Low-pass filtering

The low-pass filter used here is a Gaussian filter matched to a commonly assumed HRF with parameters $\sigma=2.98secs$ and $\mu=3secs$ (Friston et al. (1995)). The histogram for low-pass filtering demonstrates the idea of colouring, in that the whole histogram is focused to a peak centered on a $S_{\rho }$ close to that entirely due to the low-pass filtering. However, the standard deviation around this peak (standard deviation=17.6) is greater than the standard deviation of the estimator of $S_{\rho }$ (determined empirically on artificial data as standard deviation=12.9). Since the data is showing a greater variability in $S_{\rho }$ than there is in just estimating $S_{\rho }$, this suggests the requirement for local estimation of autocorrelation even when low pass filtering (colouring) is performed. Although the autocorrelation estimate is made more robust, the autocorrelation imposed by the colouring does not completely smother the intrinsic autocorrelation.