Dear Will,
thanks for the information!
> In this part of SPM the AR coefficients are estimated using a
> regression approach in which future time series values (here - the
> error time series) are regressed onto previous ones. An alternative
> approach e.g. Burg/Yule-Walker method, first computes the
> autocorrelation function of the time series and then uses the
> Yule-Walker relations to estimate the AR coefficients.
Is the regression estimation also implemented in a Bayesian way? If yes, is there a way to obtain the posterior precision of the estimates, and is any kind of (e.g. spatial smoothness) prior implemented? If not, what is the role of AR estimation in "Bayesian 1st-level"? Also, it sounds like the regression is applied to residuals – how do you make sure the deviation between error variance–covariance and residual variance–covariance doesn't bias your estimate (what's usually dealt with by ReML)? Sorry, lots of questions...
> Perhaps its the case that with higher order models (e.g. 6th order)
> there are more degrees of freedom to get less well-behaved estimates
> of the autocorrelation function (sorry to be vague). So it may be
> worth trying a lower order model (or trying some model order
> selection here - you could use spm_ar.m from the spectral toolbox to
> look at other model orders (on the GLM residuals) - with the .fm
> field returning an approximation to the model evidence).
I only looked at "Bayesian 1st-level" to have voxel-wise estimates of autocorrelation, which isn't available in "Classical 1st-level". Outside of SPM, the two approaches seem to be AR(6) and ARMA(1,1), that's why I selected order 6. Do you recommend order 3?
Best,
Carsten
> From: Carsten Allefeld <[log in to unmask]>
> Sent: 03 December 2018 18:13
> To: Guillaume Flandin
> Cc: [log in to unmask]; William Penny (PSY - Staff)
> Subject: Re: [SPM] Strange AR processes from Bayesian estimation
>
>
> Dear Guillaume & Will,
>
> I noticed a minor inaccuracy in my code transforming AR coefficients
> into autocorrelation functions. According to comparison with the
> analytic solution for specific coefficient values, I the corrected
> version has an absolute error on the order of 1e-16. The updated
> results have minor changes, but the basic observations are the same.
>
> There are 13256 in-mask voxels. Of these, 16 voxels have estimated AR
> coefficients whose roots' magnitudes are not all below 1 (describing
> nonstationary AR processes).
>
> The autocorrelation functions of the remaining 13240 voxels are shown
> superimposed in the attached autocorrelation.png. As you notice, a
> non-neglibile number of them have autocorrelation functions that do
> not decay to 0 over the course of 128 scans (= 256 s).
>
> autocorrelation_slice.png shows the autocorrelation over lags 0 to
> 127 in slice 13 (of 23), containing 1150 in-mask voxels. It is
> apparent that the voxels with long-range autocorrelation lie mainly
> at the edge of the brain mask.
>
> Guillaume, you asked for time series. timeseries.png shows the
> underlying BOLD measurements, in the top panel for the 16 voxels
> with nonstationary AR, and in the lower panel for the 71 voxels
> where the autocorrelation at lag 127 is larger than 0.1. I can't say
> that I notice anything special or common to these timeseries.
>
> For the moment I will proceed with my data analysis after excluding
> the 16 + 71 = 87 "weird" voxels. But I think it would be useful if
> you could look into this further. In particular, in my opinion an AR
> estimation method should not produce coefficients describing a
> nonstationary process. Do you agree?
>
> Thank you!
>
> Best,
> Carsten
>
>
> ----- Original Message -----
> > From: "Guillaume Flandin" <[log in to unmask]>
> > To: "Carsten Allefeld" <[log in to unmask]>
> > Cc: [log in to unmask], "William Penny (PSY)" <[log in to unmask]>
> > Sent: Wednesday, 28 November, 2018 11:35:53 AM
> > Subject: Re: [SPM] Strange AR processes from Bayesian estimation
> >
> > Dear Carsten,
> >
> > All the voxels exhibiting pathological behaviour seem to be at the
> > boundary of the brain mask so, as you say, simply discarding them
> > is
> > probably the easiest thing to do. To understand what is going on,
> > it
> > would be useful to extract and display their time series.
> > I copy this email to Will so that he can add any further comments
> > or
> > advice on the spatial noise prior.
> >
> > Best regards,
> > Guillaume.
> >
> >
> > On 27/11/2018 19:20, Carsten Allefeld wrote:
> > > Dear Guillaume,
> > >
> > > thanks for replying!
