Simultaneous Statistical Modeling of Several Large Covariance Matrices

这项研究致力于发展一个三阶段的统计模型拟合过程，包括对商业和经济、流行病学、环境监测和全球变化、生物技术和制造业(质量控制)中出现的大的多变量响应的同期协方差矩阵的模型建立、估计和诊断。相关和协方差矩阵及其谱(特征值)分解是所有经典多变量技术的基础。然而，对于时间相关性，乔列斯基分解是大多数时间序列技术的核心。在多变量统计和多变量随机波动率模型的逐级过程中，越来越多的工作隐含地依赖于Cholesky分解，或者将每个长随机向量视为变量的给定排列的“时间序列”。这项拟议的工作旨在明确和揭示使用Cholesky分解或时间序列技术来建模同时协方差矩阵的全部潜力。它包括发展简约模型，它们的估计和渐近性质，它们的实际实现和用途，着眼于确保用于计算参数的最大似然估计的迭代过程的每个阶段的协方差矩阵的正定性。实现高维无序向量的一个主要困难是变量的大量可能排列。所使用的方法和工具包括：广义线性模型、因子分析和随机效应模型、时间序列模型拟合过程、极大似然和贝叶斯估计以及数值线性代数。这项拟议的研究有可能将Cholesky分解提升为一种真正的工具，用于建模时间和同时的相关性，从而连接(统一)不同的领域，如时间序列分析、因素分析和线性结构模型、图形模型和贝叶斯协方差建模。技术和数据收集的进步使研究人员能够在更精细的时间单位内记录系统的许多特征的测量。提出这项研究的动机是在拥有大量多变量数据的情况下产生的统计问题，重点是预测、控制、分类、聚类和数据挖掘。这些任务的可靠性或错误率总是取决于对许多变量之间相关性的准确估计，以及对大协方差矩阵动态的更好理解。该提案的目标是通过对相关协方差矩阵进行建模，为分析高维数据提供一种系统和有效的方法。该提案打算使用经典和最近的统计估计和大规模计算。拟议的研究结果将在上面概述的不同领域得到应用，但将特别关注它们在理解机器和大脑中的智能方面的潜在用途。一名学生将在暑假期间参与拟议的研究。

AbstractPI: Mohsen Pourahmadi, DMS-0307055TITLE: Simultaneous Statistical Modeling of Several Large Covariance MatricesThis research focuses on the development of a three-stage statistical model-fitting process consisting of model formulation, estimation and diagnostics for contemporaneous covariance matrices of large multivariate responses arising, for example, in business and economics, epidemiology, environmental monitoring and global change, biotechnology and manufacturing (quality control). Correlation and covariance matrices and their spectral (eigenvalue) decomposition provide the basis for all classical multivariate techniques. For temporal correlations, however, the Cholesky decomposition lies at the core of most time series techniques. A growing body of work in step-down procedures in multivariate statistics and multivariate stochastic volatility models implicitly rely on the Cholesky decomposition or view each long random vactor as a "time series", for a given arrangement of the variables. The proposed work intends to make explicit and reveal the full potential of using Cholesky decomposition or time series techniques in modeling contemporaneous covariance matrices. It includes developing parsimonious models, their estimates and asymptotic properties, their practical implementation and uses, with an eye to guaranteeing the positive-definiteness of the covariance matrices at each stage of an iterative procedure used to compute the maximum likelihood estimates of the parameters. A major difficulty of the implementation for high-dimensional unordered vectors is the large number of possible arrangements of the variables. The methods and tools to be employed include: generalized linear models, factor analysis and random effects models, time-series model fitting process, maximum likelihood and Bayesian estimation and numerical linear algebra. The proposed research has the potential of elevating the Cholesky decomposition as a bona fide tool for modeling temporal and contemporaneous correlations and hence connecting (unifying) disparate areas like time series analysis, factor analysis and linear structural models, graphical models and Bayesian covariance modeling.Advances in technology and data collection have enabled researchers to record measurements on many characteristics of systems over finer units of time. The research proposed is motivated by statistical problems arising in settings where large amounts of multivariate data are available and the focus is on prediction, control, classification, clustering and data mining. The reliability or error rates of these tasks invariably hinge on the precise estimation of correlations among many variables and better understanding of the dynamics of large covariance matrices. The goal of the proposal is to provide a systematic and efficient method for analyzing high-dimensional data through modeling of the relevant covariance matrices. The proposal intends to use classical and recent statistical estimation and large-scale computing. Results obtained by the proposed research will have applications in diverse areas outlined above, however, particular attention will be paid to their potential use in understanding intelligence in machines and brains. A student will participate in the proposed research during summers.