Collaborative Research: A Unified Framework for the Investigation of Time Series Using Topological Data Analysis

传感器技术和计算机硬件的最新进展导致了对工程和自然系统的数据驱动分析和建模的转变。目前，对由此产生的数据流的研究需要广泛的专门知识，并且经常使用不能保证最佳的方法。这导致了过度定性的分析，可能会错过隐藏在信号中的信息的重要部分。该项目旨在利用和推进拓扑数据分析工具，这是一个新兴领域，专注于提供数据形状的定量测量，并阐明当前方法无法检测到的信号的不可见特征和结构。因此，结果框架将能够提供对生成数据的系统的洞察，同时为系统和过程的自动化分析提供强大的数学基础。从这项工作中获得的知识将有益于各种需要对传感器反馈信号进行分析的科学领域，如增材制造和智能药物输送系统。此外，该项目的跨学科性质将为学生在工程、应用数学和信号分析等各个领域提供丰富的合作和交叉训练机会。pi还将继续并扩大其努力，通过与数学妇女协会和女工程师协会合作，从代表性不足的群体中寻找和招收具有相关兴趣的研究生。该项目是一项合作努力，旨在通过推进和连接信号处理、动力系统和拓扑数据分析来研究时间序列分析的创新框架。研究小组将采用一种拓扑方法，利用最近发展的持久同调来研究动力系统的时间序列。具体来说，他们将(1)基于应用拓扑建立坚实的数学基础，使动力系统的研究更加稳健和定量；(2)利用拓扑数据分析从收集的时间序列中研究动力系统中底层流形拓扑的创新方法；(3)利用数值和物理实验证明和评估理论结果的有效性。该研究的理论基础特别适用于使用低维描述符而不是低维表示来检测和描述动态特征，如潜在的吸引子、混沌和自相似性。因此，这项工作能够为我们对时间序列分析的理解提供一个新的视角，特别是对于具有复杂行为的动力系统。

Recent advances in sensor technology and computer hardware have led to a shift towards data-driven analysis and modeling of engineered and natural systems. The study of the resulting streams of data currently requires extensive expertise and often utilizes methods that are not guaranteed to be optimal. This leads to an overly qualitative analysis that can miss a significant portion of the information hidden within the signal. This project aims to utilize and advance tools from topological data analysis, an emergent field focused on providing quantitative measurements of the shape of data, and to elucidate invisible features and structure of signals which current methods cannot detect. Therefore, the resulting framework will be able to provide insight into the systems that generated the data while providing a strong mathematical footing for automating the analysis of systems and processes. The knowledge gained from the work will benefit a wide variety of scientific fields where analysis from sensor feedback signals is needed such as additive manufacturing and smart drug delivery systems. Furthermore, the interdisciplinary nature of this project will offer students rich opportunities for collaboration and cross-training in a variety of areas in engineering, applied math, and signal analysis. The PIs will also continue and expand their efforts to identify and recruit graduate students from under-represented groups with relevant interests by working with the Association for Women in Mathematics and the Society for Women Engineers.This project is a collaborative effort that seeks to investigate an innovative framework for time series analysis through advancing and linking signal processing, dynamical systems, and topological data analysis. The research team will pursue a topological approach that utilizes the recently developed persistent homology to study the time series of dynamical systems. Specifically, they will (1) derive a solid mathematical foundation based on applied topology which enables a more robust and quantitative investigation of dynamical systems, (2) research innovative methods for studying the topology of the underlying manifold in dynamical systems from collected time series using topological data analysis, and (3) demonstrate and assess the validity of the theoretical results using numerical and physical experiments. The theoretical underpinning of the research is especially suitable for detecting and describing dynamical signatures such as underlying attractors, chaos, and self-similarity using lower dimensional descriptors, rather than lower dimensional representation. Therefore, this work is capable of providing a new perspective into our understanding of time series analysis particularly for dynamical systems with complex behavior.