  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.
