A Nonlinear Hierarchical Model for Longitudinal Data on Manifolds

Large longitudinal studies provide information that is particularly valuable in medical studies. A problem that must be solved in order to realize the full potential is the correlation between intra-subject measurements taken at different times. For data in Euclidean space this can be achieved with hierarchical models, i.e., models that account for intra-subject and between-subject variability at two different levels. How-ever, data from medical studies often take values in nonlinear manifolds. For such data, as a first step, geodesic hierarchical models have been developed that generalize the linear ansatz by assuming that time-induced intra-subject variations occur along a generalized straight line in the manifold. However, this assumption often does not hold, for ex-ample in periodic motions or processes with saturation. As a more general alternative we propose a hierarchical model for manifold-valued data that incorporates trends along higher-order curves, namely Bézier splines in the manifold. To this end, we present a principled way of comparing shape trends in terms of a functional-based Riemannian metric. Remark-ably, this metric allows efficient, yet simple computations by virtue of a variational time discretization that only requires solving regression problems. We validate our model using longitudinal data from the Osteoarthritis Initiative, including classification of disease progression.