Wavelet-based functional mixed models

Summary: Increasingly, scientific studies yield functional data, in which the ideal units of observation are curves and the observed data consist of sets of curves sampled on a fine grid. In this paper, we present new methodology that generalizes the linear mixed model to the functional mixed model framework, with model fitting done using a Bayesian wavelet-based approach. This method is flexible, allowing functions of arbitrary form and the full range of fixed effects structures and between-curve covariance structures available in the mixed model framework. It yields nonparametric estimates of the fixed and random effects functions as well as the various between-curve and within-curve covariance matrices. The functional fixed effects are adaptively regularized as a result of the nonlinear shrinkage prior imposed on the fixed effects ’ wavelet coefficients, and the random effect functions experience a form of adaptively regularization because of the separately estimated variance components for each wavelet coefficient. Because we have posterior samples for all model quantities, we can perform pointwise or joint Bayesian inference or prediction on the quantities of the model. The adaptiveness of the method makes it especially appropriate for modeling irregular functional data characterized by numerous local features like peaks