Collaborative Research: CDS&E: Applied Algebraic Statistics through R

应用代数几何学和统计学的接口被称为代数统计学，对实际数据分析中的新老问题有着丰富的新见解。基本的联系源于认识到，许多统计模型是或可以确定几何结构服从代数调查，使统计人员在解决统计问题时，从大量的代数工具。自从认识到这一点以来，代数工具在统计学中得到了广泛的应用，特别是在涉及交叉分类数据的情况下。尽管取得了这些进展，但在传统的数据分析统计领域中使用代数方法仍然不是主流，主要是因为这些方法涉及以前数据分析不必要的数学计算，因此无法在标准软件中使用。这项工作通过以下方式正面解决了这个问题：1）通过PI创建的附加软件包，加强数据分析师（R）和数学社区中各种软件之间的自由统计计算环境之间的联系; 2）实现外部软件支持的尖端代数统计方法的用户友好界面。R软件包algstat和支持软件包将进一步开发，加强与代数统计中使用的软件的联系，并为利用这些软件的代数统计方法提供功能和数据结构。在项目的第一年，PI及其团队将致力于LattE和4 ti 2，并将创建和改进用于对数线性，逻辑和泊松回归模型精确推断的马尔可夫基础技术。在第二年，PI和他们的团队将工作的贝尔蒂尼。与多项式方程组数值解有关的函数和数据结构将得到改进和扩展，并将考虑在遗传学中的应用。在第三年，PI和他们的团队将致力于Macaulay 2，加强其与R的连接，并使用它来增强mpoly包，并自适应地通知MCMC例程，以便在LattE和4 ti 2连接支持的指数族模型中进行精确推断。

The interface of applied algebraic geometry and statistics known as algebraic statistics abounds with fresh insight into old and new problems in practical data analysis. The fundamental connection stems from the realization that many statistical models are or can be identified with geometric structures amenable to algebraic investigation, enabling statisticians to draw from the great wealth of algebraic tools when solving statistical problems. Since this recognition, algebraic tools have found applications all over statistics, especially in contexts involving cross-classified data. Despite these advances, the use of algebraic methods in traditionally statistical areas of data analysis is still not mainstream, mostly because the methods involve kinds of mathematical computations previously unnecessary for data analyses and, consequently, not available in standard software. This work confronts this problem head-on by 1) fortifying connections between a free statistical computing environment popular among data analysts (R) and various software in the mathematics community through add-on packages created by the PIs and 2) implementing user-friendly interfaces to cutting-edge algebraic statistical methods enabled by the external software.The R package algstat and supporting packages will be further developed, strengthening connections to software used in algebraic statistics and providing functions and data structures for algebraic statistical methods that leverage those software. In year one of the project, the PIs and their teams will work on LattE and 4ti2, and Markov bases techniques for exact inference in loglinear, logistic, and Poisson regression models will be created and improved. In year two, the PIs and their teams will work on Bertini. Functions and data structures related to the numerical solution of systems of polynomial equations will be improved and expanded, and applications to phylogenetics will be considered. In year three, the PIs and their teams will work on Macaulay2, fortifying its connection to R and using it to enhance the mpoly package and adaptively inform the MCMC routines for exact inference in exponential family models enabled by the LattE and 4ti2 connections.