[Eeglablist] Two step source connectivity analysis (as implemented in SIFT)

Tue Feb 11 13:02:02 PST 2014

Sorry, I need to turn my e-mail off at night.  When I get really tired my self-auditing functions turn off and I don’t check what I’m saying as carefully as I should.

So:

1) I should have said “sphering” not “de-sphering”.  It’s also called “whitening”.

2) My description of the role of sphering was a bit off-the-cuff.

So anyway, the sphering results in an uncorrelated matrix as I said.  

The process is pretty simple.  You just take the covariance matrix of the variables and use matrix algebra to divide the data matrix by it (using an inverse of the covariance matrix).  You end up with a transform of the data matrix where the variables are uncorrelated.

EEGlab’s runica function uses sphering by default so, unless you explicitly turn it off, you’re using sphering if you use runica.

Here is the relevant portion of the runica code:

if strcmp(sphering,'on'), %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  icaprintf(verb,fid,'Computing the sphering matrix...\n');
  sphere = 2.0*inv(sqrtm(double(cov(data')))); % find the "sphering" matrix = spher()
  if ~weights,
      icaprintf(verb,fid,'Starting weights are the identity matrix ...\n');
      weights = eye(ncomps,chans); % begin with the identity matrix
  else % weights given on commandline
      icaprintf(verb,fid,'Using starting weights named on commandline ...\n');
  end
  icaprintf(verb,fid,'Sphering the data ...\n');
  data = sphere*data; % decorrelate the electrode signals by 'sphereing' them

As you can see, the output tells you when it is performing sphering.

My paper doesn’t go into the details of sphering but it’s pretty standard matrix algebra.  My paper focuses on providing a translation between ICA and PCA terminology (since they come from different academic disciplines they use different terminology) and how they interact with the special characteristics of ERP datasets.

The following book talks about sphering in the ICA context:

Hyvärinen, A., Karhunen, J., & Oja, E. (2001). Independent Component Analysis. New York: John Wiley & Sons.

The way he describes it, it’s moving partway to independence by first decorrelating everything.  So at this point you sort of have a rotation but it hasn’t taken the higher moments into account yet.

The Infomax rotation then maximizes for the parameters I mentioned last time.  As they rotate away from their initial uncorrelated state, they are likely to become correlated to some degree, in part because they are now taking the higher moments into account and the cost of doing so may be to accept some correlations and in part because the non-normality parameter has nothing to do with being uncorrelated (as far as I know) and so rotating towards it can result in an increase in correlations.

All this has been accomplished in a variable space in which they are uncorrelated.  When the data are then translated back into the original space, larger correlations may result.  So in the boundary case where no Infomax rotation has occurred, the data matrix would go from being uncorrelated to correlated when the sphering operation is reversed.  If the Infomax rotation has been fully successful at seeking independence even without the second order information (correlations), then reversing the sphering operation would not result in a correlated dataset (of activations).  If only partially successful in its rotation, then one would expect some correlations to be added back in.

The simplest demonstration is to run an ICA and see if your activations are correlated or not.  They usually will be.  So by definition, the ICA components are NOT independent.  What has happened is that they have been rotated in a manner to maximize independence (and also non-normality) and the combination of maximizing those two parameters is a set of ICA components that better correspond to the latent variables in the data (blind separation).

The key point here is that while the Infomax rotation is maximizing independence (including lack of correlations), it does not require it.  So for EEG the result is usually variables that are not fully independent, which is a good thing since brain activity should not in fact be fully independent, which means that there is no problem with performing connectivity analyses.  I think the confusion in this context comes from the term “independent”.  I think folks sometimes take that to mean “separate”, which is not the same thing as “independent”.  Two measures can be fully separated but still be correlated (e.g., because the two things they are measuring are influencing each other).  So that is how I would describe it to the reviewer in non-mathematical language.

Anyway, this is my take on things but I’m not the ICA expert that someone like Scott is so any comments or corrections welcome.

Joe

On Feb 11, 2014, at 12:52 PM, Thomas, Robin <thomasrd at miamioh.edu> wrote:

