Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be introduced. The main tool within topological data analysis (TDA) is persistent homology. Persistent homology is a method of quantifying the shape of data over a range of length scales. The required background and a method of computing persistent homology is briefly discussed in this work. Ideas from topological data analysis are then used for nonlinear dynamics to analyse some common attractors, by calculating their embedding dimension, and then to assess their general topologies. A method will also be proposed, that uses topological data analysis to determine the optimal delay for a time-delay embedding. TDA will also be applied to a Z24 Bridge case study in structural health monitoring, where it will be used to scrutinise different data partitions, classified by the conditions at which the data were collected. A metric, from topological data analysis, is used to compare data between the partitions. The results presented demonstrate that the presence of damage alters the manifold shape more significantly than the effects present from temperature.
翻译:在这项工作中,将采用一种方法,通过一个名为“表层数据分析”的专题来量化数据形状。在表层数据分析中,主要工具是持久性同质学。持久性同质学是一种在一系列长度尺度上量化数据形状的方法。在这项工作中,将简要讨论所需的背景和计算持久性同质的方法。从表层数据分析中得出的概念随后用于非线性动力学分析,以分析一些共同吸引者,计算其嵌入的维度,然后评估其一般的表层学。还将提出一种方法,即利用表层数据分析来确定在时间跨入方面的最佳延迟。TDA还将应用于结构健康监测中的Z24桥案例研究,该研究将用来根据收集数据的条件对不同数据分区进行分析。从表层数据分析中得出的指标用于对目前分区之间的数据进行显著的温度变化。所显示的结果显示的是,目前的温度变化的程度大于目前的温度变化。