As in Friston et al. (2000) we have demonstrated that when using designs such as a box car convolved with a gamma HRF, colouring can be used with minimal loss of efficiency. However, for single-event designs with randomized ISIs, jittering or just very short ISIs, colouring is much less efficient and hence prewhitening is desirable.
Prewhitening requires a robust estimator of the autocorrelation to maintain low bias. To estimate the autocorrelation or equivalently the spectral density for use in prewhitening, different techniques were considered. These were single tapering Tukey, multitapering, autoregressive model of general order and a nonparametric approach that assumes monotonicity in the autocorrelation.
Crucially, nonlinear high-pass filtering is performed as a preprocessing step to remove the worst of the non-stationary components and low frequency noise. A Tukey taper, with much greater smoothing of the spectral density than is normally recommended in the literature, performed the best when prewhitening.
Importantly, a small amount of spatial smoothing of the autocorrelation
estimates was also found to be necessary to reduce bias to close to
zero at low probability levels.
The autocorrelation was found to vary considerably between matter types, with
higher positive autocorrelation (low frequency noise) in the grey matter when compared with
the white matter. Therefore, non-linear spatial smoothing of the autocorrelation was
used, which only smoothed within matter types. Using a Tukey taper (
) along with the
non-linear spatial smoothing we were able to reduce
bias to close to zero at probability levels as low as
.