Generalized Linear Models for Large Correlation Matrices Via Partial Autocorrelations

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。本研究将以广义线性模型的精神，以偏相关作为新的无约束参数，建立大相关矩阵的统计模型。特别是，将开发计算效率高的程序来模拟随机相关矩阵，这对新的统计方法和数据挖掘算法的模拟测试、数字信号处理以及纵向数据分析中的工作相关矩阵非常感兴趣。在商业和经济、流行病学、环境监测、生物技术和光谱学中，大型相关矩阵经常出现，在这些领域，现代技术创新使以相对较低的成本收集大量数据成为可能。建模和模拟相关矩阵的三个主要困难是(I)正定性约束，(Ii)高维和(Iii)对角线输入必须等于1的附加约束。虽然Cholesky分解和其他技术可以处理(I)和(Ii)，但它们无法处理(III)。这项研究的目的是利用偏相关的基本概念，以一种不受约束和统计上可解释的方式对相关矩阵进行重新参数化。因此，将开发类似于回归和时间序列分析中常用的相关矩阵的稀疏和灵活的统计模型、数据分析和图形工具。所使用的方法和工具包括：广义线性模型理论、时间序列分析、数值线性代数、正交多项式理论和蒙特卡罗方法。提出的工作有可能将偏自相关的基本概念提升为一种真正的工具，用于以类似于其在时间序列分析、信号处理、正交多项式理论和图形模型中的公认角色的方式对标准多变量数据进行建模。它的另一个特点是将这些明显不同的领域联系起来，这体现了这项工作的跨学科性质。对高维数据分析的关注会对收集大量多变量数据的环境产生直接影响。这种环境的重要例子是金融市场、环境监测和全球变化、生物技术和制造业。研究生将参与项目的不同阶段，结果将纳入课程，并在统计领域以外的研究人员可进入的研讨会和讲习班上介绍。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This research will focus on developing statistical models for large correlation matrices in the spirit of the generalized linear models using the partial correlation as the new unconstrained parameters. In particular, computationally efficient procedures will be developed for simulating random correlation matrices which are of great interest in simulation testing of new statistical methods and data mining algorithms, digital signal processing, and working correlation matrices in the analysis of longitudinal data. Large correlation matrices arise quite often in business and economics, epidemiology, environmental monitoring, biotechnology and spectroscopy where modern technological innovations have made it possible to collect massive amount of data with relatively low cost. The three major difficulties in modeling and simulating correlation matrices are (i) the positive-definiteness constraint, (ii) the high-dimensionality and (iii) the additional constraint that its diagonal entries must equal to one. While the Cholesky decomposition and other techniques can handle (i) and (ii), they are unable to handle (iii). The proposed research intends to reparameterize a correlation matrix in an unconstrained and statistically interpretable manner using the basic concept of partial correlation. Consequently, sparse and flexible statistical models, data analytic and graphical tools for correlation matrices will be developed in analogy with those commonly used in regression and time series analysis. The methods and tools to be employed include: the theory of generalized linear models, time series analysis, numerical linear algebra, theory of orthogonal polynomials and the Monte Carlo methods. The proposed work has the potential of elevating the basic concept of partial autocorrelation as a bona fide tool for modeling standard multivariate data in a manner similar to its well-established role in time series analysis, signal processing, the theory of orthogonal polynomials and graphical models. It has the added feature of connecting these apparently disparate areas, which brings out the interdisciplinary nature of the work. The focus on high-dimensional data analysis has immediate impacts on settings where large amounts of multivariate data are collected. Important examples of such settings are financial markets, environmental monitoring and global change, biotechnology and manufacturing. Graduate students will be involved in various phases of the project, the results will be incorporated in courses and presented in seminars and workshops accessible to researchers outside the field of statistics.