On Topological Data Analysis for Structural Dynamics: An Introduction to Persistent Homology

Abstract

Topological methods can provide a way of proposing new metrics and methods of scrutinizing data, that otherwise may be overlooked. A method of quantifying the shape of data, via a topic called topological data analysis (TDA) will be introduced. The main tool of 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 are briefly discussed in this work. Ideas from topological data analysis are then used for nonlinear dynamics to analyze 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 scrutinize 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.