CIF: Medium: Collaborative Research: Tracking low-dimensional information in data streams and dynamical systems

许多具有重大社会影响的应用都是由复杂的动力系统行为模拟的，包括(生命、物理和社会科学)科学、医学、经济、法律、城市发展、国际政治和全球冲突。幸运的是，最近传感器技术的进步使人们能够以前所未有的规模观测到这些现象。不幸的是，可用数据的数量和复杂性给提取关于这些系统的有意义的信息带来了许多挑战。低维模型是理解高维信号和系统中信息的有用结构。然而，这些信息往往会随着时间的推移而变化，因此这些模型可以通过利用时间动力学来进一步改进。这个项目致力于开发新的方法来跟踪数据流和动态系统中变化的低维结构，特别是在观测可能丢失、不完整、损坏或被压缩的环境中。该项目的第一个研究目标是开发新的和实质性地改进现有的跟踪低维结构的技术，特别是将跟踪能力远远超出传统信号的跟踪能力扩展到具有内在低维结构的更一般的数据集。第二个研究目标是开发新的工具，用于跟踪系统中的低维结构，并估计时变信号和数据集的内容。第三个研究目标涉及更高维和更复杂的动态系统，目标是开发改进的方法，利用低维结构对系统识别过于复杂和高维的系统进行定量和定性分析。在第四个教育目标中，正在编制可供查阅的K-12宣传材料，以便通过在线数字图书馆进行传播。

Many applications of significant societal impact are modeled bycomplex dynamical system behavior, including the (life, physicaland social) sciences, medicine, economics, law, urban development,international politics and global conflict. Fortunately, recentadvances in sensor technology have allowed observation of thesephenomena at an unprecedented scale. Unfortunately, the volume andcomplexity of available data present many challenges to extractingmeaningful information about these systems. Low-dimensional modelsserve as a useful structure for understanding the information inhigh-dimensional signals and systems. However, this informationoften changes over time, and so these models can further beimproved by exploiting temporal dynamics. This project is concernedwith developing new methods for tracking changing low-dimensionalstructure in data streams and dynamical systems, particularly insettings where the observations may be missing, incomplete,corrupted, or compressed.The first research aim in this project is to develop new andsubstantially improve existing techniques for trackinglow-dimensional structure and, in particular, to extend trackingcapabilities far beyond conventional signals to much more generaldata sets with intrinsic low-dimensional structure. A secondresearch aim is to develop new tools for tracking low-dimensionalstructure in systems jointly with estimating the content oftime-varying signals and data sets. A third research aim isconcerned with higher-dimensional and more complex dynamicalsystems, and the goal is to develop improved methods that exploitlow-dimensional structure to perform quantitative and qualitativeanalysis in systems that are too complex and high-dimensional forsystem identification. In a fourth, educational aim, accessibleK-12 outreach materials are being developed for disseminationthrough an online digital library.