PowerPoint Presentation - Serial Correlation in AFNI

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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 aren’t 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