  Diffusion tensor imaging provides important information on tissue structure
and orientation of fiber tracts in brain white matter in vivo. It results in
diffusion tensors, which are $3\times3$ symmetric positive definite (SPD)
matrices, along fiber bundles. This paper develops a functional data analysis
framework to model diffusion tensors along fiber tracts as functional data in a
Riemannian manifold with a set of covariates of interest, such as age and
gender. We propose a statistical model with varying coefficient functions to
characterize the dynamic association between functional SPD matrix-valued
responses and covariates. We calculate weighted least squares estimators of the
varying coefficient functions for the log-Euclidean metric in the space of SPD
matrices. We also develop a global test statistic to test specific hypotheses
about these coefficient functions and construct their simultaneous confidence
bands. Simulated data are further used to examine the finite sample performance
of the estimated varying coefficient functions. We apply our model to study
potential gender differences and find a statistically significant aspect of the
development of diffusion tensors along the right internal capsule tract in a
clinical study of neurodevelopment.
