Modeling sparse longitudinal data on Riemannian manifolds.

Area of Science:

• Statistics
• Data Analysis
• Manifold Learning

Background:

• Longitudinal data collection often involves repeated measurements on Riemannian manifolds.
• Analyzing such data is complex due to sparse observations and nonlinear manifold constraints.

Purpose of the Study:

• To propose an intrinsic functional principal component analysis (iFPCA) for longitudinal Riemannian data.
• To address challenges in analyzing sparse and nonlinear manifold-constrained data.
• To enable dimension reduction and imputation of manifold-valued trajectories.

Main Methods:

• Estimating the mean curve using local Fréchet regression.
• Smoothing the covariance structure of linearized data on tangent spaces.
• Utilizing leading principal components and best linear unbiased prediction for dimension reduction and imputation.

Main Results:

• The proposed method achieves state-of-the-art convergence rates for mean and covariance function estimation.
• Demonstrated effectiveness in analyzing longitudinal brain connectivity data (symmetric positive definite matrices).
• Showcased interpretable eigenfunctions and principal component scores for emotion compositional data.

Conclusions:

• The intrinsic functional principal component analysis provides a robust framework for longitudinal Riemannian data.
• The method offers significant improvements in analyzing complex, real-world datasets, including neuroimaging and socio-economic data.
• This approach facilitates deeper insights into dynamic processes evolving on nonlinear spaces.