[Mne_analysis] Computing regression on sensor data then transforming to source space

Krieger, Donald N. kriegerd at upmc.edu
Fri Feb 21 16:42:35 EST 2014
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Thanks, Hari, for posting back and making it clear.
I agree with your points to the extent that I follow them.
I am pretty hazy about the effects of differing variances across the sensor measurements or the Betas computed from them on the MNE projections into the source space.
Because of that and because the idea of applying regression across single trials on an independent task relevant variable is quite interesting and, I think promising, I had been thinking about applying it in the source space.  If that works for a simple ANOVA, the means would represent mean neuroelectric currents and a statistical finding of a difference between two task conditions would imply that that source was differentially activated under the two conditions.  That's what fMRI is about for neurovascular activity but with MEG, differential neuroelectric activation is much closer to what we conceive the brain is doing to accomplish the task.

Our group has had success in generating 3D maps of differential activation using an even simpler chi-square test on a counting measure.  Our first results just appeared in an open access journal: Intl J Advd Comp Sci (4)1: 15-25, 2014. But parametric statistics applied to dipole amplitude estimates has far greater potential statistical power.  This discussion has spurred me to consider how to use that approach to amplify the power of those results.

Thanks again.

Regards,
 
Don
 
Don Krieger, Ph.D.
Department of Neurological Surgery
University of Pittsburgh
(412)648-9654 Office
(412)521-4431 Cell/Text


