CAREER: Algebraic Problems in Statistics and Biology

North Carolina State University, Raleigh NC

Investigators

Abstract

The long-term goal of the proposed research is to develop theoretical and computational methods to solve problems in algebraic statistics and its applications. The specific problem areas described in this proposal concern Gaussian graphical models and statistical models in evolutionary biology. The problems in the proposal have typically been open for many years and are fundamentally of an algebraic character; however, advanced algebraic techniques have rarely been applied to them. The motivation for studying these problems comes from the fact that the statistical models under investigation in the proposal arise frequently in applications, including economics, computational biology, and sociology. The specific problems that Sullivant will address for Gaussian graphical models are the identification problem, the computation and existence of maximum likelihood estimates, and the structure of covariance constraints. The specific problems that Sullivant will address in evolutionary biology are the identifiability of mixture models, the computational and combinatorial complexity of gene trees and species trees, and the comparison of speciation models to tree building algorithms. The techniques Sullivant will employ to address these problems come from computational algebra, algebraic geometry, and combinatorics. Algebra is the mathematical study of sets with operations, for example the set of all polynomials with the operations of "adding polynomials" and "multiplying polynomials". Statistics is the science of data analysis. Parametric statistics uses statistical models (families of probability distributions) as tools for analyzing data. Algebra and statistics come together in the field of algebraic statistics, which is based on the observation that many parametric statistical models are described parametrically via polynomials. Sullivant proposes to exploit the algebra/statistics connection to address a number of problems in algebraic statistics and its applications to evolutionary biology. In evolutionary biology, Sullivant and his students will study phylogenetic models, which are statistical models for determining the ancestral relationships between a collection of extant species. These models are used in applications in increasingly complicated ways, but even basic questions about the models remain open. The projects in the proposal will involve research with both graduate and undergraduate students.