Geometry and Statistics on Spaces of Dynamical Systems for Pattern Recognition in High-Dimensional Time Series

该项目的目标是开发一个数学框架，用于有效地比较从高维时间序列数据中识别的动态系统，以及用于此类数据的聚类，分类和统计分析的算法。动力系统广泛用于物理、机械、热、化学和生物过程的分析、验证和控制。然而，在许多新兴的应用中，人们还需要“比较两个进程的动态”。例如，在计算机视觉中，人们可以使用动态模型来描述人体运动的运动学和视频数据。虽然不同的人移动方式不同，但两个人执行相同任务（例如，步行）的动态模型应该比两个人执行不同任务（例如，步行与跑步）的模型“更接近”。对于商结构由群在光滑流形上的作用所定义的线性动力系统的空间，将建立一个框架。利用环境空间中的一组可有效计算的距离来定义商空间中的一组“群作用诱导距离”。这些距离将用于发展对动力系统空间进行分类、聚类和统计分析的方法。这些方法将在人类活动的运动学和视频数据上进行评估。比较动力学模型的方法的发展可以影响基础科学和整个社会。在控制理论中，这种方法会影响系统辨识和鲁棒控制。在计算机视觉中，这些技术可用于区分视频数据中的人类和人群活动，这与监控，安全，交通监控，体育报道/广播，人机交互等许多应用相关。该项目将培训工程师和科学家进行多学科研究，这些研究将需要微分几何、机器学习和计算机视觉等概念。因此，它可能会影响许多其他相关领域。该项目还将影响许多多元化的外展活动，包括正在进行的REU项目、科学与工程女性（WISE）项目和K-12外展夏令营。数据集和代码将公开提供，用于研究和教育目的。

The objective of this project is to develop a mathematical framework for efficiently comparing dynamical systems identified from high-dimensional time-series data as well as algorithms for clustering, classification, and statistical analysis of such data. Dynamical systems are widely used for the analysis, verification, and control of physical, mechanical, thermal, chemical and biological processes. However, there are many emerging applications in which one also needs to "compare the dynamics of two processes". In computer vision, for example, one can use dynamical models to describe kinematic and video data of human motion. While different people move differently, the dynamical models of two people performing the same task (e.g., walking) should be "closer" to each other than the models of two people performing different tasks (e.g., walking vs running). Framework will be developed for spaces of linear dynamical systems whose quotient structure is defined by the action of a group on a smooth manifold. A family of efficiently computable distances in the ambient space will be used to define a family of "group-action-induced distances" in the quotient space. Such distances will be used to develop methods for performing classification, clustering and statistical analysis on spaces of dynamical systems. These methods will be evaluated on kinematic and video data of human activities.The development of methods for comparing dynamical models can impact both basic science and the society at large. In control theory, such methods can impact system identification and robust control. In computer vision, such techniques can be used to discriminate human and crowd activities in video data, which is relevant to many applications in surveillance, security, traffic monitoring, sports coverage/broadcast, human-computer interaction, etc. This project will train engineers and scientist in multidisciplinary research that will need concepts from differential geometry, machine learning, and computer vision. As such, it can potentially impact many other related fields. This project will also impact many diversity outreach activities, including ongoing REU programs, the Women in Science and Engineering (WISE) program and summer camps for K-12 outreach. Datasets and code will be made publicly accessible for research and educational purposes.