  We propose a voxel-wise general linear model with autoregressive noise and
heteroscedastic noise innovations (GLMH) for analyzing functional magnetic
resonance imaging (fMRI) data. The model is analyzed from a Bayesian
perspective and has the benefit of automatically down-weighting time points
close to motion spikes in a data-driven manner. We develop a highly efficient
Markov Chain Monte Carlo (MCMC) algorithm that allows for Bayesian variable
selection among the regressors to model both the mean (i.e., the design matrix)
and variance. This makes it possible to include a broad range of explanatory
variables in both the mean and variance (e.g., time trends, activation stimuli,
head motion parameters and their temporal derivatives), and to compute the
posterior probability of inclusion from the MCMC output. Variable selection is
also applied to the lags in the autoregressive noise process, making it
possible to infer the lag order from the data simultaneously with all other
model parameters. We use both simulated data and real fMRI data from OpenfMRI
to illustrate the importance of proper modeling of heteroscedasticity in fMRI
data analysis. Our results show that the GLMH tends to detect more brain
activity, compared to its homoscedastic counterpart, by allowing the variance
to change over time depending on the degree of head motion.
