Department of Mathematics

Functional data analysis on nonlinear manifolds has drawn recent interest. We propose an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. The proposed Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fr echet mean function, and then performing a classical multivariate functional principal component analysis on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained by mapping back with exponential maps. The tangent-space approximation through functional principal component analysis is shown to be well-behaved in terms of controlling the residual variation if the Riemannian manifold has nonnegative curvature. We derive uniform convergence rates for the model components, including the mean function, covariance function, eigenfunctions, and functional principal component scores and apply the proposed methodology to obtain a novel framework for the analysis of longitudinal compositional data. This is achieved by mapping longitudinal compositional data to trajectories on the sphere, and we illustrate this approach with longitudinal fruit fly behavior patterns. The proposed Riemannian functional principal component analysis is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications that include flight trajectories, as well as in simulations. RFPCA is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores.