Computational Algebraic Methods for High-dimensional Statistical Applications

高维统计应用的计算代数方法

• 批准号: 0200888
• 负责人: Ian DINWOODIE
• 金额: $ 9.4万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-15 至 2005-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200888&HistoricalAwards=false
• 关键词: Computational; Algebraic; Methods; dimensional; Statistical

ABSTRACTProposal DMS-0200888PI Ian DinwoodieTitle "Computational algebraic methods for high-dimensional statistical applications"This project has the goal of research in algebraic methods for high-dimensional parametric statistical problems where classical asymptotic methods are not useful. Connected with the research goal is work with existing and future graduate students. The research problems are:high-dimensional Gibbs-distribution models for network reliability and traffic with incomplete data; optimization for parameter estimation in high-dimensional models with incomplete data and nonconvex log-likelihood functions; fast simulation methods including fiber walks with Markov chains for integer data tables; and multivariate exponential generating functions for computations on lattice points with the hypergeometric distribution. A range of algebraic tools will be used, including Groebner bases in commutative rings and D-modules, elimination theory, polynomial homotopy methods, and Markov Monte Carlo methods. This project will bring recent developments in computational algebra to new statistical applications where classical methods do not work. Examples of such new and challenging applications are large-scale network traffic and reliability, and large databases of tabular data such as census information where security and analysis are difficult.The algebraic tools can help to solve problems of model formulation and model fitting and statistical analysis. Many algebraic techniques have been recently developed to solve computational problems in robotics and differential equations, and these methods are very promising for statistics. The investigators will develop these algebraic methods to solve statistical applications.

题目：高维统计应用的计算代数方法本课题的目标是研究经典渐近方法无法解决的高维参数统计问题的代数方法。与研究目标相关的是与现有和未来的研究生合作。研究的问题是：数据不完全时网络可靠性和流量的高维gibbs分布模型；具有不完备数据和非凸对数似然函数的高维模型参数估计优化整型数据表的光纤马尔可夫链快速仿真方法；多维指数生成函数用于计算具有超几何分布的格点。将使用一系列代数工具，包括交换环和d模中的Groebner基、消去理论、多项式同伦方法和马尔可夫蒙特卡罗方法。这个项目将把计算代数的最新发展带到经典方法不起作用的新统计应用中。这种新的和具有挑战性的应用程序的例子是大规模的网络流量和可靠性，以及诸如人口普查信息之类的表格数据的大型数据库，其中安全性和分析都很困难。代数工具可以帮助解决模型制定、模型拟合和统计分析等问题。最近发展了许多代数技术来解决机器人和微分方程中的计算问题，这些方法在统计学中是非常有前途的。研究人员将发展这些代数方法来解决统计应用。