Collaborative Research: Graphical and Algebraic Models for Multivariate Categorical Data

拟议的研究项目通过模仿高斯模型来开发多元分类数据的模型，该模型具有可根据条件独立性的非参数概念捕获的所需模型结构。这种方法有着悠久的历史：图形对数线性模型可以通过逆协方差矩阵的零约束定义的高斯模型以这种方式导出。该项目试图极大地扩展该方法的范围。建议定义和研究列联表、类移动平均依赖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基于最近引入的AMP链图和祖先图的离散图模型的边缘无关模型。该研究的主要目标是参数化的发展，最大似然估计的有效算法的构建和实现，以及模型选择程序的调查。该项目的一个特别重点将是在分析参数空间的结构和似然函数的性质时使用计算代数的现代工具。多元统计模型试图描述大量变量之间的复杂关系。其中一类特殊的模型，称为图形模型，已经在人工智能、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。该项目提出的模型扩展了图形模型的领域，预计它们将在许多这些领域得到应用。此外，建议的方法将提供新的工具，以分析公众感兴趣的数据，如人口普查数据。研究人员还计划免费提供软件工具，作为一个更大的开源统计软件包R的一部分。

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.