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 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.

翻译：函数配准，亦称对齐，一直是函数数据分析领域的基本问题之一。经典配准方法（如Fisher-Rao对齐）侧重于估计函数间最优时间扭曲函数。近期研究中，一种关于时间扭曲的模型受到更多关注，可作为先验项与经典方法（作为似然项）结合于贝叶斯框架中。贝叶斯方法已被证明优于经典方法。然而，其时间扭曲先验模型通常基于非线性近似，可能引入不精确性与低效性。为克服这些问题，我们提出一种新的贝叶斯方法，采用可提供线性表示的先验，并能在时间扭曲上有效利用各类随机过程（高斯或非高斯）。时间扭曲计算无需线性化近似，后验分布可通过常规马尔可夫链蒙特卡洛方法获得。我们通过多个仿真算例深入研究了先验对函数配准性能的影响，证明了新框架相较于现有方法的优越性。最后，我们将新方法应用于真实数据集，获得了理想的对齐结果。