Uniform convergence of local Fréchet regression and time warping for metric-space-valued trajectories
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For real-valued functional data, it is well known that failure to separate amplitude variation from phase variation may contaminate subsequent statistical analysis and time warping methods to address this have been extensively investigated. However, much less is known about the phase variation problem for object-valued random processes that take values in a general metric space which by default does not have a linear structure. We introduce here a method to estimate warping functions by pairwise synchronization. An important starting point is uniform convergence of local Fréchet regression, which is a key result in its own right that we establish. We show how this result can be harnessed to obtain consistency and rates of convergence of the proposed warping function estimates. The finite-sample performance of the proposed warping method is evaluated in simulation studies and data illustrations, including yearly mortality distributions across countries and Zürich longitudinal growth data.
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