Dear Geraint and Saaussan,
> This is concerning temporal filtering in fmri.
> I will quote spm99b assumptions about the residuals;
>
> % For temporally correlated (fMRI) data, the algorithm implements the
> % technique of Worsley & Friston (1995). In this approach the model is
> % assummed to fit such that the residuals have (slight) short-term
> % (intrinsic) autocorrelation given by Vi. (I.e. Residuals e =
> % sqrt(Vi)*e', where e' is an (unobserved) white noise time series).
> % The data and model are then temporally filtered prior to model
> % fitting by a linear filter given by matrix K, giving model K*Y =
> % K*X*beta + K*e. K=inv(sqrt(Vi)) corresponds to pre-whitening (leaving
> % residuals e'), K=I to no temporal smoothing (appropriate for
> % independent data when Vi=I), and K=S (a temporal filter matrix - see
> % spm_make_filter) for band-pass filtering.
> %
> % The autocorrelation in the filtered time series is then K*Vi*K'
> % Standard results for serially correlated regression can then be used
> % to produce variance estimates and effective degrees of freedom
> % corrected for the temporal auto-correlation. Note that these
> % corrections give non-integer degrees of freedom."
>
> I got the following results for different temporal filtering schemes ;
>
> 1-if i don't include temporal filtering in my model (neither low pass
> nor high pass), i can use spm_regions.m to plot the spectral density of
> the residuals R=xY.y-xY.Y at a particular voxel. I get a certain
> graph.
>
> 2-If i include high pass filtering and hrf low pass, the spectral
> density for R (at the same voxel) can be plot again.
>
> 3-If i include high pass filter, and guassian filter with FWHM=TR, i
> can still plot R.
>
> 4-If i use a butterworth band pass filter with hamming window, i get
> a spectrum that is clearly far from the original (non-filtered) spectrum.
>
> 5-However, if i RE_APPLY a (guassian lowpass& high pass) filter on the
> "butterworth filtered data", i get exactly the same residual values and
> residual spectrum as if i applied the (guassian lowpass& high pass)
> filter on the original data.
>
> In a way, i can include any intermediate filter between the original
> data and the final (guassian lowpass& high pass) filter, and this won't
> change the final estimates.
>
> I think this is because all of the mentioned filters above have the
> characteristics of "(slight) short-term
> % (intrinsic) autocorrelation given by Vi. (I.e. Residuals e =
> % sqrt(Vi)*e', where e' is an (unobserved) white noise time series).".
> Thus, the final filter is the dominant filter.
>
> I might be mistaken here in my understanding. Please correct me if i'm
> wrong.
>
> Therefore comes the natural question;
>
> -If the final filter is the dominant one, can i use multiple filters
> (as intermediate filters) to refine my data?? This in a way, will give
> me the flexibility to get rid of the noise without messing with the
> original data variance estimate.
I think your point is that the autocorrelation function (and its
Fourier Transform - the spectral density) are largely determined by the
dominant filter. I am sure you are right. In fact this is the
motivation for filtering in SPM: By imposing a correlation structure
on the time-series (with K) the mismatch between the assumed
autocorrelations K*Vi(assumed)*K' and the actual correlation matrix
K*Vi(actual)*K' is minimised. This is important because the mismatch
introduces bias into the estimate of error variance. Applying
intermediate filters (say S) will not really be beneficial because
(usually) K*Vi*K' ~ K*S*Vi*S'*K'. In fact the real problem reduces to
finding the best form for K (i.e. K*S). This form is usually a
compromise between high efficieny for parameter estimation where K
should ~ inv(sqrt(Vi)) and low bais in estimating standard error, where
K ~ stringent band-pass filter.
We are currently revising a submitted paper on this issue which I will
make available in the new year.
With best wishes - Karl
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|