Bayesian Function-on-Scalars Regression for High-Dimensional Data

We develop a fully Bayesian framework for function-on-scalars regression with many predictors. The functional data response is modeled nonparametrically using unknown basis functions, which produces a flexible and data-adaptive functional basis. We incorporate shrinkage priors that effectively remov...

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Bibliographic Details
Published in:Journal of computational and graphical statistics 2020-07, Vol.29 (3), p.629-638
Main Authors: Kowal, Daniel R., Bourgeois, Daniel C.
Format: Article
Language:English
Subjects:
Actigraphy data
Algorithms
Basis functions
Bayesian analysis
Computer simulation
Decision theory
Factor model
Feature selection
Functionals
MCMC
Model accuracy
Parameter estimation
Regression models
Scalars
Shrinkage
Variable selection
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Summary:We develop a fully Bayesian framework for function-on-scalars regression with many predictors. The functional data response is modeled nonparametrically using unknown basis functions, which produces a flexible and data-adaptive functional basis. We incorporate shrinkage priors that effectively remove unimportant scalar covariates from the model and reduce sensitivity to the number of (unknown) basis functions. For variable selection in functional regression, we propose a decision theoretical posterior summarization technique, which identifies a subset of covariates that retains nearly the predictive accuracy of the full model. Our approach is broadly applicable for Bayesian functional regression models, and unlike existing methods provides joint rather than marginal selection of important predictor variables. Computationally scalable posterior inference is achieved using a Gibbs sampler with linear time complexity in the number of predictors. The resulting algorithm is empirically faster than existing frequentist and Bayesian techniques, and provides joint estimation of model parameters, prediction and imputation of functional trajectories, and uncertainty quantification via the posterior distribution. A simulation study demonstrates improvements in estimation accuracy, uncertainty quantification, and variable selection relative to existing alternatives. The methodology is applied to actigraphy data to investigate the association between intraday physical activity and responses to a sleep questionnaire. Supplementary materials for this article are available online.
ISSN:1061-8600
1537-2715
DOI:10.1080/10618600.2019.1710837