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 estimate 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 EEG
signals and ozone concentrations. 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
