A Note on Bivariate Smoothing for Two-Dimensional Functional Data

In this paper we study a bivariate smoothing approach for estimating multiple functional parameters for functional data with a two-dimensional domain. We present a penalized regression framework for smoothing with the purpose to: (a) facilitate the estimation of the smooth overall bivariate mean function of two-dimensional functional data, (b) enable the estimation of the functional effect of a scalar covariate, (c) accommodate completely or incompletely sampled data, (d) implement the fitting approach using available statistical software, and (e) construct pointwise approximate confidence intervals for multiple bivariate functional parameters. Implementation results from simulation studies show that these methods perform very well in practice. We illustrate the usefulness of the bivariate smoothing approach to several real datasets, including applications to electricity demand.