RI: Small: Statistical Modeling of Dynamic Covariance Matrices

合适的动态协方差矩阵模型在几个应用领域非常有用，例如在文本挖掘和主题建模中，人们可以研究主题之间不断发展的相关性；在从股票/债券回报到利率和货币的金融数据中，跟踪不断变化的协方差的重要性已得到广泛承认；在环境信息学中，研究大气和生物圈中不同变量之间的动态协方差趋势。在这些域中，仅仅计算每个时间步的样本协方差是不够的；目标是发现协方差结构的演化过程中可能存在的任何趋势。本项目介绍并研究了一种新的动态Wishart模型（DWMs），它具有与卡尔曼滤波器相同的图形模型结构，但跟踪协方差矩阵的演变而不是状态向量。类似于在卡尔曼滤波器中使用多变量高斯，该模型在协方差矩阵上使用Wishart和逆Wishart分布族。与卡尔曼滤波不同，分析封闭形式滤波在dwm中可能不可行，但模型仍然具有足够的结构来允许有效的近似推理算法。该项目侧重于dwm背景下滤波，平滑和相关问题的近似推理；开发合适的数值稳定递归更新，以防止正确定性的数值损失；并研究dwm的推广，包括用于跟踪复杂协方差动态的混合模型。有效的协方差跟踪算法的发展将允许动态系统的建模，其中状态真正代表多个实体之间的变化关系。该研究的关键贡献在于利用现有的动态潜状态向量文献，为动态潜协方差矩阵创建同样强大的方法。这种转变将对文本分析和主题建模、金融数据分析、社会网络分析、环境信息学和其他几个领域的应用产生直接影响，并将产生新的机会，将这些学科的研究人员和学生聚集在一起，从而扩大计算机科学的参与。

Suitable models for dynamic covariance matrices can be extremely useful in several application domains, such as in text mining and topic modeling, where one can study the evolving correlation between topics; in financial data ranging from stock/bond returns to interest rates and currencies, where the paramount importance of tracking evolving covariances has been widely acknowledged; in environmental informatics to study trends in dynamic covariance among disparate variables from the atmosphere as well as the biosphere. In such domains, it is not sufficient to simply compute the sample covariance at each time step; the goal is to discover any trends there may be in the evolution of the covariance structure. This project introduces and investigates a novel family of Dynamic Wishart Models (DWMs), which has the same graphical model structure as the Kalman filter, but tracks evolution of covariance matrices rather than state vectors. Similar to the use of multivariate Gaussians in Kalman filters, the models use the Wishart and inverse Wishart family of distributions on covariance matrices. Unlike Kalman filters, an analytic closed form filtering may not be possible in DWMs, but the models still have enough structure to allow efficient approximate inference algorithms. The project focuses on approximate inference for filtering, smoothing, and related problems in the context of DWMs; develop suitable numerically stable recursive updates in order to prevent numerical loss in positive definiteness; and investigate generalizations of DWMs including mixture models for tracking complex covariance dynamics. The development of effective tracking algorithms for covariances will permit the modeling of dynamic systems where the states really represent the varying relationships between multiple entities. The key contribution of the research is in leveraging the existing literature of dynamic latent state vectors to create equally powerful methods for dynamic latent covariance matrices. Such a transformation will have direct impact on applications in text analysis and topic modeling, financial data analysis, social network analysis, environmental informatics, and several other domains, and will spawn new opportunities for bringing together researchers and students across these disciplines, thereby broadening participation in computer sciences.