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的高斯模型的代数和几何结构：彩色高斯图（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模型是数学遗传学中连续性状进化的高斯模型。然后，研究人员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STEM学科的本科生，特别是那些教育资源有限的代表性不足的群体。 研究人员将使用代数，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矩阵或浓度矩阵，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）计算找到给定数据的统计模型的最大似然估计（MLE）的复杂性，依赖于代数和几何的工具，例如在代数簇上优化，交集理论和多面体几何。本研究的主要目的是：（1）确定系统树中影响BMT模型最大似然度的特征，以及与BMT模型消失理想代数度的关系;（2）对具有环面结构的CGG模型进行分类;也就是说，对于复曲面消失理想或者对于在变量的适当线性变化之后变成复曲面的消失理想，（3）给出了CGG模型的极大似然度公式和具有MLD的CGG模型的极大似然估计函数公式。研究者强调CGG模型具有复曲面簇的代数结构，因为复曲面统计模型的二元方程可以用于将计算固定在模型的MLD上，产生马尔可夫基，有助于假设检验算法，与复曲面模型相关的多面体对于研究MLE的存在是有用的。该奖项反映了NSF的法定使命，并被认为值得支持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。

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.