Collaborative Research: Graphical and Algebraic Models for Multivariate Categorical Data

提出的研究项目通过模仿具有期望的模型结构的高斯模型来开发多变量分类数据的模型，该模型结构可以根据条件独立性的非参数概念来捕捉。这种方法有很长的历史：用逆协方差矩阵的零约束定义的高斯模型可以用这种方法推导出图形对数线性模型。该项目寻求极大地扩大该方法的范围。在最近引入的AMP链图和祖先图的基础上，定义和研究了列联表、具有移动平均相依结构的离散值时间序列、具有离散响应变量的看似不相关的回归以及离散图模型的边际独立性模型。这项研究的主要目标是开发参数，构建和实施有效的最大似然估计算法，并研究模型选择的程序。该项目的一个特别重点将是使用计算代数中的现代工具来分析参数空间的结构和似然函数的性质。多元统计模型寻求描述一大组变量之间的复杂关系。一类特殊的模型，称为图形模型，在人工智能、生物信息学、生物学、流行病学和语音识别等领域得到了广泛的应用。该项目中提出的模型扩展了图形模型的领域，预计它们将在许多领域中得到应用。此外，拟议的方法将为分析人口普查数据等公共利益数据提供新的工具。研究人员还计划将软件工具作为更大的开放源码统计软件包R的一部分免费提供。

The proposed research project develops models for multivariate categoricaldata by mimicking Gaussian models with a desired model structure that canbe captured in terms of the non-parametric concept of conditionalindependence. This method has a long history: graphical log-linear modelscan be induced in this way by Gaussian models defined by zero constraintson the inverse covariance matrix. The project seeks to greatly extend thescope of the approach. It is proposed to define and study marginalindependence models for contingency tables, discrete-valued time serieswith moving average-like dependence structure, seemingly unrelatedregressions with discrete response variables, and discrete graphicalmodels based on the recently introduced AMP chain graphs and ancestralgraphs. The main objectives of the study are development ofparameterizations, construction and implementation of efficient algorithmsfor maximum likelihood estimation, and investigation of procedures formodel selection. A particular focus of the project will be on employingmodern tools from computational algebra in the analysis of the structure ofparameter spaces and properties of likelihood functions.Multivariate statistical models seek to describe the complex relationshipsbetween a large set of variables. A particular class of such models,called graphical models, has found wide-spread application in fields likeartificial intelligence, bio-informatics, biology, epidemiology, andspeech recognition. The models proposed in the project extend the realmof graphical models and it is anticipated that they will be applied inmany of these fields. Moreover, the proposed methodology will provide newtools for the analysis of data of public interest such as census data.The researchers also plan to make software tools freely available as partof a larger open source statistical software package called R.