Robust, Adaptive Functional Regression in Functional Mixed Model Framework

Functional data are increasingly encountered in scientific studies, and their high dimensionality and

complexity lead to many analytical challenges. Various methods for functional data analysis have been

developed, including functional response regression methods that involve regression of a functional response on

univariate/multivariate predictors with nonparametrically represented functional coefficients.

In existing methods, however, the functional regression can be sensitive to outlying curves and outlying regions

of curves, so is not robust. In this paper, we introduce a new Bayesian method, robust functional mixed models (R-FMM),

for performing robust functional regression within the general functional mixed model framework, which includes multiple continuous or categorical predictors and random effect functions accommodating potential between-function correlation

induced by the experimental design. The underlying model involves a hierarchical scale mixture model

for the fixed effects, random effect and residual error functions. These modeling assumptions across curves result in robust

nonparametric estimators of the fixed and random effect functions which down-weight outlying curves and

regions of curves, and produce statistics that can be used to flag global and local outliers. These assumptions also lead to distributions across wavelet coefficients that have outstanding sparsity and adaptive shrinkage properties, with

great flexibility for the data to determine the sparsity and the heaviness of the tails.

Together with the down-weighting of outliers, these within-curve properties lead to fixed and random effect

function estimates that appear in our simulations to be remarkably adaptive in their ability to remove spurious features yet retain true features of the functions. We have developed general code to implement this fully Bayesian method that is automatic, requiring the user to only provide the functional data and design matrices. It is efficient enough to handle large data sets, and yields posterior samples of all model parameters that can be used to perform desired Bayesian estimation and inference. Although we present details for a specific implementation of the R-FMM using

specific distributional choices in the hierarchical model, 1D functions, and wavelet transforms, the method can be

applied more generally using other heavy-tailed distributions, higher dimensional functions

(e.g. images), and using other invertible transformations as alternatives to wavelets.