New Geometric Methods of Mixture Models for Interactive Visualization

该研究项目将通过新颖的数学工具来探索混合模型的精确几何形状，从而扩展混合建模的理论基础，用于统计学习。在理论结果的基础上，研究人员将开发新的聚类、降维、变量选择和时间分析方法。这些方法将为复杂数据的交互式可视化和数据汇总开辟有前途的道路。一套统计工具将作为技术骨干集成到一个新的可视化系统中。应用于非常大规模，高维，和时间演变的数据将被探索。在理论统计学、计算统计学和信息可视化方面具有互补背景的主要研究人员也将与宾夕法尼亚州立大学多个部门的同事合作，使用现实世界的数据集测试他们的方法和原型系统。在极端天气预报、制造工程设计等对我们的日常生活有直接和巨大影响的众多科学和工程领域，研究人员面临着在维度、数据类型、统计依赖和时间变化方面具有巨大复杂性的海量数据。可视化在支持分析复杂数据方面发挥了重要作用。可视化系统帮助用户增加可用的空间和认知资源，改进搜索，增强模式识别，并最终理解抽象现象。该研究项目旨在从根本上推进可视化系统的数学核心。研究人员采用概率框架对数据建模，特别是混合模型。混合建模为汇总数据和自动从数据中提取模式提供了高度灵活和理论上坚实的基础。本项目将发展混合建模的理论和算法，并利用它们来构建新的统计学习和数据挖掘技术。这些统计方法将彻底改变可视化系统的设计方式，提供更多的功能和更好的功能。将开发和分发用于高级统计学习方法和交互式可视化的软件包，供公众使用。对数据可视化和建模技术的研究有望影响科学、工程和商业的广泛领域。在飓风预报和工程设计中的应用对我们的日常生活有着深远的影响。

This research project will extend the theoretical foundations of mixture modeling for statistical learning by novel mathematical tools that can probe into the precise geometry of mixture models. Based on the theoretical results, the investigators will develop new approaches to clustering, dimension reduction, variable selection, and temporal analysis. These methods will open promising paths for interactively visualizing complex data and for data summarization. A suite of statistical tools will be integrated as the technical backbone into a new visualization system. Applications to very large-scale, high dimensional, and temporally evolving data will be explored. The principal investigators, with complementary backgrounds in theoretical statistics, computational statistics, and information visualization, will also work with colleagues across multiple departments at Penn State University to test their methods and prototype systems using real-world data sets.In a plethora of scientific and engineering areas with direct and tremendous impacts on our everyday life, such as extreme weather prediction and manufacturing engineering design, researchers are facing gigantic amount of data with great complexity in terms of dimensionality, data types, statistical dependence, and temporal variations. Visualization has played important roles in support of analyzing complex data. Visualization systems help users increase available spatial and cognitive resources, improve searching, enhance pattern recognition, and ultimately make sense of abstract phenomena. This research project aims at fundamentally advancing the mathematical core of visualization systems. The investigators take a probabilistic framework to model data, specifically the mixture model. Mixture modeling provides a highly flexible and theoretically solid basis for summarizing data and automatically extracting patterns from data. This project will develop theories and algorithms for mixture modeling and exploit them to construct new statistical learning and data mining techniques. These statistical methods will thoroughly change the ways visualization systems are designed, offering more functions as well as better functions. Software packages for advanced methods of statistical learning and interactive visualization will be developed and distributed for public use. The proposed research on data visualization and modeling techniques are expected to affect a wide range of fields in science, engineering, and commerce. The applications to hurricane forecast and engineering design can deeply influence our daily life.