  Although there are many methods for functional data analysis (FDA), little
emphasis is put on characterizing variability among volatilities of individual
functions. In particular, certain individuals exhibit erratic swings in their
trajectory while other individuals have more stable trajectories. There is
evidence of such volatility heterogeneity in blood pressure trajectories during
pregnancy, for example, and reason to suspect that volatility is a biologically
important feature. Most FDA models implicitly assume similar or identical
smoothness of the individual functions, and hence can lead to misleading
inferences on volatility and an inadequate representation of the functions. We
propose a novel class of FDA models characterized using hierarchical stochastic
differential equations. We model the derivatives of a mean function and
deviation functions using Gaussian processes, while also allowing covariate
dependence including on the volatilities of the deviation functions. Following
a Bayesian approach to inference, a Markov chain Monte Carlo algorithm is used
for posterior computation. The methods are tested on simulated data and applied
to blood pressure trajectories during pregnancy.
