  The existence of characteristic structure, or shape, in complex data sets has
been recognized as increasingly important for mathematical data analysis. This
realization has motivated the development of new tools such as persistent
homology for exploring topological invariants, or features, in large data sets.
In this paper we apply persistent homology to the characterization of gas
plumes in time dependent sequences of hyperspectral cubes, i.e. the analysis of
4-way arrays. We investigate hyperspectral movies of Long-Wavelength Infrared
data monitoring an experimental release of chemical simulant into the air. Our
approach models regions of interest within the hyperspectral data cubes as
points on the real Grassmann manifold $G(k, n)$ (whose points parameterize the
$k$-dimensional subspaces of $\mathbb{R}^n$), contrasting our approach with the
more standard framework in Euclidean space. An advantage of this approach is
that it allows a sequence of time slices in a hyperspectral movie to be
collapsed to a sequence of points in such a way that some of the key structure
within and between the slices is encoded by the points on the Grassmann
manifold. This motivates the search for topological features, associated with
the evolution of the frames of a hyperspectral movie, within the corresponding
points on the Grassmann manifold. The proposed mathematical model affords the
processing of large data sets while retaining valuable discriminatory
information. In this paper, we discuss how embedding our data in the Grassmann
manifold, together with topological data analysis, captures dynamical events
that occur as the chemical plume is released and evolves.
