Exploration, Modeling and Inference for Complex Data Objects

该提案旨在开发新的统计学习工具，以应对理解人口水平变化、提取特征和从一组复杂数据对象中获取知识的挑战性问题。面向对象的数据分析（OODA）是函数式数据分析的一个分支，其中数据分析的基本元素是曲线。OODA的基本元素是复杂的数据对象，包括树结构对象。在医学图像分析中，当医学研究的焦点涉及分支结构的变化时，发现树结构对象对于数据表示是有效的。拟议的工作是由人脑动脉系统的数据集驱动的，显然将对涉及树结构对象群体的许多其他科学领域产生影响。复杂的数据对象，如树，一般的图形，网络和形状的分析，对方法的发展提出了严峻的挑战，因为传统的多元数据和功能数据的统计模型依赖于在欧几里德空间或向量空间的线性运算。因此，它需要在一个全新的统计范式中开发新的和非传统的技术，以从数据对象中提取模式和信息。拟议的工作是针对解决一些基本问题，包括一维表示在树空间。这个项目的第一个目标是提供数据探索和总结的工具。接下来，研究者将研究概率分布（混合模型），它可以用作统计推断的基础。研究者将进一步研究树型结构数据的建模，以解释树型结构协变量与数值响应之间和/或数值协变量与树型结构响应之间的关系。基于核的方法和逻辑回归将在树空间中实现分类。过去二十年来，科学和技术中高度复杂的数据收集过程激发了对复杂数据对象的研究。拟议的工作将开辟一个新的统计研究领域，奠定了基础，并丰富了面向对象的数据分析工具包。研究者将继续对人脑动脉数据实施新开发的建模程序，并帮助改进现有的脑肿瘤诊断程序。这也将对面向对象的数据分析产生重大影响，在各个科学领域之间开展跨学科研究。预计这一提议产生的想法和方法将超越分析人脑动脉数据的激励性例子，并将为研究人员提供更深入的数据收集学科的见解。

This proposal aims to develop new statistical learning tools geared towards the challenging problem of understanding population level variation, extracting features and gaining knowledge from a set of complex data objects. Object Oriented Data Analysis (OODA) is an outgrowth of Functional Data Analysis, in which the basic elements of data analysis are curves. The basic elements of OODA are complex data objects including tree-structured objects. In medical image analysis, tree-structured objects are found to be efficient for data representation when the focus of the medical study involves variation in branching structures. The proposed work is driven by a data set of human brain artery systems and will clearly have an impact on many other scientific fields involving populations of tree-structured objects. Analysis of complex data objects, such as trees, general graphs, networks and shapes, poses serious challenges towards methodological development since traditional statistical models for multivariate data and functional data rely on linear operations in Euclidean spaces or vector spaces. Thus, it requires the development of novel and nontraditional techniques in a whole new statistical paradigm for extracting patterns and information from data objects. The proposed work is targeted to address some fundamental issues, including one-dimensional representation in tree space. The first goal of this project is to provide tools for data exploration and summarization. Next, the investigator will study probability distributions (mixture models), which can be used as the basis of statistical inference. The investigator will further study modeling of tree-structured data to explain the relationship between tree-structured covariates and numerical response, and/or between numerical covariates and tree-structured response. Kernel based methods and logistic regression will be implemented for classification in tree spaces.Highly sophisticated data collection processes in science and technology from the last two decades motivate the study of complex data objects. The proposed work will open up a new area of statistical research, lay down a foundation and enrich the toolkit available for the analysis of object oriented data. The investigator will continue to implement newly developed modeling procedures to the human brain artery data, and help to improve existing brain tumor diagnosis procedure. This will also have a major impact on object oriented data analysis by developing interdisciplinary research among various scientific fields. It is expected that the ideas and methods resulting from this proposal will go beyond the motivating example of analyzing human brain artery data, and will provide researchers deeper insights in the discipline where the data were collected.