Robust, Adaptive Functional Regression in Functional Mixed Model Framework

Abstract

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.

A Unified Analysis of Structured Sonar-Terrain Data Using Bayesian Functional Mixed Models

“…Note that both Gfmm and Rfmm are not methodologically new. The general methods have been proposed by Morris and Carroll (2006) and Zhu et al (2011); extensions and applications can be found in Zhu et al (2012), Martinez et al (2013), Lee and Morris (2016), among others. The purpose of this section is not to propose a brand new model, but to outline the FMMs in the context of sonar-terrain data analysis, with the goal of integrating them in the unified analytical framework in Section 4.…”

“…The prior for $B_{\mathit{lk}}^{\ast }$ is set similarly with the prior for $G_{\mathit{alk}}^{\ast }$. These formulations are equivalent to setting double exponential (DE) distributions for the residuals, the random effects, and the “slab” part of the fixed effects, which has the effect of accommodating heavier-tailed behavior (non-Gaussianity) in data and downweighting the effect of outlying curves or outlying regions (Zhu et al, 2011). …”

“…Guo (2002) appears to be the first to outline the mixed model framework for functional data. Morris and Carroll (2006) proposed a Gaussian process FMM under the Bayesian setup, and their approach was further extended by Zhu et al (2011) to incorporate data with non-Gaussian behaviors. Despite its success in modeling various data structures (Morris et al, 2008; Davidson, 2009; Morris et al, 2011; Lancia et al, 2015), FMM has been primarily used to characterize the effects of covariates.…”

“…In addition to the numerous benefits mentioned above, it is worth highlighting the contributions of this paper beyond existing works such as Zhu et al (2011) and Zhu et al (2012). First, unlike those existing works which focus on a specific model or approach, this paper proposes a rather different framework that integrates multiple models and analytical approaches.…”

Functional Nonlinear Mixed Effects Models for Longitudinal Image Data

“…Many parametric mixed effects models including both fixed and random effects are the predominant approach for characterizing both the temporal correlations and random individual variations. Although there is a great interest in the analysis of functional data with various levels of hierarchical structures [11, 18, 7], only a handful of them [6, 17, 21] focused on the development of linear mixed models for longitudinal image data. Recently, there was some attempt on the development of hierarchical geodesic models on diffeomorphism for longitudinal shape analysis [15].…”