EAGER: Characterizing Regime Shifts in Data Streams using Computational Topology - the Mathematics of Shape

时间序列数据出现在广泛的工程系统中，包括网络流量、机床上的振动传感器、反应堆安全壳上的声学传感器以及许多其他示例。开发高效和有效的方法来表征这些数据中的模式在工程、商业和其他领域中具有广泛的实用性。用于表征这些流中的模式的方法可用于检测网络上的恶意软件攻击、正在降级的车床轴承或反应堆中即将发生的安全壳故障。常见的挑战包括可观测性-传感器昂贵或难以部署，或者当它们干扰被检查的行为时的情况-以及高信息量，噪声和快速状态转换。这个EARLY探索性研究资助（EAGER）项目的最终目标是使用计算拓扑学，形状的基本数学，来应对这些挑战。形状也许是结构的最粗略的概念，并且可以特别鲁棒地抵抗信号的污染。本研究的具体目标是开发新的方法来识别和分类的时间模式与政权转移的数据流。时间序列分析的拓扑方法与机器学习和流挖掘社区的标准方法不同，后者通常使用概率方法并通常隐含地假设线性。该项目旨在提取非线性结构，不一定在回归或光谱方法中可见。实际上，状态偏移不需要对应于信号的频率内容的变化，但是仍然可以被表示为同调的偏移（例如，Betti数）。一个目标是开发技术有用的工程师和科学家的早期系统故障的检测或从隐藏的原因状态变化的快速评估。现有的计算拓扑算法通常需要很长的计算时间，特别是对于许多维度的大型数据集。 然而，由于并非所有这些变量都是可观察的，因此可能必须从部分测量值重建完整的动态-例如，使用称为延迟坐标嵌入的过程。该项目旨在基于不完整的部分嵌入快速评估贝蒂数。一个新颖的方面是，动力学在单纯复形上产生了一个多值映射，一个“见证映射”。"算法中多个参数的选择将基于持久的同源性，以前只为静态数据集和单个参数的分析而开发。最终的目标是强大的和快速的政权检测有限的数据流从一个“黑匣子”的来源。

Time-series data arise in a wide array of engineered systems, including network traffic, vibration sensors on machine tools, acoustic sensors on reactor containment vessels, and many other examples. The development of efficient and effective methods to characterize the patterns in such data has widespread utility in engineering, commerce and other fields. Methods for characterizing patterns in these streams could be used to detect malware attacks on a network, a lathe bearing that is degrading, or an impending containment failure in a reactor. Common challenges include observability - situations when sensors are expensive or difficult to deploy, or when they perturb the behavior under examination - as well as high information content, noise, and rapid regime shifts. The ultimate goal of this EArly-Grant for Exploratory Research (EAGER) project is to use computational topology, the fundamental mathematics of shape, to deal with these challenges. Shape is perhaps the roughest notion of structure and can be particularly robust to contamination of the signal. The specific goal of this study is to develop new methods for identifying and categorizing the temporal patterns associated with the regime shifts a stream of data. A topological approach to time series analysis is distinct from standard methods of the machine learning and stream-mining communities, which typically use probabilistic approaches and often implicitly assume linearity. This project seeks to extract nonlinear structure not necessarily visible in a regresssion or spectral approach. Indeed, a regime shift need not correspond to a change in the frequency content of a signal, but could nevertheless be represented as a shift in the homology (e.g., Betti numbers) of the embedded signal. A goal is to develop techniques useful to engineers and scientists for the detection of incipient system failure or rapid evaluation of state changes from hidden causes. Existing algorithms of computational topology often require lengthy computations, especially for large data sets in many dimensions. However, since not all of those variables may be observable, one may have to reconstruct the full dynamics from partial measurements--e.g., using the process called delay-coordinate embedding. This project seeks rapid evaluation of Betti numbers based on incomplete, partial embeddings. A novel aspect is that the dynamics gives rise to a multivalued map on a simplicial complex, a "witness map." Selection of multiple parameters in the algorithms will be based on persistent homology, previously developed only for the analysis of static data sets and for a single parameter. The ultimate goal is robust and rapid regime detection for a limited data stream from a "black-box" source.