Statistics on the space of trajectories for longitudinal data analysis

Statistical analysis of longitudinal data is a significant problem in Biomedical imaging applications. In the recent past, several researchers have developed mathematically rigorous methods based on differential geometry and statistics to tackle the problem of statistical analysis of longitudinal neuroimaging data. In this paper, we present a novel formulation of the longitudinal data analysis problem by identifying the structural changes over time (describing the trajectory of change) to a product Riemannian manifold endowed with a Riemannian metric and a probability measure. We present theoretical results showing that the maximum likelihood estimate of the mean and median of a Gaussian and Laplace distribution respectively on the product manifold yield the Fréchet mean and median respectively. We then present efficient recursive estimators for these intrinsic parameters and use them in conjunction with a nearest neighbor (NN) classifier to classify MR brain scans (acquired from the publicly available OASIS database) of patients with and without dementia.