A Robust Bayesian Approach to Function Registration in ℝ^1
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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 functional alignment methods cannot handle. A pair of functions can also have multiple optimal alignment solutions which is not addressed in current literature. In this paper, we present a flexible Bayesian approach to functional alignment which appropriately accounts for noise in the data without any pre-smoothing necessary. Additionally, by running parallel MCMC chains, our method can account for multiple optimal alignments via the multi-modal posterior distribution of the warping functions. To most efficiently sample the warping functions, our approach relies on the ∞-HMC sampling algorithm described in Beskos et al. (2017), a modification of the standard Hamiltonian Monte Carlo to be well-defined on the infinite-dimensional Hilbert space. We apply this novel and flexible Bayesian alignment method to both simulated data and real data to show its efficiency to handle noisy functions and successfully account for multiple optimal alignments in the posterior, characterizing the uncertainty surrounding the warping functions.
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