Computational Algebraic Methods for High-dimensional Statistical Applications

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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.