This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of data, from the homology of a simplicial complex, calculated over a range of values. The required background theory and a method of computing persistent homology is presented here, with applications specific to structural health monitoring. These results allow for topological inference and the ability to deduce features in higher-dimensional data, that might otherwise be overlooked. A simplicial complex is constructed for data for a given distance parameter. This complex encodes information about the local proximity of data points. A singular homology value can be calculated from this simplicial complex. Extending this idea, the distance parameter is given for a range of values, and the homology is calculated over this range. The persistent homology is a representation of how the homological features of the data persist over this interval. The result is characteristic to the data. A method that allows for the comparison of the persistent homology for different data sets is also discussed.

翻译：本文旨在讨论一种量化数据“形状”的方法,其方法称为“表层数据分析”。 地形数据分析中的主要工具是持久性同质学;这是测量数据形状的一种手段,它来自简化复合物的同质学,根据一系列数值计算。 此处介绍了所需的背景理论和计算持久性同质学的方法, 以及结构健康监测的具体应用。 这些结果允许在结构健康监测中得出可能被忽视的较高维度数据中的表层推论和能力。 为特定距离参数的数据构建了一个简单复合体。 这个复杂的复合体将数据点在本地的相近性信息编码。 从这个简单复杂体中可以计算出单一同性值。 扩展这个概念, 为一系列数值提供距离参数, 并计算同质学。 持久性同质学代表了数据在这一间隔中持续存在的同性特征。 结果是数据的特征。 一种可以比较不同数据集的持久性同质学的方法也得到了讨论。