RI: Small: Statistical Modeling of Dynamic Covariance Matrices

动态协方差矩阵的适当模型可在若干应用领域中极其有用，如文本挖掘和主题建模，人们可以在其中研究专题之间不断变化的相关性；在从股票/债券回报到利率和货币的各种金融数据中，跟踪不断变化的协方差的最重要作用已得到广泛承认；在环境信息学中，用于研究来自大气层和生物圈的不同变量之间动态协方差的趋势。在这样的领域中，简单地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变中可能存在的任何趋势。该项目引入并研究了一类新的动态Wishart模型(DWM)，它具有与卡尔曼滤波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中使用多变量高斯分布，该模型使用协方差矩阵上的Wishart和逆Wishart分布族。与卡尔曼滤波不同，解析闭合形式的滤波在离散余弦模型中可能是不可能的，但模型仍然有足够的结构来允许高效的近似推理算法。该项目专注于在离散多维模型的背景下对过滤、平滑和相关问题进行近似推理；开发适当的数值稳定的递归更新以防止正确定性中的数值损失；以及研究离散多维模型的一般性，包括用于跟踪复杂协方差动态的混合模型。协方差有效跟踪算法的发展将允许对动态系统进行建模，其中状态真实地表示多个实体之间的变化关系。这项研究的主要贡献是利用现有的动态潜在状态向量文献来创建同样强大的动态潜在协方差矩阵的方法。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，并将为这些学科的研究人员和学生提供新的机会，从而扩大对计算机科学的参与。

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.