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

代数统计学是应用代数几何与统计学的交叉领域，它对实际数据分析中的新老问题有着独到的见解。这种基本联系源于这样一种认识，即许多统计模型是或可以被识别为适合代数研究的几何结构，使统计学家能够在解决统计问题时利用大量的代数工具。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背景下。尽管有这些进步，代数方法在数据分析的传统统计领域的使用仍然不是主流，主要是因为这些方法涉及以前对数据分析不必要的各种数学计算，因此，在标准软件中不可用。这项工作直面这个问题：1)通过pi创建的附加包，加强在数据分析师(R)中流行的免费统计计算环境与数学界各种软件之间的连接；2)实现由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和他们的团队将在LattE和4ti2上工作，并且将创建和改进用于在线性，逻辑和泊松回归模型中进行精确推理的马尔可夫基础技术。第二年，pi和他们的团队将研究Bertini。与多项式方程组数值解相关的函数和数据结构将得到改进和扩展，并将考虑应用于系统发育。在第三年，pi和他们的团队将在Macaulay2上工作，加强它与R的连接，并使用它来增强mpoly包，并自适应地通知MCMC例程，以便在由LattE和4ti2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.