> From [log in to unmask] Tue Oct  3 15:00:48 2000
> User-Agent: Microsoft-Outlook-Express-Macintosh-Edition/5.02.2022
> Date: Tue, 03 Oct 2000 09:58:26 -0400
Dear Geoff,

> I posted a message about a week ago regarding the method used to fit an
> AR(1) model to the residuals of the GLM for the purposes of estimating
> the serial correlations in the data. I wanted to restate the question
> in perhaps a more provocative way and ask again if anyone would care to
> comment.

In fact SPM99 does not fit an AR(1) model to the residuals (see below)
but to the raw data (before filtering and model fitting).  The ensuing
(conservative) estimate includes the correlations due to activations
but these are trivial when averaged over all voxels.

> Consider that there are two ways to remove the effects of low-frequency
> noise from a time-series. First, one could apply exogenous smoothing to
> the data with a notch filter, which would remove power from (e.g.) the
> lowest 10 frequencies. Alternatively, one could include a set of 10
> sines and cosines as nuisance covariates, designed to model power at
> these 10 low frequencies.  In both cases, the residuals would have no
> power at the low frequencies. My question is this: does the model of
> intrinsic correlation need to differ between the cases for valid
> inference?

Both these approaches to removing low frequnecy components are equivalent,
where the high-pass filter matrix R is the residual forming matrix of
the nuisance components X.

		y - X*(pinv(X)*y) = R*y,   R = 1 - X*(pinv(X)

The model of intrinsic correlations V should be the same irrespective
of filtering but the correlations among the residuals are now simply
R*V*R' = R*V (R is idempotent).  Because R*V has a non-stationary form
(i.e. it is not a Toepltz matrix and is difficult to deal with
computationally) we have chosen not to esitmate the correlations using
the residuls but to use the raw data instead and estimate V.

A proper (less conservative but still valid) estimation of V, in the
context of activations, would require an iterative parameter
re-estimation procuedure (e.g. REML).  We are currently evaluating a
number of these.

I hope this helps - Karl



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