Applications of Algebraic Geometry to Multivariate Gaussian Models

本项目旨在深入分析应用中常用的两种主要类型高斯模型的代数和几何结构：彩色高斯图形（CGG）模型和布朗运动树（BMT）模型。CGG模型用于模拟随机变量之间的相互作用，考虑可能的相似性状，BMT模型是数学系统发育中连续性状进化的高斯模型。然后研究者将使用这些信息来计算每个模型的最大似然估计问题的复杂性，这是分析数据时的一个关键问题。该项目的一些要素将涉及主修STEM学科的本科生，特别是那些来自教育资源有限、代表性不足的群体的学生。研究者将使用代数、几何、组合学和符号计算来更好地理解统计模型，并在最大似然估计（MLE）问题上取得进展。对于高斯模型，首先要识别高斯分布及其协方差或浓度矩阵，并分析在这些矩阵上消失的多项式。统计模型的最大似然度（MLD）用于计算给定数据的统计模型的最大似然估计（MLE）的复杂性，它依赖于代数和几何工具，如代数变量优化、交理论和多面体几何。本项目旨在解决的具体问题是：(1)确定系统发育树中影响其BMT模型的最大似然度及其与BMT模型消失理想的代数度的连接的特征；(2)对具有环形结构的CGG模型进行分类；(3)找到CGG模型的最大似然度和MLD为1的CGG模型的最大似然估计函数的公式。研究者强调CGG模型具有一个环变量的代数结构，因为一个环统计模型的二项式方程可以用来加快模型的MLD的计算，产生马尔可夫基，有助于假设检验算法，并且与环模型相关的多面体对研究MLE的存在性是有用的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The present project aims to do an in depth analysis on the algebraic and geometric structure of two main types of Gaussian models that are commonly chosen in applications: colored Gaussian graphical (CGG) models and Brownian motion tree (BMT) models. CGG models are for modeling interactions among random variables, taking in consideration possible similar traits, and BMT models are Gaussian models for the evolution of continuous traits in mathematical phylogenetics. The investigator will then use this information to compute the complexity of the maximum likelihood estimate problem for each model, a key issue when analyzing data. Some elements of the project will involve undergraduate students majoring in STEM disciplines, especially those from underrepresented groups with limited educational resources. The investigator will use algebra, geometry, combinatorics, and symbolic computations to better understand statistical models and make advancements on their maximum likelihood estimate (MLE) problem. For Gaussian models this starts by identifying Gaussian distributions with their covariance or concentration matrices and analyzing the polynomials vanishing on these matrices. The maximum likelihood degree (MLD) of a statistical model, which computes the complexity of finding the maximum likelihood estimate (MLE) of a statistical model for given data, relies on tools from algebra and geometry such as optimizing over an algebraic variety, intersection theory and polyhedral geometry. Specific questions that this project aims to answer are: (1) determine features in a phylogenetic tree that affect the maximum likelihood degree of its BMT model and connections to the algebraic degree of the vanishing ideal for the BMT model, (2) classify CGG models with toric structure; that is, with toric vanishing ideal or with vanishing ideal that turns toric after an appropriate linear change of variables, (3) find formulas for the maximum likelihood degree of CGG models and for the maximum likelihood estimate function of CGG models with MLD one. The investigator emphasizes CGG models with the algebraic structure of a toric variety because the bimonial equations of a toric statistical model can be used to fasten computations on the MLD of the model, produce Markov bases, contribute in hypothesis testing algorithms, and the polytope associated to the toric model is useful for studying the existence of MLE.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.