Bayesian adaptive selection of basis functions for functional data representation

Abstract

Considering the context of functional data analysis, we developed and applied a new Bayesian approach via the 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 ones 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.

Keywords:

• Bayesian inference
• functional data
• functional data analysis
• basis selection
• latent variable

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, by the Natural Sciences and Engineering Research Council of Canada (NSERC), grant number RGPIN-2019-05915, and FAPESP grant 2018/04654-9.