Computational Algebraic Methods for High-dimensional Statistical Applications

本项目的目标是研究用于高维参数统计问题的代数方法，其中经典的渐近方法是不起作用的。与研究目标相关的是与现有和未来的研究生合作。所研究的问题包括：不完全数据下网络可靠性和通信量的高维Gibbs分布模型；不完全数据和非凸对数似然函数下高维模型中参数估计的优化；整数数据表的包括马尔可夫链的光纤行走的快速模拟方法；超几何分布格点计算的多元指数母函数。将使用一系列代数工具，包括交换环和D-模中的Groebner基、消元理论、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的统计应用中。这些新的和具有挑战性的应用的例子是大规模的网络流量和可靠性，以及大型表格数据数据库，如人口普查信息，这些数据库的安全性和分析都很困难。代数工具可以帮助解决模型建立、模型拟合和统计分析的问题。最近发展了许多代数技术来解决机器人学和微分方程中的计算问题，这些方法在统计学中非常有前途。研究人员将开发这些代数方法来解决统计应用问题。

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.