Wellcome Department of Imaging Neuroscience, University College London
Subject: Modeling & Analysis
Abstract
A key issue in the analysis of fMRI time-series is the concern that successive samples are serially correlated. Two recent approaches which address this issue make use of the Expectation-Maximisation (EM) algorithm (Friston et al. (2002a), Worsley et al (2002).). The EM algorithm provides estimates of the model `parameters' and `hyperparameters'. The parameters refer to the regression coefficients in a General Linear Model (GLM) and the hyperparameters refer to error variances and the parameters of Autoregressive (AR) models (Worsley et al. (2002)) or Autoregressive Moving Average (ARMA) models (Friston et al. (2002a)).
The EM framework provides estimates of the full probability distribution over the model parameters but only gives point estimates for the hyperparameters. The variability of the hyperparameter estimates cannot therefore be taken into account when making statistical inferences about the underlying hemodynamics. This is an inherent flaw in existing instantiations of the EM approach which may result in inflated false-positive rates (Friston et al. (2002b)). Friston et al. (2002a) overcome this by pooling hyperparameter estimates over the whole brain. As there are so many voxels, the variance of the resulting hyparameter estimates becomes negligible. A similar solution is given by Worsley et al. who pool hyperparameter estimates over local regions. This `pooling trick' assumes that the hyperparameter values are constant over the whole brain or over local regions.
In this paper we offer an alternative solution based on the Variational Bayesian (VB) framework which we use for inference in GLMs with AR error processes. VB was developed in the machine learning community and is a generalisation of the EM approach. VB has the advantage of taking into account the variability of hyperparameter estimates and this is achieved with little additional computational effort. Further, VB allows for automatic selection of AR model order. VB works by approximating the true posterior density with a factorised density. For the GLM-AR model the fidelity of this approximation is verified via Gibbs sampling. Results are shown on simulated data and on data from an event-related fMRI experiment.
References
K. Friston, D. Glaser, R. Henson and J. Ashburner (2002a). Classical and Bayesian Inference in Neuroimaging: Variance component estimation in fMRI, Submitted to Neuroimage.
K. Friston, S. Kiebel, C. Phillips and J. Ashburner (2002b). Classical and Bayesian Inference in Neuroimaging: Empirical Bayes models for PET, Submitted to Neuroimage.
K. J. Worsley and C. H. Liao and J. Aston and V. Petre and G. H. Duncan and F. Morales and A. C. Evans (2002). A General Statistical Analysis for fMRI Data, NeuroImage, 15(1).