Functional data registration is a necessary processing step for many applications. The observed data can be inherently noisy, often due to measurement error or natural process uncertainty; which most functional alignment methods cannot handle. A pair of functions can also have multiple optimal alignment solutions, which is not addressed in current literature. In this paper, a flexible Bayesian approach to functional alignment is presented, which appropriately accounts for noise in the data without any pre-smoothing required. Additionally, by running parallel MCMC chains, the method can account for multiple optimal alignments via the multi-modal posterior distribution of the warping functions. To most efficiently sample the warping functions, the approach relies on a modification of the standard Hamiltonian Monte Carlo to be well-defined on the infinite-dimensional Hilbert space. This flexible Bayesian alignment method is applied to both simulated data and real data sets to show its efficiency in handling noisy functions and successfully accounting for multiple optimal alignments in the posterior; characterizing the uncertainty surrounding the warping functions.

功能数据注册是许多应用程序的必要处理步骤。观察到的数据可能具有固有的噪声，通常是由于测量误差或自然过程的不确定性；大多数功能对齐方法无法处理。一对函数也可以有多个最优对齐解决方案，这在当前文献中没有解决。在本文中，提出了一种灵活的贝叶斯函数对齐方法，它适当地考虑了数据中的噪声，而无需任何预平滑。此外，通过运行并行 MCMC 链，该方法可以通过扭曲函数的多模态后验分布来考虑多个最佳对齐。为了最有效地采样翘曲函数，该方法依赖于标准哈密顿蒙特卡罗的修改，以便在无限维希尔伯特空间上明确定义。这种灵活的贝叶斯对齐方法适用于模拟数据和真实数据集，以显示其处理噪声函数的效率，并成功地考虑了后验中的多个最佳对齐；表征围绕翘曲函数的不确定性。