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Abstract:
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Functional data registration is a necessary processing step for many applications. The observed data can be inherently noisy, often due to measurement error or natural process uncertainty, which most existing functional alignment methods cannot handle. It is also the case that there could be multiple optimal alignments for a pair of functions which is not demonstrated in current literature. In this paper, we present a robust Bayesian approach to pairwise and multiple functional alignment which appropriately accounts for noise in the data without any preprocessed smoothing necessary. Additionally, by running parallel MCMC chains, our method can also account for multiple optimal alignments, resulting in multi-modal posterior distribution of the warping functions. To most efficiently sample the warping functions, our approach relies on the $\infty$-HMC sampling algorithm described in Beskos et al. (2017), a modification of the standard Hamiltonian Monte Carlo (developed to be well-defined on the infinite-dimensional Hilbert space. We show that this approach is robust to two common challenges with functional data: noisy functions and multiple possible optimal alignments. We apply this
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