On topological data analysis for SHM; an introduction to persistent homology

TL;DR

This paper introduces topological data analysis, specifically persistent homology, as a method to quantify the shape of data for structural health monitoring, enabling the detection of features in complex, high-dimensional datasets.

Contribution

It provides an accessible introduction to persistent homology, its computation, and its application to structural health monitoring data analysis.

Findings

Persistent homology captures data features across multiple scales.

Method enables comparison of different datasets' topological features.

Application demonstrates potential for detecting structural anomalies.

Abstract

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.