Registration of multivariate functional data involves handling of both cross-component and cross-observation phase variations. Allowing for the two phase variations to be modelled as general diffeomorphic time warpings, in this work we focus on the hitherto unconsidered setting where phase variation of the component functions are spatially correlated. We propose an algorithm to optimize a metric-based objective function for registration with a novel penalty term that incorporates the spatial correlation between the component phase variations through a kriging prediction of an appropriate phase random field. The penalty term encourages the overall phase at a particular location to be similar to the spatially weighted average phase in its neighbourhood, and thus engenders a regularization that prevents over-alignment. Utility of the registration method, and its superior performance compared to methods that fail to account for the spatial correlation, is demonstrated through performance on simulated examples and two multivariate functional datasets pertaining to electroencephalogram signals and ozone concentration functions. The generality of the framework opens up the possibility for extension to settings involving different forms of correlation between the component functions and their phases.

多变量函数数据的配准涉及跨组件和跨观察相位变化的处理。允许将两个相位变化建模为一般微分同胚时间扭曲，在这项工作中，我们重点关注迄今为止未考虑的设置，其中分量函数的相位变化在空间上相关。我们提出了一种算法来优化基于度量的目标函数，以使用新颖的惩罚项进行配准，该惩罚项通过适当相位随机场的克里格预测结合了分量相位变化之间的空间相关性。惩罚项鼓励特定位置的整体相位与其邻近区域的空间加权平均相位相似，从而产生防止过度对准的正则化。通过模拟示例以及与脑电图信号和臭氧浓度函数有关的两个多元函数数据集的性能，证明了配准方法的实用性及其优于无法考虑空间相关性的方法的性能。该框架的通用性为扩展到涉及组件功能及其阶段之间不同形式的相关性的设置提供了可能性。