¥ 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