Equivalent source estimation of scalp potential fields contaminated by heteroscedastic and correlated noise

The customary ordinary least squares (OLS) approach to the estimation of equivalent sources of scalp potential fields relies on the assumption that noise in the potential measurements has an equal variance and is uncorrelated over leads. It is shown that this assumption is likely to be violated in practice, for instance by the use of a common reference lead. We describe tests to detect these violations and we propose several versions of an alternative estimation method called iterated generalised least squares (IGLS), which accounts for heteroscedastic or correlated noise by incorporating an estimate of the covariance matrix of the noise derived from single trial OLS residuals. Simulation results indicate that these alternatives give a considerable increase in the accuracy of both the parameter and the standard error and confidence interval estimates. The proposed tests and methods are finally integrated into a stepwise approach to equivalent source estimation, which incorporates in addition a test on the goodness of fit of the model, an assessment of the confidence intervals of the parameters and a powerful test of differences between experimental conditions. This stepwise approach is applied to the modelling of equivalent sources of early visual potentials elicited in a spatial attention task.