Exploring and detecting complex multivariate dependencies through sparse graphical models

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。当探索参数模型中许多变量之间的复杂关系时，协方差参数是感兴趣的自然参数。目前的高维协方差估计方法主要是利用基于套索的方法对协方差矩阵或其逆进行正则化或置零。虽然非常有用，但这些方法并没有解决文献中一些明显的空白。首先，众所周知，lasso和类似的惩罚方法产生稀疏模型和估计器-然而，在文献中没有对正则化协方差估计器或随机矩阵的谱特性进行正式的承诺，这些特性在图形模型中自然出现。这一差距在文献将被解决。第二，最近受到广泛关注的一类重要模型是所谓的协方差图模型。这些模型在多变量分布中编码边际独立性，因此可以产生更简洁的表示。对于这类模型，还没有一个全面的贝叶斯推理和模型选择框架。这类重要的问题将在本项目中进行研究。需要协方差正则化估计的原始理由之一是协方差矩阵在均值估计问题中具有特征，并且在为均值构建置信区间时（例如在MANOVA中），或者在回归中，但是在回归的特定需求所需的协方差正则化领域中相对很少有工作。本文采用了一个广义框架，研究了使用解释变量的协方差矩阵进行回归的优点，从而为获得比标准方法建议的回归系数更好的估计提供了见解。近年来，来自基因组、金融、环境、营销（以及其他）应用的高通量数据的可用性产生了对分析高维数据的方法和工具的迫切需求。理解数据中所有复杂的关系，制定正确的模型和开发推理程序是当今统计学家面临的主要挑战之一，也是那些在应用领域工作的人。该项目旨在解决在探索高维的多变量依赖关系时出现的一些紧迫问题。作为一个具体的应用，本项目开发的方法将用于了解癌症研究和心血管医学中基因的相互联系，同时保持先前开发方法的统计严谨性和易于解释性。因此，这种性质的项目将具有广泛的应用，因为理解许多变量或参与者之间的关系是许多科学学科共同的努力。这项拟议的工作虽然植根于统计学原理，但却是跨学科的，涉及与生物医学科学家、工程师和环境科学家的合作。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The covariance parameter is the natural parameter of interest when exploring complex relationships between many variables in parametric models. Current methodology on high dimensional covariance estimation has focused on regularizing or putting zeros in the covariance matrix or its inverse using methods based on the lasso. Though very useful, these methods do not address some of the glaring gaps in the literature. First it is well known that lasso and similar penalization methods yield sparse models and estimators - yet a formal undertaking of the spectral properties of regularized covariance estimators or those of random matrices that arise naturally in graphical models is not available in the literature. This gap in the literature will be addressed. Second, an important class of models that have recently received much attention are the so-called covariance graph models. These models encode marginal independences in multivariate distributions and thus can yield more parsimonious representations. A comprehensive framework for Bayesian inference and model selection for this class of models is not available. This important class of problems is investigated in this project. One of the original justifications for the need for covariance regularized estimation is that the covariance matrix features in the mean estimation problem, and when constructing confidence intervals for the mean (for instance in MANOVA), or in regression yet there is relatively very little work in the area of covariance regularization required for the specific needs of regression. A generalized framework which investigates the merits of using the covariance matrix of the explanatory variables for regression purposes is undertaken, thereby providing insights into obtaining better estimators for regression coefficients than those suggested by standard methods. In recent years, the availability of high-throughput data from genomic, finance, environmental, marketing (among other) applications has created an urgent need for methodology and tools for analyzing high-dimensional data. Making sense of all the many complex relationships that are in the data, formulating correct models and developing inferential procedures is one of the major challenges facing statisticians today, and also those working in applied fields. This project proposes to tackle some of the pressing questions that arise when exploring multivariate dependencies in high dimensions. As a concrete application, the methodology developed in this project will be used to understand the interconnectedness of genes in cancer studies and cardiovascular medicine, while maintaining the statistical rigor and ease of interpretability of previously developed methods. Hence a project of this nature will have widespread applications, as understanding relationships between many variables or players is an endeavor that is common to many scientific disciplines. The proposed work, though rooted in the principles of statistics, is interdisciplinary, and involves collaborations with biomedical scientists, engineers and the environmental scientists.