Bayesian smoothing and estimation of functional data with approximation by basis functions

In this paper, we propose conducting posterior inferences to a Bayesian hierarchical model with approximation by basis functions, with the purpose of smoothing all functional observations simultaneously as well as estimating the functional mean and covariance, when there are multiple signals observed at random grids. In order to apply the model on functional data with extremely high-dimensionality and functional data observed on random grids, we approximate the infinite dimensional model on a finite dimensional space of basis functions. Posterior inferences to all parameters in the Bayesian hierarchical model --- including smooth functional signals, mean and covariance --- are conducted through the posterior coefficient samples of the basis functions. Numerical experiments with simulated data (both stationary and nonstationary) demonstrate that our method with approximation by basis functions generates similar results as the same Bayesian hierarchical model without approximation (i.e., with posterior references on observation grids) in the workable-common-grid case. Moreover, both simulation and real case studies show that our method provides the best estimates of the signals, mean and covariance functions, in the random-grid case and high dimensional common-grid case. Further, more accurate estimates of the mean and covariance functions are obtained than if one smooths the individual observations and computes a mean and covariance from the individually smoothed estimates of the signals. This basis function approximation method also shows how one can interpolate a covariance matrix to obtain a covariance function.