Iterative testing procedures and high-dimensional scaling limits of extremal random structures

在过去的十年中，网络和网络模型在包括经济学、神经科学、基因组学和生物医学在内的各个领域的应用和重要性日益增加。这些领域的工作推动了有关复杂网络建模和推理的统计研究的增加。pi将在统计网络研究中追求几个新的方向，一个关键主题是在复杂，随机和几何网络理论的概率论方面的最新工作的应用和扩展。特别是，pi将开发迭代测试方法，以识别大型数据集中的关系变化，并增强将遗传变异与基因表达的全球变化联系起来的基因组研究的能力。他们将扩展现有的概率技术，为迭代测试过程提供理论支持，并解决有关大型高维数据集特征之间复杂关联的推断的更广泛的统计问题。方法的开发和应用将与北卡罗来纳大学基因组学、生物医学和社会学的研究人员合作进行，PI和联合PI与他们有长期的合作关系。在很大程度上，由于网络在各个领域的使用和重要性日益增加，统计界已经开展了大量的工作，致力于测试和估计高维数据集中变量之间的关联问题。在此统计活动的同时，概率组合学领域的最新发展显著提高了我们对离散随机结构的理解，这些结构捕获了高维物体的关联。pi将为基于关联的推理问题带来许多这样的概率工具。特别是，pi将开发并实现一个迭代测试过程，该过程可以识别图中自关联的顶点集，以及高维数据集中自关联的变量集。在迭代测试程序的框架内，他们将为几个应用问题开发计算效率的方法：挖掘两个样本研究中的块相关差异，并在研究中识别相互相关的变量组，其中每个样本都使用两个或多个测量平台进行评估。作为后一个问题的一个特例，他们将开发工具来增强将局部遗传变异与基因表达的全球变化联系起来的基因组研究的力量。所提出的研究的第二个组成部分是适应和扩展概率组合学中的现有技术，为迭代测试过程提供支持理论，并解决有关相关性测试和估计的更广泛的统计问题。这些方法的开发和应用将与北卡罗来纳大学基因组学、生物医学和社会学的研究人员合作进行，PI和co-PI与他们有长期合作。

Over the past ten years, networks and network models have seen increasing use and importance in a variety of fields, including economics, neuroscience, genomics, and biomedicine. Work in these fields has driven an increase in statistical research concerning modeling of, and inference about, complex networks. The PIs will pursue several new directions in statistical network research, a key theme being the application and extension of recent work in probability on the theory of complex, random and geometric networks. In particular, the PIs will develop iterative testing methods to identify relational changes in large data sets, and to enhance the power of genomic studies that link genetic variation to global changes in gene expression. They will extend existing probabilistic techniques to provide theoretical support for the iterative testing procedure, and to address broader statistical questions concerning inference about complex associations between the features of large, high dimensional data sets. Methodological development and application will be carried out in cooperation with researchers in genomics, biomedicine, and sociology at UNC, with whom the PI and co-PI have long standing collaborations. Motivated in large part by the increasing use and importance of networks in a variety of fields, there has been a great deal of work in the statistics community devoted to the problem of testing and estimating associations between variables in high dimensional data sets. Concurrent with this statistical activity, recent developments in the fields of probabilistic combinatorics have significantly advanced our understanding of discrete random structures that capture the association of high-dimensional objects. The PIs will bring a number of these probabilistic tools to bear on association based inference problems. In particular, the PIs will develop and implement an iterative testing procedure that identifies self-associated sets of vertices in a graph, and self-associated sets of variables in a high dimensional data set. Within the framework of the iterative testing procedure they will develop computationally efficient methods for several applied problems: mining of block correlation differences in two sample studies, and identifying groups of mutually correlated variables in studies where each sample is assessed with two or more measurement platforms. As a special case of the latter problem, they will develop tools to enhance the power of genomic studies that link local genetic variation to global changes in gene expression. A second component of the proposed research is to adapt and extend existing techniques in probabilistic combinatorics to provide supporting theory for the iterative testing procedure, and to address broader statistical questions concerning the testing and estimation of correlations. Development and application of the methods will be carried out in cooperation with researchers in genomics, biomedicine, and sociology at UNC, with whom the PI and co-PI have long standing collaborations.