着造福社会。调查员通过与来自各个领域的本科生和研究生，特别是代表性不足的群体密切合作，将研究和教育结合起来。此外，还将通过介绍、辅导和会议以及互联网传播成果。

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.