[FieldTrip] Different way of calculating the covariance for LCM
Jean-Michel Badier
jean-michel.badier at univmed.fr
Thu Mar 24 22:08:46 CET 2011
Thanks Michael and Yuval,
That clarify the theoretical points.
Clearly the point 2 and 3 of my original message would give the same
results.
Jean-Michel
Le 24/03/11 16:58, Yuval Harpaz a écrit :
> Thanks Michael for the detailed explanation. sorry for
> misunderstanding you before, making you work hard to explain yourself.
> thanks again, yuval
>
> On 24 March 2011 16:24, Michael Wibral <michael.wibral at web.de
> <mailto:michael.wibral at web.de>> wrote:
>
> Dear Yuval,
>
> you are completely right with what you wrote but you might have
> misunderstood my post (as I say exactly the same):
>
> What I wrote is exactly this: concatenation of trials followed by
> cov computation or trial wise cov computation followed by
> averaging is the same, BUT different from averaging and then
> computing cov (just as you point out).
>
> Also note that i did NOT write that cov computation is linear BUT
> bi-linear (see also the wikipedia article on covariance), hence
> you have to honor distributivity in both arguments (and since you
> average both arguments, that's were the problem comes in!).
>
> Let's denote cov computation by C<.|.>, then
>
> (1a) C<a+b|c>=(C<a|c> + C<b|c> ) (this is linearity in
> first argument)
> (1b) C<a+b|c+d>=(C<a|c> + C<b|c> + C<a|d> + C<b|d> ) (this
> is bi-linearity when considering both arguments)
>
> concatenating signal vectors a, b and c,d into …[a b] and [c d] means:
> (1c) C<[a b]|[c d]> = (C<a|c> + C<b|d> )
>
> For numerical factors 1/m, 1/n as they arise in averaging the
> following holds:
> C<1/m*a| 1/n*b> =1/m*1/m*C<a|b>
>
>
> (i.e. concatenation or single trial cov computation and later
> averaging are the same. This is because the dot product that is
> used to compute the covariance for two signals vectors reads like
> this: [a b c d] * [e f g h]=ae+bf+cg+dh=[a b] * [e f] + [c
> d]*[gh], so whether you multiply the vectors in pieces and later
> add results or rather multiply them as a whole doesn't matter.
> What matters is that a, b, ... stay exactly the same in both cases)
>
>
> Now to trial averging and cov computation (see 3) versus cov
> computation and averaging afterwards (see 4):
>
> for two trials a,b and two channels c1,c2:
>
> (4)
> C<AVG(c1a|c1b)|AVG(c2a,c2b)>=1/2*1/2*(C<c1a|c2a>+C<c1b|c2b>+C<c1a|c2b>+C<c1b|c2a>)
> (here things get multiplied across trials, the two factors 1/2
> come from each separate averaging in the first and second argument)
> (5) AVG(C<c1a|c2a>,C<c1b|c2b>) =1/2*(C<c1a|c2a>+C<c1b|c2b>)
> (here things do only get multiplied within trials, we have only
> the outer averaging, hence only a factor of 1/2)
>
> Michael
>
>
>
> ------------------------------------------------------------------------
> *Von:* "Yuval Harpaz" <yuvharpaz at gmail.com
> <mailto:yuvharpaz at gmail.com>>
> *Gesendet:* Mar 24, 2011 10:41:02 AM
>
> *An:* "Email discussion list for the FieldTrip project"
> <fieldtrip at donders.ru.nl <mailto:fieldtrip at donders.ru.nl>>
> *Betreff:* Re: [FieldTrip] Different way of calculating the
> covariance for LCM
>
> Dear Michael, Jean-Michel and FieldTrip users.
> To my experience and understanding it does matter if you first
> asses the covariance and then average or the other way around.
