<div dir="ltr">Thanks Michael for the detailed explanation. sorry for misunderstanding you before, making you work hard to explain yourself.<br>thanks again, yuval<br><br><div class="gmail_quote">On 24 March 2011 16:24, Michael Wibral <span dir="ltr"><<a href="mailto:michael.wibral@web.de">michael.wibral@web.de</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin: 0pt 0pt 0pt 0.8ex; border-left: 1px solid rgb(204, 204, 204); padding-left: 1ex;"><div><div style="min-height: 200px;">Dear Yuval,<br><br> you are completely right with what you wrote but you might have misunderstood my post (as I say exactly the same):<br>
<br>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).<br>
<br>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!).<br>
<br>Let's denote cov computation by C<.|.>, then<br><br>(1a) C<a+b|c>=(C<a|c> + C<b|c> ) (this is linearity in first argument)<br>(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)<br>
<br>concatenating signal vectors a, b and c,d into …[a b] and [c d] means:<br>(1c) C<[a b]|[c d]> = (C<a|c> + C<b|d> ) <br><br>For numerical factors 1/m, 1/n as they arise in averaging the following holds:<br>
C<1/m*a| 1/n*b> =1/m*1/m*C<a|b><br><br><br>(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)<br>
<br><br>Now to trial averging and cov computation (see 3) versus cov computation and averaging afterwards (see 4):<br><br>for two trials a,b and two channels c1,c2:<br><br>(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)<br>
(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)<br><br>Michael<br>
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<b>Von:</b> "Yuval Harpaz" <<a href="mailto:yuvharpaz@gmail.com" target="_blank">yuvharpaz@gmail.com</a>><br><b>Gesendet:</b> Mar 24, 2011 10:41:02 AM<div><div></div><div class="h5"><br><b>An:</b> "Email discussion list for the FieldTrip project" <<a href="mailto:fieldtrip@donders.ru.nl" target="_blank">fieldtrip@donders.ru.nl</a>><br>
<b>Betreff:</b> Re: [FieldTrip] Different way of calculating the covariance for LCM<br><br>
<div>Dear Michael, Jean-Michel and FieldTrip users.<br>To my experience and understanding it does matter if you first asses the covariance and then average or the other way around.<br>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.<br>
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.<br><br>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<br>
1. running it with cfg.keeptrials='yes'; then I averaged (across trials) the 3d matrix to get 2d matrix.<br>2. running it without keeping trials; it gives a 2d covariance matrix.<br>3. averaging the data with this command: mD1st=ft_timelockanalysis([],D1st). no filters baseline correction or anything.<br>
then I ran timelockanalysis again to compute the covariance (no keeptrials of course).<br><br>the result is that 1. and 2. gave the same result but not 3.<br><br>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.<br>
<br>If I am wrong here I will appreciate further help with this issue.<br><br>thanks<br>yuval<br><br><br>
<div class="gmail_quote">On 23 March 2011 21:27, Michael Wibral <span><<a href="mailto:michael.wibral@web.de" target="_blank">michael.wibral@web.de</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin: 0pt 0pt 0pt 0.8ex; border-left: 1px solid rgb(204, 204, 204); padding-left: 1ex;"><br> Dear Fieldtrip users interested in covariance computation,<br> dear Yuval,<br><br> I would like to my opinion on covariance computation up for discussion here.<br>
<br> 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,<br>
<br> (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.<br>
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.<br>
<br> (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.<br>
<br> 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).<br>
<br> 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).<br><br><br> Please let me know If I overlooked something important.<br>
Michael<br><br><br><br><br> ---------------------<br> Von: "Jean-Michel Badier" <<a href="mailto:jean-michel.badier@univmed.fr" target="_blank">jean-michel.badier@univmed.fr</a>><br> Gesendet: Mar 23, 2011 9:42:53 AM<br>
An: "Email discussion list for the FieldTrip project" <<a href="mailto:fieldtrip@donders.ru.nl" target="_blank">fieldtrip@donders.ru.nl</a>><br> Betreff: Re: [FieldTrip] Different way of calculating the covariance for LCM<br>
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<div><br><br> Thanks Yuval,<br><br><br> Le 22/03/11 10:55, Yuval Harpaz a écrit :<br><br> So just run the commands on an unaveraged dataset.<br><br><br><br> Yes but that would be correct if there was only one trial in the data set (see the message from Luisa).<br>
<br><br><br> Another option to consider is the one used by Dr. Robinson when performing SAMerf (we have his tool [here], works for our 4D machine).<br><br><br> Thanks for it I will test it.<br><br><br> 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.<br>
<br><br> On 22 March 2011 10:43, Jean-Michel Badier <[<a href="mailto:jean-michel.badier@univmed.fr" target="_blank">jean-michel.badier@univmed.fr</a>]> wrote:<br><br><br> Dear Yuval,<br><br> I have to admit that I did not look at the matlab routines.<br>
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).<br>
<br> Jean-Michel<br><br> Le 22/03/11 05:47, Yuval Harpaz a écrit :<br><br><br><br> Dear Jean Michel<br> 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.<br>
<br> Yuval<br><br><br> On 21 March 2011 22:58, Jean-Michel Badier <[<a href="mailto:jean-michel.badier@univmed.fr" target="_blank">jean-michel.badier@univmed.fr</a>]> wrote:<br><br><br> Dear fieldtrip users,<br>
<br> There are different ways of estimating the covariance for LCMV calculation.<br> If I am correct:<br><br> 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).<br>
<br> 2. Estimate the covariance from the average of the covariance rather than the covariance of the average using cfg.keeptrials = "yes"<br>
<br> 3. Estimate the covariance from the whole trials concatenated together.<br> Is there an easy way to do that in fieldtrip (beside create a new data set of one trial constituted of all the trials)?<br>
<br> Thanks<br><br> Jean-Michel<br><br><br> -- Jean-Michel Badier PhD<br> Laboratoire de MagnétoEncéphaloGraphie INSERM U751. Aix Marseille Université 33 (0)4 91 38 55 62 [<a href="mailto:jean-michel.badier@univmed.fr" target="_blank">jean-michel.badier@univmed.fr</a>]<br>
Service de Neurophysiologie Clinique. CHU Timone 264 Rue Saint-Pierre, 13005 Marseille-France<br>
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<div>Jean-Michel Badier<br><br> Laboratoire de MagnétoEncéphaloGraphie<br> INSERM U751. Aix Marseille Université<br> 33 (0)4 91 38 55 62<br> [<a href="mailto:jean-michel.badier@univmed.fr" target="_blank">jean-michel.badier@univmed.fr</a>]<br>
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