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

Abstract

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 theoretic 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.

Integer-Valued Functional Data Analysis for Measles Forecasting

“…Models and are designed to reproduce the classical setting for functional data analysis, where real‐valued functional data are modeled as noisy observations of a smooth function , commonly via a basis expansion. For maximal flexibility, we model the basis functions as smooth yet unknown functions (subject to identifiability constraints), which produces a data‐adaptive functional basis (Kowal, ; Kowal and Bourgeois, ). For modeling measles counts, learning corresponds to learning the intrayear seasonalities, with year‐specific weights given by the coefficients (see Section ).…”

“…For the autoregressive coefficient in , we assume the prior to constrain for stationarity, and select and for measles and simulated data. These choices were successfully applied for Gaussian functional data in Kowal () and Kowal and Bourgeois ().…”

“…While many options exist for the basis functions in , we follow the highly flexible approach of Kowal et al . () for unknown with the computational improvements in Kowal () and Kowal and Bourgeois (). Modeling each basis function as unknown (a) produces a data‐adaptive functional basis and (b) incorporates the uncertainty about into the posterior distribution.…”

“…For smoothness, we assume the prior , where is a known roughness penalty matrix, such as for the second derivative of , and is an unknown smoothing parameter. Details on the construction of and , including useful reparametrizations, are given in Kowal (). For identifiability, we enforce the matrix orthonormality constraint , where is the basis matrix and is the basis function evaluated at the observation points .…”

“…For identifiability, we enforce the matrix orthonormality constraint , where is the basis matrix and is the basis function evaluated at the observation points . The matrix orthonormality constraint, coupled with the ordering implied by the MGP prior, is sufficient for identifiability, and may be leveraged to improve computational efficiency in sampling the coefficients (Kowal, ; Kowal and Bourgeois, ).…”