Simultaneous Statistical Modeling of Several Large Covariance Matrices

多个大协方差矩阵的同时统计建模

• 批准号: 0307055
• 负责人: Mohsen Pourahmadi
• 金额: $ 8.21万
• 依托单位: Northern Illinois University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0307055&HistoricalAwards=false
• 关键词: Simultaneous; Statistical; Modeling; Several; Large

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.

摘要PI：Mohsen Pourahmadi，DMS-0307055职务：几个大协方差矩阵的同时统计建模本研究侧重于开发一个三阶段的统计模型拟合过程，包括模型制定，估计和诊断，用于商业和经济，流行病学，环境监测和全球变化等领域的大多元响应的同期协方差矩阵，生物技术和制造业（质量控制）。相关和协方差矩阵及其谱（特征值）分解为所有经典的多变量技术提供了基础。然而，对于时间相关性，Cholesky分解是大多数时间序列技术的核心。在多元统计和多元随机波动率模型的逐步下降过程中，越来越多的工作隐含地依赖于Cholesky分解或将每个长随机向量视为“时间序列”，对于给定的变量安排。拟议的工作旨在明确和揭示使用Cholesky分解或时间序列技术建模同期协方差矩阵的全部潜力。它包括开发简约模型，其估计和渐近性质，其实际实施和使用，着眼于保证在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。将采用的方法和工具包括：广义线性模型、因素分析和随机效应模型、时间序列模型拟合过程、最大似然和贝叶斯估计以及数值线性代数。拟议的研究有可能提升Cholesky分解作为一个真正的工具建模时间和同期的相关性，从而连接（统一）不同的领域，如时间序列分析，因子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对系统许多特性的测量。提出的研究的动机是在大量的多变量数据的设置中出现的统计问题，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许多变量之间的相关性的精确估计和更好地理解大型协方差矩阵的动态。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而有效的方法。该提案打算使用经典和最近的统计估计和大规模计算。拟议的研究所获得的结果将在上述各个领域中得到应用，但是，将特别关注它们在理解机器和大脑智能方面的潜在用途。一名学生将在夏季参加拟议的研究。