CAREER: Scalable methods for discovering multivariate dependencies in high dimensional data.

该建议旨在开发原则性的方法，发现多元依赖关系，迎合超高的维度设置。一个共同的主题，团结所提出的方法是可扩展性和识别其局限性。一个流行的方法来识别稀疏逆协方差矩阵是通过惩罚似然方法。我们提出了一种新的方法来解决惩罚高斯对数似然比它的竞争对手快了许多数量级。该提案的第二个研究部分调查有限样本中阈值矩阵的统计特性，以期获得一种可高度扩展的正定协方差估计方法。该项目的第三个研究方面调查量化估计的图形网络模型的可变性和不确定性。一种方法，利用凸伪似然制定的图形模型选择问题的介绍。这允许开发具有理论保障的高度可扩展的不确定性量化方法。该项目的第四个研究方面审查了在气候变化领域的应用程序中使用前三个子组件中提出的方法，其中需要高维协方差估计。该提案也有一个重要的教学和推广的组成部分，其目的是介绍统计数据，以有抱负的年轻科学家在他们的本科和研究生学习的各个阶段，从各种应用，包括基因组学，环境科学和其他高通量数据的可用性，创造了一个迫切需要的方法和工具，用于分析高维数据。提取和理解数据中的许多复杂关系和多变量依赖关系，并开发原则性的推理程序，是统计学家和数据科学家面临的主要挑战之一。本项目中提出的理论和方法学工作是由地球和环境科学，基因组学和癌症研究以及社会科学等不同领域的应用和跨学科合作所推动的。例如，在基因组学中，人们通常有兴趣知道各种基因是如何关联的，以及这些关联在实验组（患病组）和对照组之间有何不同。基因调控网络也是研究疾病演变的重要工具。在气候变化辩论的背景下，对地球仪上不同点的温度进行建模需要对这些变量之间的关系进行简化建模。在材料科学和工程中，建模相关性也很自然地出现，人们有兴趣了解当新材料产生时，不同的原子粒子如何相互作用。 因此，所提出的用于估计非常高维设置中的相关性的项目将具有广泛的应用，因为理解许多变量之间的关联/关系是许多科学学科共同的奋进。拟议的工作，虽然牢牢扎根于统计科学，是非常跨学科的，并涉及统计人员/数据科学家和生物医学科学家，工程师和地球科学家之间的合作和伙伴关系。

This proposal aims to develop principled methods for discovering multivariate dependencies which cater to ultra high dimensional settings. A common theme that unites the proposed methods is scalability and identification of their limitations. A popular approach to identifying sparse inverse covariance matrices is through penalized likelihood methods. We propose a novel approach for solving the penalized Gaussian log-likelihood that is faster than its competitors by many orders of magnitude. The second research component in the proposal investigates the statistical properties of thresholded matrices in finite samples, with a view to obtaining a positive definite covariance estimation method which is highly scalable. The third research aspect of the project investigates quantifying the variability and uncertainty of estimated graphical network models. A methodology that takes advantage of a convex pseudo-likelihood formulation of the graphical model selection problem is introduced. This allows for the development of a highly scalable uncertainty quantification method with theoretical safeguards. The fourth research aspect of the project examines the use of the methodology proposed in the previous three sub-components to an application in the area of climate change, where high dimensional covariance estimation is required. The proposal also has a significant teaching and outreach component which aims to introduce statistics to aspiring young scientists at various stages of their undergraduate and graduate studies.The availability of high-throughput data from various applications, including genomics, environmental sciences and others, has created an urgent need for methodology and tools for analyzing high dimensional data. Extracting and making sense of the many complex relationships and multivariate dependencies in the data and developing principled inferential procedures is one of the major challenges facing statisticians and data scientists. The theoretical and methodological work proposed in this project is motivated by applications and interdisciplinary collaborations in fields as diverse as the earth and environmental sciences, genomics and cancer research, and the social sciences. In genomics for instance, one is often interested to know how various genes are associated, and how these associations differ between an experimental (diseased) and control group. Gene regulatory networks also serve as important tools to study the evolutions of diseases. In the context of the climate change debate, modeling temperature at different points on the globe requires parsimonious modeling of the way in which these variables are related. Modeling correlations also arises naturally in material sciences and engineering where one is interested in seeing how different atomic particles interact when new materials are produced. Hence the proposed project for estimating correlations in very high dimensional settings will have widespread applications, since understanding associations/relationships between many variables is an endeavor that is common to many scientific disciplines. The proposed work, though firmly rooted in the statistical sciences, is very much interdisciplinary, and involves collaborations and partnerships between statisticians/data scientists and biomedical scientists, engineers and earth scientists.