  We propose a model for functional data registration that compares favorably
to the best methods of functional data registration currently available. It
also extends current inferential capabilities for unregistered data by
providing a flexible probabilistic framework that 1) allows for functional
prediction in the context of registration and 2) can be adapted to include
smoothing and registration in one model. The proposed inferential framework is
a Bayesian hierarchical model where the registered functions are modeled as
Gaussian processes. To address the computational demands of inference in
high-dimensional Bayesian models, we propose an adapted form of the variational
Bayes algorithm for approximate inference that performs similarly to MCMC
sampling methods for well-defined problems. The efficiency of the adapted
variational Bayes (AVB) algorithm allows variability in a predicted registered,
warping, and unregistered function to be depicted separately via bootstrapping.
Temperature data related to the el-ni\~no phenomenon is used to demonstrate the
unique inferential capabilities for prediction provided by this model.
