4
A Tiny Amount of Mathematics
¥ White noise estimate of variance:
H N = number of time points; i = time index
H m = number of fit parameters
H N m = degrees of freedom (DOF) = how many equal-variance independent random values are left after the time series is fit with m regressors
oOLSQ assumption is that each of the N noise values in the data time series are equal-variance and independent (AKA white noise)
¥ If noise values arenÕt independent, then N m is too large an estimate of DOF, so variance estimate is too small
¥ Two possible solutions are:
1)Adjust variance estimate (and so the t- and F-values) to allow for too few DOF
2)Come up with a different variance estimator that has all N m DOF possible
oRequires estimating the temporal correlation structure of the noise as well
oOnce temporal correlation matrix is known, use Generalized Least Squares (GLSQ; AKA pre-whitening) to estimate b parameter vector
oGLSQ is consistent and should produce b-values with smaller variance than OLSQ
¥ Solution #2 is what 3dREMLfit implements