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Stochastic Volatility Regression for Functional Data Dynamics

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.