Collaborative Research: Tree Structured Object Oriented Data Analysis

这项建议是为了研究如何发展对树状结构对象集合的总体水平的理解和洞察。也就是说，目标是分析作为树的数据点群体中的变化，包括分支结构的变化。虽然这一目标本质上是统计的，但它远远超出了现有统计方法的范围。因此，这里提出的工作开辟了统计研究的一个全新领域。这项工作是由神经外科医生合作者收集的人类大脑动脉树的特定示例数据集推动的，该合作者已经并将继续告知所选择的研究方向和采取的步骤。虽然这个鼓舞人心的例子是人脑的血管生成，但还有许多其他背景将受到这里开发的新方法的影响，下面讨论。最接近拟议研究的统计领域是目前活跃的功能数据分析领域，其中统计分析的原子是曲线(而不是更典型的数字或矢量)。这个抽象的概念被Wang和Marron(2007)扩展到面向对象的数据分析(OODA)，其中原子变成了更复杂的各种类型的对象，包括树形结构的对象。面向对象的官方发展援助提出了一些新的重大数据分析挑战。应对这些挑战将需要开发全新类型的统计方法。即使是简单的统计概念，如人口平均值，也不是一目了然的。更深层次的性质，特别是关于平均值的变化的量化，要具有更大的挑战性。Wang和Marron(2007)的结果是在最优化问题方面形成统计概念的第一步。一个主要的限制是不清楚如何为现实的数据集计算这些有用的解决方案。Aydin等人(2008)在这个方向上取得了重大突破，发明了一些显然难以解决的优化问题的线性时间解决方案，这使得动脉树数据集的实际Ooda可以到达现代计算设施的范围。拟议的工作是对更深层次的分析，这需要发明强大的新方法来理解变异。具体地说，当前仅限于拓扑的分析将扩展到全节点属性数据类型，这将使得能够同时研究其他类型的变化(例如，在树枝粗细和位置上)，将探索针对树结构对象群体的全新区分领域，并将在复杂树数据对象的可视化方面进行创新。这些更深入的分析有望产生深刻的新解剖学结果，涉及对称性和对协变量(如年龄)的依赖，这是目前用于分析树数据的简单总结所无法获得的。这项研究的驱动数据集是100多名患者的集合？磁共振血管成像(MRA)脑动脉树。这个拟议的项目将开发新的统计方法，以便从这组树中提取有用的信息。将针对人群层面对人脑解剖学的重大新见解。这项研究的总体目标是开发表征正常脑动脉结构的方法。这本身就是一个重要的科学解剖学目标，具有重大医学应用的潜力。例如，这项工作为基于血管的脑肿瘤(具有异常动脉树)的诊断提供了可能性。动脉树分析应该会通过比现有技术更早的诊断来提高当前癌症治疗的成功率。未来的医学应用有望远远超出人脑动脉的驱动问题，扩展到许多其他类型的被广泛研究的解剖结构，如肺部的呼吸道、神经系统和各种类型的收集管道系统。此外，这项工作预计将在许多其他领域推动新的想法，这些领域自然会遇到树作为数据对象，例如文本挖掘(其中一种标准技术是将语法结构表示为树)、遗传学中的系统发育树以及社会和计算机网络的分析。最后，这项工作预计将对数学产生影响，刺激那里的新思想的发展，例如在优化和图论方面。

This proposal is for research on finding ways of developing population level understanding of, and insights about, a collection of tree structured objects. I.e. the goal is the analysis of the variation, including variation in branching structure, in a population of data points that are trees. While this goal is statistical in nature, it is very far beyond the reach of existing statistical methods. Thus an entire new area of statistical research is opened up by the work proposed here. This work is driven by a particular example data set of human brain artery trees, collected by a neuro-surgeon collaborator, who has, and will continue to, inform the research directions chosen, and the steps taken. While this motivating example is vasculature of the human brain, there are many other contexts which will be impacted by the new methods developed here, discussed below. The closest statistical area to the proposed research is the currently active area of Functional Data Analysis, in which the atoms of the statistical analysis are curves (instead of the more typical numbers or vectors). This abstract concept was extended to Object Oriented Data Analysis (OODA), by Wang and Marron (2007), where the atoms become more complicated objects of various types, including tree structured objects. OODA presents a number of major new data analytic challenges. Addressing these challenges will require the development of totally new types of statistical methods. Even simple statistical concepts, such as the population mean, are not straightforward to develop. Deeper properties, in particular the quantitation of variation about the mean, are far more challenging. The results of Wang and Marron (2007) were a first pass at formulating statistical concepts in terms of optimization problems. A major limitation was that it was unclear how to compute useful solutions of these for realistic data sets. Aydin et al (2008) achieved a major breakthrough in this direction by inventing linear time solutions to some of these apparently intractable optimization problems, which brings practical OODA of the artery tree data set within reach of modern computational facilities. The proposed work is on much deeper analyses, which requires the invention of powerful new approaches to understanding variation. In particular, the current topology-only analyses will be extended to full nodal attribute data types that will enable simultaneous study of other types of variation as well (e.g. in branch thickness and location), the entirely new area of discrimination for populations of tree structured objects will be explored, and innovations in the visualization of complex tree data objects will be made. These deeper analyses are expected to yield deep new anatomical results, involving symmetry and dependence on covariates such as age, that are unavailable from the simple summaries currently being used to analyze tree data.The driving data set for this research is a collection of over 100 patients? magnetic resonance angiographic (MRA) brain artery trees. This proposed project will develop new statistical methods for extracting useful information from this collection of trees. Major new population-level insights on human brain anatomy will be targeted. The big picture goal of this research is the development of methods for characterizing normal brain artery structure. As well as being an important scientific anatomical goal in itself, this has potential for major medical applications. For example, this work provides the potential for a vascular-based diagnosis of brain tumors (which have aberrant arterial trees). Arterial tree analysis should improve the success of current cancer treatments through earlier diagnosis than is available using current techniques. Future medical applications are expected to extend well beyond the driving problem of human brain arteries to many other types of widely-studied anatomical structures, such as airways in the lung, the nervous system and various types of collection duct systems. In addition, this body of work is expected to drive new ideas in many other areas which naturally encounter trees as data objects, such as text mining (where a standard technique is representation of grammatical structures as trees), phylogenetic trees in genetics, and the analysis of social and computer networks. Finally this work is expected to have an impact on mathematics by stimulating the development of new ideas there, e.g. in optimization and graph theory.