  Due to developments in instruments and computers, functional observations are
increasingly popular. However, effective methodologies for flexibly estimating
the underlying trends with valid uncertainty quantification for a sequence of
functional data (e.g. functional time series) are still scarce. In this work,
we develop a locally adaptive smoothing method, called functional horseshoe
smoothing, by introducing a shrinkage prior to the general order of differences
of functional variables. This allows us to capture abrupt changes by making the
most of the shrinkage capability and also to assess uncertainty by Bayesian
inference. The fully Bayesian framework allows the selection of the number of
basis functions via the posterior predictive loss. We provide theoretical
properties of the model, which support the shrinkage ability. Also, by taking
advantage of the nature of functional data, this method is able to handle
heterogeneously observed data without data augmentation. Simulation studies and
real data analysis demonstrate that the proposed method has desirable
properties.
