  Functional data are defined as realizations of random functions (mostly
smooth functions) varying over a continuum, which are usually collected with
measurement errors on discretized grids. In order to accurately smooth noisy
functional observations and deal with the issue of high-dimensional observation
grids, we propose a novel Bayesian method based on the Bayesian hierarchical
model with a Gaussian-Wishart process prior and basis function representations.
We first derive an induced model for the basis-function coefficients of the
functional data, and then use this model to conduct posterior inference through
Markov chain Monte Carlo. Compared to the standard Bayesian inference that
suffers serious computational burden and unstableness for analyzing
high-dimensional functional data, our method greatly improves the computational
scalability and stability, while inheriting the advantage of simultaneously
smoothing raw observations and estimating the mean-covariance functions in a
nonparametric way. In addition, our method can naturally handle functional data
observed on random or uncommon grids. Simulation and real studies demonstrate
that our method produces similar results as the standard Bayesian inference
with low-dimensional common grids, while efficiently smoothing and estimating
functional data with random and high-dimensional observation grids where the
standard Bayesian inference fails. In conclusion, our method can efficiently
smooth and estimate high-dimensional functional data, providing one way to
resolve the curse of dimensionality for Bayesian functional data analysis with
Gaussian-Wishart processes.
