🤖 AI Summary
To address the challenges of basis function selection and modeling observation error correlations in functional data smoothing, this paper proposes a fast Bayesian basis selection method that supports both univariate and multivariate functional data jointly. It is the first to explicitly model the correlation structure of observational errors within a Bayesian framework. The method employs variational inference combined with an EM algorithm, replacing conventional MCMC sampling to achieve substantial computational speedup without compromising estimation accuracy. Extensive evaluations on simulated data and real-world applications—including motorcycle acceleration and Canadian weather datasets—demonstrate significant improvements in adjusted R², accurate recovery of basis coefficients, optimal basis sets, and within-curve correlation patterns. An open-source R implementation is publicly available.
📝 Abstract
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.