论文标题
部分可观测时空混沌系统的无模型预测
On topological data analysis for SHM; an introduction to persistent homology
论文作者
论文摘要
本文旨在通过一种称为拓扑数据分析的方法来讨论一种量化数据“形状”的方法。拓扑数据分析中的主要工具是持续的同源性。这是一种从简单复合物的同源性测量数据形状的一种方法，该方法是在一系列值范围内计算出来的。此处介绍了所需的背景理论和一种计算持续同源性的方法，并具有针对结构健康监测的应用。这些结果允许拓扑推理和在高维数据中推断功能的能力，否则可能会被忽略。 为给定距离参数的数据构建了一个简单复合物。该复合物编码有关数据点局部接近性的信息。可以从这个简单复合物中计算出奇异的同源性值。扩展此想法，为一系列值提供了距离参数，并且在此范围内计算同源性。持续的同源性是在此间隔内如何持续存在数据的同源特征的一种表示。结果是数据的特征。还讨论了一种允许对不同数据集进行持续同源性比较的方法。
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.