> Joe,
> 
> I am somewhat confused about step #2.  If I performed an ica per
> eeglab's process, would such de-sphering and sphering be a part of
> that process already or something that would need to be done
> "manually"?  Is this process explained in your paper or another paper
> (what is it exactly that is done)?
> 
> - Robin
> 
> On Mon, Feb 10, 2014 at 6:20 PM, Joseph Dien <jdien07 at mac.com> wrote:
>> This is a really great question that highlights some widespread ambiguities
>> in how ICA tends to be discussed in the EEG field.
>> 
>> To bullet point it:
>> 
>> 1) ICA (specifically thinking about Infomax here) separates components using
>> an algorithm that maximizes non-normality of the variables (Central Limit
>> Theorem says a mix should be more normal so non-normal variables are more
>> likely to not be a mix) and also to minimize the degree to which one
>> variable or its higher powers can predict another variable (independence).
>> 
>> 2) While this makes sense in principle, in practice the second order moments
>> (correlations) tend to swamp the algorithm, making it not work, so it is
>> common practice to desphere the variables (remove the correlations), run the
>> ICA, and then sphere them again (put the correlations back in).
>> 
>> 3) So although the ICA algorithm maximizes independence, the EEG variables
>> resulting from a typical ICA procedure are NOT independent.  They are
>> correlated.  This is a good thing since it does not make sense for brain
>> processes to always be orthogonal to each other.  Clearly they should in
>> fact be able to influence each other.  A mathematical procedure that imposed
>> such an artificial constraint would be unlikely to yield accurate results.
>> So we get the benefit of separated components as well as flexibility for
>> them to be correlated.  This means that ICA-derived measures of sources can
>> absolutely reflect influences on each other (in principle, without
>> addressing the specifics of SIFT).
>> 
>> Of course, there are subtleties here about the nature of how one source
>> might influence another and so for this to work the nature of the influence
>> would need to be more apparent in the second-order moments than the higher
>> order moments, but I think this is not an unreasonable assumption.  Of
>> course, this discussion only applies if you did indeed desphere/sphere in
>> your ICA procedure.
>> 
>> I go into some discussion of how ICA is applied to ERPs and how it compares
>> to PCA rotations in my 2007 paper, where I also demonstrate that in the
>> temporal domain (time points), Promax (an oblique rotation that also allows
>> for correlated components) yields better results with a simulated dataset,
>> although ICA (Infomax) does a better job in the spatial domain (channels).
>> 
>> Dien, J., Khoe, W., & Mangun, G. R. (2007). Evaluation of PCA and ICA of
>> simulated ERPs: Promax versus Infomax rotations. Human Brain Mapping, 28(8),
>> 742-763.
>> 
>> Interestingly, a similar conclusion (Langers, 2009) has been made for fMRI
>> datasets (although here ICA works better for temporal and Promax works
>> better for spatial, for reasons that make sense in view of the discussion in
>> my 2007 paper).
>> 
>> Cheers!
>> 
>> Joe
>> 
>> 
>> 
>> On Feb 10, 2014, at 7:57 AM, Bethel Osuagwu <b.osuagwu.1 at research.gla.ac.uk>
>> wrote:
>> 
>> Hi
>> I am not an expert but I just want to give my own opinion!
>> 
>> I do not think that temporal independence of two variables (A and B) violets
>> causality between them as implemented in SIFT. In fact if  say A=sunrise and
>> B=ice-cream-sale, then the ICA in EEGLAB should find that A is maximally
>> temporaly independent from B. However we know there is causal flow from A to
>> B.
>> 
>> This is what I think, but I wait to be corrected so that I can learn!
>> 
>> Thanks
>> Bethel
>> ________________________________________
>> From: eeglablist-bounces at sccn.ucsd.edu [eeglablist-bounces at sccn.ucsd.edu] On
>> Behalf Of IMALI THANUJA HETTIARACHCHI [ith at deakin.edu.au]
>> Sent: 07 February 2014 01:27
>> To: mullen.tim at gmail.com
>> Cc: eeglablist at sccn.ucsd.edu
>> Subject: [Eeglablist] Two step source connectivity analysis (as implemented
>> in SIFT)
>> 
>> Hi Tim and the list,
>> 
>> I am just in need of a clarification regarding the ICA source reconstruction
>> and the subsequent MVAR -based effective connectivity analysis using the
>> components, which is the basis of the SIFT toolbox. I was trying to use this
>> approach in my work but was questioned on the validity using ICA and
>> subsequent MVAR analysis by my colleagues.
>> 
>> "When using independent component analysis (ICA), we assume the mutual
>> independence
>> of underlying sources, however when we try to estimate connectivity between
>> EEG sources,
>> we implicitly assume that the sources may be  influenced by each other. This
>> contradicts the
>> fundamental assumption of mutual independence between sources in ICA [Cheung
>> et al., 2010, Chiang et al., 2012, Haufe et al., 2009 ]. "
>> 
>> So due to this reason different approaches such as MVARICA,
>> CICAAR(convolution ICA+MVAR),  SCSA and state space-based methods have been
>> proposed as ICA+MVAR based source connectivity analysis techniques.
>> 
>> 
>> ·         So, how would you support the valid use of SIFT ( ICA+MVAR as a
>> two-step procedure) for the source connectivity analysis?
>> 
>> 
>> ·         If I argue that I do not assume independent sources but rely on
>> the fact that ICA will decompose the EEG signals and output 'maximally
>> independent' sources and then, I subsequently model for the dependency, will
>> you agree with me? How valid would my argument be?
>> 
>> It would be really great to see different thoughts and opinions.
>> 
>> Kind regards
>> 
>> Imali
>> 
>> 
>> Dr. Imali Thanuja Hettiarachchi
>> Researcher
>> Centre for Intelligent Systems research
>> Deakin University, Geelong 3217, Australia.
>> 
>> Mobile : +61430321972
>> Email: ith at deakin.edu.au<mailto:ith at deakin.edu.au>
>> Web :www.deakin.edu.au/cisr<http://www.deakin.edu.au/cisr>
>> 
>> [cid:image001.jpg at 01CF23FF.F8259940]
>> 
>> 
>> 
>> 
>> 
>> 
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>> 
>> 
>> --------------------------------------------------------------------------------
>> 
>> Joseph Dien,
>> Senior Research Scientist
>> Maryland Neuroimaging Center
>> University of Maryland
>> 
>> E-mail: jdien07 at mac.com
>> Phone: 202-297-8117
>> http://joedien.com
>> 
>> 
>> 
>> 
>> 
>> 
>> 
>> 
>> 
>> 
>> 
>> 
>> 
>> 
>> 
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> 
> 
> 
> -- 
> PLEASE NOTE NEW EMAIL:  thomasrd at MiamiOH.edu
> 
> Robin D. Thomas, Ph.D.
> Full Professor and Director
> Center for Human Psychophysiology
> Program in Brain and Cognitive Science
> Miami University, Oxford, OH 45056

--------------------------------------------------------------------------------

Joseph Dien,
Senior Research Scientist
Maryland Neuroimaging Center
University of Maryland 

E-mail: jdien07 at mac.com
Phone: 202-297-8117
http://joedien.com

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