> -----Original Message-----
> From: Hari Bharadwaj [mailto:hari at nmr.mgh.harvard.edu]
> Sent: Friday, February 21, 2014 9:36 AM
> To: Krieger, Donald N.
> Cc: mne_analysis at nmr.mgh.harvard.edu
> Subject: Re: [Mne_analysis] Computing regression on sensor data then
> transforming to source space
> 
> Hi Donald,
>     Just to clarify, I did not intend to suggest that the betas in sensor and
> source would be equal or equivalent... Apologies for any confusion caused..
> What I meant in (A) below, was that the following would give identical
> results if the regression is linear:
>  (1) computing beta values on sensor data (1 sensor at a time separately )
> and computing the inverse solution with these beta values to get "source
> beta" values..
> (2) computing inverse solution of the original MEG data and then computing
> "source beta" values directly by fitting the regression (1 source at a
> time)...The "source betas" in the two cases are what I think will be identical.
> 
> Regarding the noise variance, the discussion that I articulated  was with the
> problem of source estimation in mind. Hence it was about specifying the
> noise covariance matrix for MNE (which was Denis's concern, I think) rather
> than about statistical inference on the betas. The assertion is that if the
> regression/ANOVA model is simple enough such that the sensor betas are
> linear combinations across trials (or trial groups) AND if the weights for the
> linear combination are normalized, then the noise covariance in the betas is
> the same as the noise covariance of the original trials (or trial groups) that
> went into its calculation (and hence the originally planned scaling of noise
> covariance for MNE purposes would still be appropriate).  I too, only
> thought about the regression/ANOVA as being done on one sensor at a time
> or one source at a time... Even with this relatively simple "mass univariate"
> approach, if the model being fit is complex or has many terms the
> relationship between the variance of the noise in the beta values and the
> variance of the noise in original MEG/EEG data gets complicated and inverse
> estimation could suffer.
> 
> How to do parametric inference in source or sensor space by correctly
> accounting for across sensor or across source variance inhomogeneities is a
> whole issue on its own and is likely one with many complexities. If I
> understand correctly, that is the question you have posed?
> 
> Best,
> Hari
> 
> Hari Bharadwaj
> PhD Candidate, Biomedical Engineering,
> Auditory Neuroscience Laboratory
> Boston University, Boston, MA 02215
> 
> Martinos Center for Biomedical Imaging,
> Massachusetts General Hospital
> Charlestown, MA 02129
> 
> hari at nmr.mgh.harvard.edu
> Ph: 734-883-5954
> 
> > On Feb 21, 2014, at 7:26 AM, "Krieger, Donald N." <kriegerd at upmc.edu>
> wrote:
> >
> > Hi everyone,
> >
> > First with regards the continuing discussion of regression models which
> include both scaled and categorical variables, etc.: There are general
> purpose open sources statistical packages, e.g. R, which implement ANOVA,
> ANCOVA, MANCOVA, etc. in the general linear model format, i.e. the one
> we are discussing.  It therefore might be worth evaluating one or more of
> them for development of python wrappers which use them to do the
> regressions we are discussing.
> >
> > Second with regards both the equivalent variance discussion and the
> assertion of the equivalence of computing the Betas in sensor or source
> space: Perhaps I am misunderstanding.  I had just assumed that the
> regression on an independent variable was to be for one sensor at a time.
> Thus in the statistical framework the problem could be handled as a
> MANOVA with the sensor measures being multiple dependent variables or
> as a repeated measures ANOVA where the sensor measures are repeated
> measures.  In neither case though would regression on the independent
> variable against a source be conceptually equivalent or give the same
> answer. i.e. against a linear combination of the sensors,
> >
> > For the MANOVA approach, the regression against each sensor is handled
> separately.  So I think that at least the differences in variances across the
> sensors don't matter.  What do you think?  I don't write with the voice of
> authority here and would like to understand this.
> >
> > Don
> >
> > Don Krieger, Ph.D.
> > Department of Neurological Surgery
> > University of Pittsburgh
> > (412)648-9654 Office
> > (412)521-4431 Cell/Text
> >
> > ________________________________________
> > From: mne_analysis-bounces at nmr.mgh.harvard.edu
> > [mne_analysis-bounces at nmr.mgh.harvard.edu] on behalf of Hari
> Bharadwaj
> > [hari at nmr.mgh.harvard.edu]
> > Sent: Thursday, February 20, 2014 10:57 AM
> > To: Denis-Alexander Engemann
> > Cc: mne_analysis at nmr.mgh.harvard.edu
> > Subject: Re: [Mne_analysis] Computing regression on sensor data then
> > transforming to source space
> >
> > Hi Denis et al.,
> >    It appears to me that there are two separate issues being confused
> > here and perhaps there will be some clarity if we talk through it:
> >
> > (A) Whether to compute "betas" in sensor-space or source-space:
> >    This is not really a difficult question within the MNE or other
> > linear inverse solution framework. Because, for a given inverse
