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We consider the problem of nonparametric registration of
functional data that have been subjected to random deformation (warping)
of their time scale. The separation of this phase variation
("horizontal" variation) from the amplitude variation ("vertical"
variation) is crucial in order to properly conduct further analyses,
which otherwise can be severely distorted. We determine precise
conditions under which the two forms of variation are identifiable,
under minimal assumptions on the form of the warp maps. We show that
these conditions are sharp, by means of counterexamples. We then present
a nonparametric registration method based on a "local variation
measure", which bridges the registration problem of functional data and
the problem of optimal transportation. The method is proven to
consistently estimate the warp maps from discretely observed data,
without requiring any penalization or tuning on the warp maps
themselves. This circumvents the problem of over/underregistration often
encountered in practice. Similar results hold in the presence of
measurement error, with the addition of a preprocessing smoothing step.
A detailed theoretical investigation of the strong consistency and the
weak convergence properties of the resulting functional estimators is
carried out including the rates of convergence. We also give a
theoretical study of the impact of deviating from the identifiability
conditions, quantifying it in terms of the spectral gap of the amplitude
variation. Numerical experiments demonstrate the good finite sample
performance of our method, and the methodology is further illustrated by
means of a data analysis.
This work is joint with Victor M. Panaretos. |