> > >
> > >> These results were obtained with which spatial noise prior
> > >> option?
> > >> The
> > >> interface lists five options: UGL, GMRF, LORETA, Tissue-type and
> > >> Robust.
> > >
> > > I used the default, "UGL". Which one would you recommend?
> > >
> > > Options in detail:
> > > fmri_est.method.Bayesian.space.volume.block_type = 'Slices';
> > > fmri_est.method.Bayesian.signal = 'UGL';
> > > fmri_est.method.Bayesian.ARP = 6;
> > > fmri_est.method.Bayesian.noise.UGL = 1;
> > > fmri_est.method.Bayesian.LogEv = 'No';
> > > fmri_est.method.Bayesian.anova.first = 'No';
> > > fmri_est.method.Bayesian.anova.second = 'No';
> > > fmri_est.method.Bayesian.gcon = struct('name', {}, 'convec', {});
> > >
> > >> Could you show a map of where the voxels you are concerned about
> > >> are?
> > >
> > > Attached are plots of the absolute value of the autocorrelation
> > > at
> > > lags 0 to 127 in a middle slice (#13).
> > > Comparison with the coregistered T1 indicates that they are
> > > located
> > > mainly at the outer edge of gray matter (maybe meninges), but
> > > also
> > > frontally and posteriorly slightly into the longitudinal fissure.
> > >
> > > That suggests I should simply exclude these voxels from further
> > > analysis.
> > >
> > > Do you have a suggestion which criterion to use?
> > > The data-based threshold (2.02e-9) discards more than half of the
> > > brain, and any other threshold seems arbitrary.
> > >
> > > Best,
> > > Carsten
> > >
> > >
> > >>
> > >> Best regards,
> > >> Guillaume.
> > >>
> > >>
> > >> On 27/11/2018 17:11, Carsten Allefeld wrote:
> > >>> Hello all,
> > >>>
> > >>> I'm interested in getting local estimates of temporal
> > >>> autocorrelation in SPM, and for that purpose used Bayesian
> > >>> 1st-level estimation.
> > >>> The fMRI data I used to test that have 3 sessions of 128 scans
> > >>> at
> > >>> a
> > >>> TR of 2 s and 64x64x23 voxels of size 4x4x4 mm, unsmoothed, of
> > >>> which approximately 13,000 are within brain.
> > >>>
> > >>> I then extracted the AR coefficients (order 6) for the first
> > >>> session (Sess1_AR_0001.nii to Sess1_AR_0006.nii) and used the
> > >>> Yule–Walker equations iteratively to obtain the corresponding
> > >>> autocorrelation function across lags 0 to 128.
> > >>>
> > >>> The results are strange (see attached plot):
> > >>> – In 16 voxels the AR coefficients describe a non-stationary
> > >>> process. After excluding them:
> > >>> – At lag 127, 7906 voxels have an autocorrelation > 1e-6, 1651
> > >>> voxels > 1e-3, and 113 voxels > 0.1.
> > >>> – The largest negative autocorrelation at lag 127 is -2.02e-9.
> > >>> If
> > >>> I
> > >>> take that as an indicator of numerical/estimation precision,
> > >>> there
> > >>> are 8185 voxels where the autocorrelation at lag 127 is
> > >>> different
> > >>> from 0 (> +2.02e-9).
> > >>>
> > >>> This makes me suspect that the AR estimation in "Bayesian
> > >>> 1st-level" is not very reliable. Is there something I might
> > >>> have
> > >>> done wrong?
> > >>>
> > >>> Is there a recommended postprocessing for the AR coefficients
> > >>> or
> > >>> autocorrelation functions?
> > >>> I thought about tapering à la FSL, or clustering as a crude
> > >>> form
> > >>> of
> > >>> spatial regularization.
> > >>> Or should I simply exclude voxels with unbelievably long-range
> > >>> autocorrelation?
> > >>>
> > >>> Thank!
> > >>>
> > >>> Best,
> > >>> Carsten
> > >>>
> > >>
> > >> --
> > >> Guillaume Flandin, PhD
> > >> Wellcome Centre for Human Neuroimaging
> > >> UCL Queen Square Institute of Neurology
> > >> London WC1N 3BG
> > >>
> >
> > --
> > Guillaume Flandin, PhD
> > Wellcome Centre for Human Neuroimaging
> > UCL Queen Square Institute of Neurology
> > London WC1N 3BG
> >
>
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