Robust Bayesian general linear models

Abstract

We describe a Bayesian learning algorithm for Robust General Linear Models (RGLMs). The noise is modeled as a Mixture of Gaussians rather than the usual single Gaussian. This allows different data points to be associated with different noise levels and effectively provides a robust estimation of regression coefficients. A variational inference framework is used to prevent overfitting and provides a model order selection criterion for noise model order. This allows the RGLM to default to the usual GLM when robustness is not required. The method is compared to other robust regression methods and applied to synthetic data and fMRI.

Introduction

Neuroimaging data contain a host of artifacts arising from ‘physiological noise’, e.g. subject respiration, heartbeat or movement of the head, eye, tongue or mouth or ‘non-physiological noise’, e.g. EEG electrodes with poor electrical contact, spikes in fMRI or extracranial magnetic sources in MEG. The presence of these artifacts can severely compromise the sensitivity with which we can detect the neuronal sources we are interested in. The optimal processing of artifactual data is therefore an important issue in neuroimaging analysis and a number of processing methods have been proposed. One approach is visual inspection and removal of trials deemed to contain artefacts. In the analysis of Event-Related Potentials (ERPs), however, this can lead to up to a third of the trials being removed. Because the statistical inferences that follow are based on fewer data points, this results in a loss of sensitivity.

In fMRI, signal processing methods exist for the removal of k-space spikes (Zhang et al., 2001, Greve et al., 2006), and Exploratory Data Analysis (EDA) methods have been proposed for removal of outliers in the context of mass-univariate modeling (Luo and Nichols, 2003). Alternatively, Independent Component Analysis (ICA) can be used to isolate ‘noise sources’ and remove them from the data (Jung et al., 1999). This is, however, a non-automatic process and will typically require user intervention to disambiguate the discovered components. In fMRI, autoregressive (AR) modeling can be used to downweight the impact of periodic respiratory or cardiac noise sources (Penny et al., 2003). More recently, a number of approaches based on robust regression have been applied to imaging data (Wager et al., 2005, Diedrichsen and Shadmehr, 2005). These approaches relax the assumption underlying ordinary regression that the errors be normally (Wager et al., 2005) or identically (Diedrichsen and Shadmehr, 2005) distributed. In Wager et al. (2005), for example, a Bisquare or Huber weighting scheme corresponds to the assumption of identical non-Gaussian errors. The method was applied to group-level fMRI analysis and was found to lead to more sensitive inferences.

Interestingly, Wager et al. (2005) tested a number of standard robust estimation methods by generating data from a known mixture process as this was thought to capture the essence of signal embedded in artefactual data. In this paper we take this idea one step further and develop an optimal robust estimation procedure for the case of mixture errors.

Specifically, we propose a Robust General Linear Model (RGLM) framework in which the noise is modeled with a Mixture of Gaussians. This allows different data points to be associated with different noise levels and provides a robust estimation of regression coefficients via a weighted least squares approach. Data points associated with high noise levels are downweighted in the parameter estimation step. Moreover, a Bayesian estimation framework (Attias, 2000) is used to prevent model overfitting and provides a model order selection criterion for noise model order. This allows selection of the usual GLM, i.e. a noise mixture with a single component, when an outlier model is not appropriate. This work is based on a similar algorithm for robust estimation of autoregressive processes (Roberts and Penny, 2002).