A trajectory is a spatial representation of a time series of states determined either by a value for each state or a procedure by which the pairwise distances between the states is used to create a location in some space for each state. In this study, the states are the covariance matrices at different age centers, and the distances are the Riemannian (geodesic) distances in the non-Euclidean space of positive definite symmetric (PDS) matrices, as described above. Two different approaches are then used to describe the trajectories as determined by the pairwise distances between the matrices. The first uses multidimensional scaling (MDS) applied to the set of geodesic distances between all pairs of the covariances matrices of each type to provide a Euclidean approximation suitable for large scale description and analysis. The second works directly in the Riemannian space to characterize the progression and deviation of the observed trajectory from a trajectory along a geodesic determined by two matrices on the trajectory, providing a measure of the “non-linearity” of the observed trajectory. The analyses based on direct calculations in the non-Euclidean space of the covariance matrices are based on ideas in Fletcher et al. (2004) and Fletcher and Joshi (2007).
