Bayesian Adaptive Selection of Basis Functions for Functional Data Representation

TL;DR

This paper introduces a Bayesian method using a Gibbs sampler for adaptive basis function selection in functional data analysis, effectively determining the number and type of basis functions while quantifying uncertainty.

Contribution

It presents a novel Bayesian approach with Bernoulli latent variables for adaptive basis selection, applicable to multiple curves and capable of handling data variability and uncertainty.

Findings

Accurately estimates basis coefficients.

Effectively identifies true basis functions.

Performs well compared to LASSO methods.

Abstract

Considering the context of functional data analysis, we developed and applied a new Bayesian approach via Gibbs sampler to select basis functions for a finite representation of functional data. The proposed methodology uses Bernoulli latent variables to assign zero to some of the basis function coefficients with a positive probability. This procedure allows for an adaptive basis selection since it can determine the number of bases and which should be selected to represent functional data. Moreover, the proposed procedure measures the uncertainty of the selection process and can be applied to multiple curves simultaneously. The methodology developed can deal with observed curves that may differ due to experimental error and random individual differences between subjects, which one can observe in a real dataset application involving daily numbers of COVID-19 cases in Brazil. Simulation studies show the main properties of the proposed method, such as its accuracy in estimating the coefficients and the strength of the procedure to find the true set of basis functions. Despite having been developed in the context of functional data analysis, we also compared the proposed model via simulation with the well-established LASSO and Bayesian LASSO, which are methods developed for non-functional data.