Intrinsic Riemannian functional data analysis for sparse longitudinal observations

A new framework is developed to intrinsically analyze sparsely observed Riemannian functional data. It features four innovative components: a frame-independent covariance function, smooth vector bundle termed bundle, parallel transport and metric on the bundle. The introduced intrinsic function links estimation of structure smoothing problems that involve raw observations derived from data, while provides rigorous mathematical foundation for formulating such problems. together make it possible measure fidelity fit function. They also play critical role in quantifying quality estimators As an illustration, based proposed framework, we develop local linear estimator its theoretical properties provide numerical demonstration via simulated real data sets. feature makes applicable not only Euclidean submanifolds but manifolds without canonical ambient space.