> > operator (let's call it M), computing betas in sensor or source space
> > should lead to identical results unless the statistical model being
> > fitted to compute the betas is somehow non-linear.
> >
> > (B) Choice of the noise model used to compute the operator M:
> >    This issue is more subtle and important and does not depend on the
> > sequence in which you do things in (A). One has to consider this
> > question regardless of whether he/she is doing stats in sensor-space
> > or source space. By projecting things to source-space first, one
> > doesn't become immune to an inappropriate choice of the noise model.
> >   If the noise covariance employed corresponds to that of
> > single-trials (eg., if you choose to scale it with nave = 1 because
> > you are projecting single trial data), the map would be unnecessarily
> > smooth even if later you combine 100s of trials to show you final
> > "map" of interest. The prior (the minimum norm part) will be given
> > much more weight and the data fit will be given lot less weight than
> > is appropriate. Thus, one should employ a noise-covariance scaling
> > *with the foresight* of what analysis is planned, especially in the
> > context of regressions.
> >   I think there is one special case in the context of
> > regressions/ANOVA that occurs frequently and is easy to handle. This
> > is the case where "betas" are computed with the weights that are
> > normalized (these weights are sometimes called "contrasts" in stats
> parlance)..
> > That is if: beta = sum_over_i{w_i * x_i} where x_i is the mean from
> > the i^th group of trials, and if sum_over_i { w_i**2} == 1, and the
> > x_i's come from the same number of trials, then the beta values have
> > the same noise variance as the x_i's. To take the simplest example of
> > this case, if x_1 and x_2 are evoked responses for conditions 1 and 2
> > (each with nave = 100 trials), then if you compute your beta values as
> > the difference divided by sqrt(2), then scaling the noise-covariance
> > by nave = 100 is appropriate when computing M. In practice, if all the
> > conditions that go into the stats have the same number of trials, and
> > if the stats employs orthonormal contrasts, then the default MNE
> > scaling of nave = #trials works well..
> >
> > Hope this helps..
> >
> > Hari
> >
> >
> >
> >> On Thu, February 20, 2014 5:48 am, Denis-Alexander Engemann wrote:
> >> Hi everyone,
> >>
> >> this is really an interesting and productive discussion which I enjoy
> >> following very much. And it's especially timely and relevant, since
> >> we plan to support high-level functions for single trial regression
> >> in MNE-Python in the future. Tal Linzen has recently started drafting
> >> examples and first functions, cf.
> >> https://github.com/mne-tools/mne-python/pull/1034.
> >>
> >> Since the there were a few open questions for Teon's use case we
> >> decided to postpone adding direct support for projecting beta-maps.
> >> However, if we could use the collective knowledge and experience
> >> distributed over this mailing list, it might be possible to clarify
> >> those questions and add support this use case directly in MNE-Python
> >> rather soon.
> >>
> >> My main concern with the beta-map projection approach is that the
> >> noise structure (and the units) as reflected in the noise covariance
> >> computed on raw M/EEG signals don't necessarily match with the noise
> >> structure present in the beta maps. This would mean a model mismatch.
> >> To tackle this issues different options might be possible, such as
> >> using an identity matrix as noise covariance or estimating the noise
> >> covariance on whitened single trials.
> >>
> >> Input on both the work-in-progress on MNE-Python and the conceptual
> >> issues is highly appreciated.
> >>
> >> Cheers,
> >> Denis
> >>
> >> On Thu, Feb 20, 2014 at 7:57 AM, Alexandre Gramfort
> >> <alexandre.gramfort at telecom-paristech.fr> wrote:
> >>> hi Don,
> >>>
> >>> thanks a lot for sharing these insights.
> >>>
> >>> Although I get the idea of what you suggest, I am not sure I would
> >>> be able to perfectly replicate this analysis.
> >>> Do you happen to have a script you could share?
> >>>
> >>> Also can you give the full ref from which the figure is extracted?
> >>>
> >>> thanks again
> >>>
> >>> Best,
> >>> Alex
> >>>
> >>>
> >>> On Wed, Feb 19, 2014 at 8:30 PM, Krieger, Donald N.
> >>> <kriegerd at upmc.edu>
> >>> wrote:
> >>>> Dear Teon,
> >>>>
> >>>>
> >>>>
> >>>> You have raised several interesting questions on which I would like
> >>>> to expand.
> >>>>
> >>>> Hari responded to several technical issues, viz. (1) constraints on
> >>>> what you do to retain the validity of your subsequent projection
> >>>> into source space and (2) weighting the regression to compensate
> >>>> for unequal numbers of trials for different levels of the
> >>>> independent variable.
> >>>>
> >>>>
> >>>>
> >>>> Here are some points about the meaning of what you are doing and
> about
> >>>> the
> >>>> technical issues.
> >>>>
> >>>> (1)    If the independent variable is scaled rather than a 0/1 dummy,
> >>>> i.e.
> >>>> has multiple numeric levels, then your regression is asking a specific
> >>>> quantitative question about the amplitude of the magnetic
> field/source,
> >>>> i.e.
> >>>> is the amplitude a linear function of the independent variable?  If for
