Fast Bayesian Basis Selection for Functional Data Representation with Correlated Errors

Functional data analysis finds widespread application across various fields. While functional data are intrinsically infinite-dimensional, in practice, they are observed only at a finite set of points, typically over a dense grid. As a result, smoothing techniques are often used to approximate the observed data as functions. In this work, we propose a novel Bayesian approach for selecting basis functions for smoothing one or multiple curves simultaneously. Our method differentiates from other Bayesian approaches in two key ways: (i) by accounting for correlated errors and (ii) by developing a variational EM algorithm, which is faster than MCMC methods such as Gibbs sampling. Simulation studies demonstrate that our method effectively identifies the true underlying structure of the data across various scenarios, and it is applicable to different types of functional data. Our variational EM algorithm not only recovers the basis coefficients and the correct set of basis functions but also estimates the existing within-curve correlation. When applied to the motorcycle and Canadian weather datasets, our method demonstrates comparable, and in some cases superior, performance in terms of adjusted $R^2$ compared to regression splines, smoothing splines, Bayesian LASSO and LASSO. Our proposed method is implemented in R and codes are available at https://github.com/acarolcruz/VB-Bases-Selection.