A Nonlinear Hierarchical Model for Longitudinal Data on Manifolds

TL;DR

This paper introduces a hierarchical model for analyzing longitudinal manifold-valued data, extending geodesic models to include Bézier spline trends, enabling better capture of complex temporal patterns in medical studies.

Contribution

It proposes a novel hierarchical model using Bézier splines on manifolds, with an efficient Riemannian metric for comparing shape trends, applicable to medical longitudinal data.

Findings

Successfully modeled complex trends in osteoarthritis data

Enabled classification of disease progression using the new model

Validated the approach on real longitudinal medical data

Abstract

Large longitudinal studies provide lots of valuable information, especially in medical applications. A problem which must be taken care of in order to utilize their full potential is that of correlation between intra-subject measurements taken at different times. For data in Euclidean space this can be done with hierarchical models, that is, models that consider intra-subject and between-subject variability in two different stages. Nevertheless, data from medical studies often takes values in nonlinear manifolds. Here, 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 is often not the case (e.g., periodic motion or processes with saturation). We propose a hierarchical model for manifold-valued data that extends this to include trends along higher-order curves, namely B\'ezier splines in the manifold. To this end, we present a principled way of comparing shape trends in terms of a functional-based Riemannian metric. Remarkably, this metric allows efficient, yet simple computations by virtue of a variational time discretization requiring only the solution of regression problems. We validate our model on longitudinal data from the osteoarthritis initiative, including classification of disease progression.