> First, as far as my understanding goes, if we have some brain
> activity such as alpha, which is not phase locked to the event
> then in the covariance will not 'catch' too much covariability
> there after averaging, compared to unaveraged data where it
> should. if the process of calculating covariance and averaging
> id linear then I must be wrong here.
> anyway, beamforming on averaged and unaveraged base covariance
> matrices give different results in fieldtrip. this is because
> the covariance matrices are not the same with or without
> averaging.
>
> I ran the following test to check it all. I took the same
> structure as in the example above, and I tested the covariance
> matrix after
> 1. running it with cfg.keeptrials='yes'; then I averaged
> (across trials) the 3d matrix to get 2d matrix.
> 2. running it without keeping trials; it gives a 2d covariance
> matrix.
> 3. averaging the data with this command:
> mD1st=ft_timelockanalysis([],D1st). no filters baseline
> correction or anything.
> then I ran timelockanalysis again to compute the covariance
> (no keeptrials of course).
>
> the result is that 1. and 2. gave the same result but not 3.
>
> It seems to me that the process is not linear, at least in
> fieldtrip, and that one doesn't need to specify keeptrials for
> unaveraged data because it calculates the covariance for every
> trial and then averages the cov matrices.
>
> If I am wrong here I will appreciate further help with this issue.
>
> thanks
> yuval
>
>
> On 23 March 2011 21:27, Michael Wibral <michael.wibral at web.de
> <mailto:michael.wibral at web.de>> wrote:
>
>
> Dear Fieldtrip users interested in covariance computation,
> dear Yuval,
>
> I would like to my opinion on covariance computation up
> for discussion here.
>
> Covariance is a (bi-)linear measure (like an inner
> (scalar) product) and should in principle be commutative
> mathematically with other linear procedures as long as you
> respect the distributative law from elementary math. Hence,
>
> (1) averaging single trial covariances or computing the
> covariance of concatenated trials should give the same
> result if a am not mistaken. In these two approaches
> samples from your two time course are in the end
> multiplied with each other one by one. Whichever route of
> these two you take does not matter at all.
> This holds as long as you treat baseline correction and
> filtering (!) exactly the same way in both cases -
> otherwise you'll additionally get the baseline covariance
> over trials as a term in the covariance matrix or
> filtering related differences. This latter point maybe
> important if you post hoc decide on a band width to
> confine your beamformer analysis to and do not go back to
> the raw data for the band pass filtering.
>
> (2) In contrast, when you compute covariance from a
> precomputed trial average then every sample at time t in
> one trial on channel A will in the course of the
> calculations be implicitely multiplied with samples at
> time t at EVERY other trial on channel B. So this
> covariances focuses on covariance structures that are
> consistent irrespective of the correct pairings of trials.
>
> It's a little bit like first computing the power spectrum
> and averaging to get total activity (induced+evoked) -
> this would correspond to procedure (1) , or first
> averaging and then computing spectral power in order to
> get the power of evoked activity alone - this would
> correspond to procedure (2).
>
> This said, when you're interested in the sources of all
> task related oscillatory activity - not only the activity
> phase locked to a stimulus you should pick option (1).
>
>
> Please let me know If I overlooked something important.
> Michael
>
>
>
>
> ---------------------
> Von: "Jean-Michel Badier" <jean-michel.badier at univmed.fr
> <mailto:jean-michel.badier at univmed.fr>>
> Gesendet: Mar 23, 2011 9:42:53 AM
> An: "Email discussion list for the FieldTrip project"
> <fieldtrip at donders.ru.nl <mailto:fieldtrip at donders.ru.nl>>
> Betreff: Re: [FieldTrip] Different way of calculating the
> covariance for LCM
>
>
> Thanks Yuval,
>
>
> Le 22/03/11 10:55, Yuval Harpaz a écrit :
>
> So just run the commands on an unaveraged dataset.
>
>
>
> Yes but that would be correct if there was only one trial
> in the data set (see the message from Luisa).
>
>
>
> Another option to consider is the one used by Dr.
