The Shape of Data: A New Way to Detect Critical Shifts in System Performance

该项目是关于一种新的方法来检测和分类传感器测量模式的变化。 例如，桥台上的振动传感器输出的细微变化可能表明该结构中正在形成裂缝。 识别这些变化可能是一个真实的挑战，因为现代传感器可以生成如此多的信息，而且速度如此之快。 现有的方法使用统计学的数学：平均值，变异性等。 本项目使用拓扑学的技术，即与形状有关的数学分支，来检测这类变化，本项目将开发评估和描述非平稳时间序列数据中的时间模式的技术。该项目将计算同源性的各个方面-例如，流的前几个Betti数-以获得其政权的签名。为了减轻传统技术的计算负担，这个项目使用了一个简单的复杂的基础上，一个小的，有代表性的一组“地标”，与其余的数据，“证人”，定义的连接。新技术包括一个泛化的常用的假近邻方法来评估重建的拓扑结构的保真度，有效地选择地标和证人关系的选择，通过多参数的嵌入式数据的持久同源性的结构分类，和证人复杂的地图上的发展，以获得每个政权的动态签名。这些目标包括有效地检测政权转移，发展一个目录内的政权签名，并检测和减轻噪音。例如，分类技术将适用于检测制造系统中的故障模式、网络上的恶意软件攻击、早期心脏问题或即将发生的遏制故障。

This project is about a new way to detect and classify shifts in the patterns of measurements taken by sensors. Subtle shifts in the output from a vibration sensor on a bridge abutment, for instance, can indicate that a crack is developing in that structure. Identifying these shifts can be a real challenge because modern sensors can generate so much information, and so quickly. Existing approaches to this use the mathematics of statistics: averages, variability, and the like. This project uses techniques from topology, the branch of mathematics that is concerned with shape, to detect these kinds of shifts.This project will develop techniques to assess and characterize temporal patterns in nonstationary time-series data. The project will compute aspects of the homology---e.g., the first few Betti numbers---of the stream "on the fly" to obtain a signature of its regime. To mitigate the computational burden of traditional techniques, this project uses a simplicial complex based on a small, representative set of "landmarks," with the remaining data, the "witnesses," defining the connections. New techniques include a generalization of the oft-used false near-neighbor method to evaluate the fidelity of the topology of the reconstruction, efficient selection of landmarks and choice of witness relation, classification of structure through multi-parameter persistent homology of embedded data, and development of a map on the witness complex to obtain a dynamical signature of each regime. The goals include efficient detection of regime shifts, the development of a catalog of signatures within regimes, and the detection and mitigation of noise. The classification techniques will be applicable, for example, to the detection of failure modes in manufacturing systems, malware attacks on a network, incipient heart problems, or an impending containment failure.