  In many applications, smooth processes generate data that is recorded under a
variety of observation regimes, such as dense, sparse or fragmented
observations that are often contaminated with error. The statistical goal of
registering and estimating the individual underlying functions from discrete
observations has thus far been mainly approached sequentially without formal
uncertainty propagation, or in an application-specific manner. We propose a
unified Bayesian framework for simultaneous registration and estimation, which
is flexible enough to accommodate inference on individual functions under
general observation regimes. Our ability to do this relies on the specification
of strongly informative prior models over the amplitude component of function
variability. We provide two strategies for this critical choice: a data-driven
approach that defines an empirical basis for the amplitude subspace based on
training data, and a shape-restricted approach when the relative location and
number of local extrema is well-understood. The proposed methods build on
elastic functional data analysis, which separately models amplitude and phase
variability inherent in functional data. We emphasize the importance of
uncertainty quantification and visualization of these two components as they
provide complementary information about the estimated functions. We validate
the framework using simulations and real applications to medical imaging and
biometrics.
