A new framework for Bayesian function registration

Function registration, also referred to as alignment, has been one of the fundamental problems in the field of functional data analysis. Classical registration methods such as the Fisher-Rao alignment focus on estimating optimal time warping function between functions. In recent studies, a model on time warping has attracted more attention, and it can be used as a prior term to combine with the classical method (as a likelihood term) in a Bayesian framework. The Bayesian approaches have been shown improvement over the classical methods. However, its prior model on time warping is often based on a nonlinear approximation, which may introduce inaccuracy and inefficiency. To overcome these problems, we propose a new Bayesian approach by adopting a prior which provides a linear representation and various stochastic processes (Gaussian or non-Gaussian) can be effectively utilized on time warping. No linearization approximation is needed in the time warping computation, and the posterior can be obtained via a conventional Markov Chain Monte Carlo approach. We thoroughly investigate the impact of the prior on the performance of functional registration with multiple simulation examples, which demonstrate the superiority of the new framework over the previous methods. We finally utilize the new method in a real dataset and obtain desirable alignment result.