Computational Algebraic Methods for High-dimensional Statistical Applications

摘要提案DMS-0200888 PI Ian Dinwoodie题目“高维统计应用的计算代数方法“该项目的目标是研究经典渐近方法不适用的高维参数统计问题的代数方法。与研究目标相连的是与现有和未来的研究生合作。研究的问题是：高维吉布斯分布模型的网络可靠性和流量与不完整的数据;优化参数估计的高维模型与不完整的数据和非凸对数似然函数;快速模拟方法，包括纤维行走与马尔可夫链的整数数据表;和多元指数生成函数的计算格点与超几何分布。将使用一系列代数工具，包括交换环和D-模中的Groebner基，消除理论，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。 这种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，以及安全和分析困难的表格数据的大型数据库，如人口普查信息，代数工具可以帮助解决模型制定和模型拟合和统计分析的问题。近年来，许多代数方法被用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。

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.