> Robinson when performing SAMerf (we have his tool [here],
> works for our 4D machine).
>
>
> Thanks for it I will test it.
>
>
> The idea is to calculate the covariance on all trials,
> calculate weights by this covariance (keep filter
> in LCMV) and then apply these weights on the
> averaged data. I found it useful because the
> covariance is better for longer datasets, and the
> averaging in the end increases the signal to noise
> ratio. I do not know exactly how to do it in
> fieldtrip.
>
>
> On 22 March 2011 10:43, Jean-Michel Badier
> <[jean-michel.badier at univmed.fr
> <mailto:jean-michel.badier at univmed.fr>]> wrote:
>
>
> Dear Yuval,
>
> I have to admit that I did not look at the matlab
> routines.
> In item 2 I suppose that the covariance is calculated for
> each trial then averaged. In item 3 I would
> like to calculate the covariance from all
> the signal (the trials being concatenated).
>
> Jean-Michel
>
> Le 22/03/11 05:47, Yuval Harpaz a écrit :
>
>
>
> Dear Jean Michel
> As far as I know you can do it on an averaged
> data structure (item 1) or do the same with
> the data structure before
> averaging (3). I did not
> understand what you meant by 2.
>
> Yuval
>
>
> On 21 March 2011 22:58,
> Jean-Michel Badier <[jean-michel.badier at univmed.fr
> <mailto:jean-michel.badier at univmed.fr>]> wrote:
>
>
> Dear fieldtrip users,
>
> There are different ways of estimating the
> covariance for LCMV calculation.
> If I am correct:
>
> 1. As suggested in one of the tutorial one
> can apply the calculation of the
> covariance directly on the average
> data (for the different
> periods of interest that are
> at least a base line and the
> period of interest).
>
> 2. Estimate the covariance from the
> average of the covariance rather than the
> covariance of the average using
> cfg.keeptrials = "yes"
>
> 3. Estimate the covariance from the whole
> trials concatenated together.
> Is there an easy way to do that in
> fieldtrip (beside create a new data set of
> one trial constituted of all the
> trials)?
>
> Thanks
>
> Jean-Michel
>
>
> -- Jean-Michel Badier PhD
> Laboratoire de
> MagnétoEncéphaloGraphie INSERM U751. Aix
> Marseille Université 33 (0)4 91 38
> 55 62 [jean-michel.badier at univmed.fr
> <mailto:jean-michel.badier at univmed.fr>]
> Service de
> Neurophysiologie Clinique. CHU
> Timone 264 Rue
> Saint-Pierre, 13005
> Marseille-France
>
>
>
>
>
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> --
>
> Y.Harpaz
>
> a link to the BIU MEG lab:
> [http://faculty.biu.ac.il/~goldsa/index.html
> <http://faculty.biu.ac.il/%7Egoldsa/index.html>]
>
>
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> --
>
> Y.Harpaz
>
> a link to the BIU MEG lab:
> [http://faculty.biu.ac.il/~goldsa/index.html
> <http://faculty.biu.ac.il/%7Egoldsa/index.html>]
>
>
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> Jean-Michel Badier
>
> Laboratoire de MagnétoEncéphaloGraphie
> INSERM U751. Aix Marseille Université
> 33 (0)4 91 38 55 62
> [jean-michel.badier at univmed.fr
> <mailto:jean-michel.badier at univmed.fr>]
>
> Service de Neurophysiologie Clinique.
> CHU Timone
> 264 Rue Saint-Pierre, 13005 Marseille-France
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Jean-Michel Badier
Laboratoire de MagnétoEncéphaloGraphie
INSERM U751. Aix Marseille Université
33 (0)4 91 38 55 62
_jean-michel.badier at univmed.fr_ <mailto:jean-michel.badier at univmed.fr>
Service de Neurophysiologie Clinique. CHU Timone
264 Rue Saint-Pierre, 13005 Marseille-France
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