Discovering Sparse Covariance Structures in High Dimensions

该项目的重点是发现和利用数据中的稀疏结构，以提高高维协方差矩阵的估计。协方差矩阵在许多数据分析方法中起着关键作用，包括主成分分析，判别分析，多变量分析中的均值推断，以及图形模型中的独立性和条件独立关系的推断。随机矩阵理论的发展表明，传统的估计，样本协方差，在高维表现不佳。 现有的替代估计的研究，包括以前的工作的PI，主要集中在情况下，有一个概念的距离或排序的变量指标（时间序列，纵向数据，空间数据，光谱等）。 然而，在许多应用中，这种排序是不可用的：例如，遗传学、金融、社会和经济数据。 这个项目开发了几种方法来构造正则化稀疏估计，这些估计对于协方差矩阵及其逆矩阵的变量排列都是不变的。 这些方法的主要组成部分是阈值处理、鼓励稀疏性的平滑惩罚、置换不变损失函数、自适应权重和流形投影，以发现变量的潜在结构化重新排序。 分析结果建立的一致性和收敛速度的建议估计在高维充分发展。这些理论结果在高维需要的工具，是不同的标准渐近分析，并有几个在现有的文献。开发了计算这些估计量所需的有效优化算法，重点是计算成本随着维度尽可能缓慢地增长。 一些估计进行设计的计算成本非常低，而其他人需要计算的巧妙性，在真正的高维是可行的。 通过PI的跨学科合作，在模拟和一些应用程序中对拟议的方法进行了广泛的测试。在现代世界收集的大量数据正在为统计人员带来新的挑战。 迫切需要新的理论和实践方法来处理高维数据，以及大量需要估计高维协方差矩阵作为数据分析一部分的应用：金融，遗传学，光谱学，遥感，气候研究，脑成像，语音识别等。 PI正在与化学家合作研究骨骼的拉曼光谱，与海洋学家合作使用光谱数据进行海洋遥感，与气候科学家合作进行温度建模，并与生物统计学家合作研究一种新型的蛋白质水平基因表达技术。 PI还积极地工作在无线传感器网络的统计信号处理领域，其中空间协方差估计是重要的，并且具有许多安全应用。对该项目开发的估计高维协方差的新方法进行了理论分析，并在这些应用程序中进行了测试和验证，反过来，该项目在后期发展的方向受到应用程序的问题和需求的影响。 该项目还有助于在现代统计的一个重要领域教育研究生。

This project focuses on discovering and exploiting sparse structures in the data to improve estimation of covariance matrices in high dimensions. The covariance matrix plays a key role in many data analysis methods, including principal component analysis, discriminant analysis, inference about the means in multivariate analysis, and inference about independence and conditional independence relationships in graphical models. Advances in random matrix theory have shown that the traditional estimator, the sample covariance, performs poorly in high dimensions. The existing research on alternative estimators, including previous work of the PI, focuses mostly on the situation when there is a notion of distance or ordering for the variable indexes (time series, longitudinal data, spatial data, spectroscopy, etc). However, there are many applications where such ordering is not available: for example, genetics, financial, social and economic data. This project develops several methods for constructing regularized sparse estimators that are invariant to variable permutations, both for the covariance matrix and its inverse. The main building blocks of the methods are thresholding, smooth penalties that encourage sparsity, permutation-invariant loss functions, adaptive weights, and manifold projections to discover potential structured re-orderings of the variables. Analytical results establishing consistency and convergence rates of the proposed estimators in high dimensions are fully developed. These theoretical results in high dimensions require tools that are different from standard asymptotic analysis, and there are few available in the existing literature. Efficient optimization algorithms needed to compute these estimators are developed, with the emphasis on the computational cost growing as slowly as possible with dimension. Some of the estimators proposed carry a very low computation cost by design, while others require computational ingenuity to be feasible in really high dimensions. The proposed methodology is tested extensively, both in simulations and on a number of applications through the PI's interdisciplinary collaborations.Massive amounts of data collected in the modern world are creating new challenges for statisticians. There is an urgent need for new theoretical and practical methods that deal with high-dimensional data, and a vast number of applications where high-dimensional covariance matrices need to be estimated as part of data analysis: finance, genetics, spectroscopy, remote sensing, climate studies, brain imaging, speech recognition, and many others. The PI has ongoing collaborations with chemists on Raman spectroscopy of bone, with oceanologists on using spectral data for remote ocean sensing, with climate scientists on temperature modeling and with a biostatistician on a new type of gene expression technology that works at protein level. The PI also works actively in the area of statistical signal processing by wireless sensor networks, where spatial covariance estimation is important, and which has many security applications. The new methodology for estimating high-dimensional covariances developed in this project is analyzed theoretically and tested and validated in these applications, and in turn, the directions in which the project develops at later stages are influenced by the issues and needs of the applications. The project also contributes to educating graduate students in an important area of modern statistics.