  In many modern applications, discretely-observed data may be naturally
understood as a set of functions. Functional data often exhibit two confounded
sources of variability: amplitude (y-axis) and phase (x-axis). The extraction
of amplitude and phase, a process known as registration, is essential in
exploring the underlying structure of functional data in a variety of areas,
from environmental monitoring to medical imaging. Critically, such data are
often gathered sequentially with new functional observations arriving over
time. Despite this, most available registration procedures are only applicable
to batch learning, leading to inefficient computation. To address these
challenges, we introduce a Bayesian framework for sequential registration of
functional data, which updates statistical inference as new sets of functions
are assimilated. This Bayesian model-based sequential learning approach
utilizes sequential Monte Carlo sampling to recursively update the alignment of
observed functions while accounting for associated uncertainty. As a result,
distributed computing, which is not generally an option in batch learning,
significantly reduces computational cost. Simulation studies and comparisons to
existing batch learning methods reveal that the proposed approach performs well
even when the target posterior distribution has a challenging structure. We
apply the proposed method to three real datasets: (1) functions of annual
drought intensity near Kaweah River in California, (2) annual sea surface
salinity functions near Null Island, and (3) PQRST complexes segmented from an
electrocardiogram signal.