> >>>> example the variable takes values n and 2n, you are asking: "Is the
> >>>> amplitude for the "2n" trials double what it is for the "n" trials?
> >>>>
> >>>> (2)    I think it's reasonable to assume that many of the sources
> >>>> contributing to the magnetic field have nothing to do with the task.
> >>>> Although you are working with single trial data, your regression across
> >>>> the
> >>>> trials is collapsing the data in a generalized version of averaging.
> >>>> That
> >>>> helps attenuate the contributions to the field of unrelated sources.
> >>>> But if
> >>>> (a) there was a way up front to define regions of interest within the
> >>>> brain
> >>>> which you think are involved in the task, and (b) if the linear
> >>>> hypothesis
> >>>> you are testing is true, you should do better by doing the projection
> >>>> first
> >>>> and then doing the regression on the vertices within one ROI at a time.
> >>>>  In
> >>>> that way you take advantage of the signal space separation capabilities
> >>>> of
> >>>> your projection operation to isolate the sources you think are
> >>>> involved.  If
> >>>> you want to get formal statistics from your regression, you must find a
> >>>> way
> >>>> to adjust the degrees of freedom since presumably the source
> estimates
> >>>> from
> >>>> nearby vertices lack independence.
> >>>>
> >>>> (3)    Multidimensional regression: I presume that you are doing your
> >>>> regression for a single time point, tau.  Or perhaps you are averaging
> >>>> the
> >>>> amplitude values centered on the peak.  In either case you get a single
> >>>> number for each magnetic field sensor for each trial.  Instead you
> >>>> could use
> >>>> multiple points about the center of a peak and use a low order
> >>>> polynomial of
> >>>> tau multiplied by your original independent variable.  Note that
> >>>> averaging
> >>>> is equivalent to using a zero-order polynomial.   If you use say 21
> >>>> data
> >>>> points centered on the peak, you increase your degrees of freedom by
> >>>> quite a
> >>>> lot.  Of course your 21 data points lack independence but you still are
> >>>> using more information to do the regression.
> >>>>
> >>>> (4)     The more important additional variable is along the time axis
> >>>> for
> >>>> the sequence of trials.  If you use a polynomial function for that, any
> >>>> non-zero Beta other than the zero-order one represents a
> >>>> nonstationarity in
> >>>> your measurements.  This is rarely assessed but with humans doing a
> >>>> task is
> >>>> always a concern and it's interesting too.  The attached figure
> >>>> illustrates
> >>>> ideas (3) and (4) with evoked potential data.
> >>>>
> >>>>
> >>>>
> >>>> I hope I'm understanding you correctly and that this is helpful.
> >>>>
> >>>>
> >>>>
> >>>> Regards,
> >>>>
> >>>>
> >>>>
> >>>> Don
> >>>>
> >>>>
> >>>>
> >>>> Don Krieger, Ph.D.
> >>>>
> >>>> Department of Neurological Surgery
> >>>>
> >>>> University of Pittsburgh
> >>>>
> >>>> (412)648-9654 Office
> >>>>
> >>>> (412)521-4431 Cell/Text
> >>>>
> >>>>
> >>>>
> >>>> From: mne_analysis-bounces at nmr.mgh.harvard.edu
> >>>> [mailto:mne_analysis-bounces at nmr.mgh.harvard.edu] On Behalf Of
> Teon
> >>>> Brooks
> >>>> Sent: Wednesday, February 19, 2014 12:34 AM
> >>>> To: mne_analysis at nmr.mgh.harvard.edu
> >>>> Subject: [Mne_analysis] Computing regression on sensor data then
> >>>> transforming to source space
> >>>>
> >>>>
> >>>>
> >>>> Hi MNE listserv,
> >>>>
> >>>>
> >>>>
> >>>> I have single-trial data that I would like to regress a predictor
> >>>> (let's say
> >>>> word frequency) on it and then compute a source estimate. I'm
> planning
> >>>> to
> >>>> use mne-python to do this computation. I was wondering if I could do
> >>>> the
> >>>> regression over single trial sensor data first, get the beta values for
> >>>> each
> >>>> sensor over time, and then compute the source estimate as if it were
> an
> >>>> evoked object.
> >>>>
> >>>>
> >>>>
> >>>> My presumption is that it should be fine if the source transformation
> >>>> is
> >>>> linear. The other option would be to source transform the data then do
> >>>> the
> >>>> regression but the problem with doing this first is that computing the
> >>>> source estimates is more demanding on memory (say about 1000 trials
> >>>> with the
> >>>> around 5000 sources over 600-800ms of time). It would be more
> efficient
> >>>> if
> >>>> this computation could be done first if it is not computationally ill.
> >>>>
> >>>>
> >>>>
> >>>> What are your thoughts?
> >>>>
> >>>>
> >>>>
> >>>> Best,
> >>>>
> >>>> --
> >>>>
> >>>> teon
> >>>>
> >>>>
> >>>>
> >>>>
> >>>> _______________________________________________
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> >
> >
> > --
> > Hari Bharadwaj
> > PhD Candidate, Biomedical Engineering,
> > Boston University
> > 677 Beacon St.,
> > Boston, MA 02215
> >
> > Martinos Center for Biomedical Imaging,
> > Massachusetts General Hospital
> > 149 Thirteenth Street,
> > Charlestown, MA 02129
> >
> > hari at nmr.mgh.harvard.edu
> > Ph: 734-883-5954
> >
> >
> > _______________________________________________
> > Mne_analysis mailing list
> > Mne_analysis at nmr.mgh.harvard.edu
> > https://mail.nmr.mgh.harvard.edu/mailman/listinfo/mne_analysis
> >
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> >




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