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Thanks Michael and Yuval,<br>
<br>
That clarify the theoretical points.<br>
Clearly the point 2 and 3 of my original message would give the same
results.<br>
<br>
Jean-Michel<br>
<br>
Le 24/03/11 16:58, Yuval Harpaz a écrit :
<blockquote
cite="mid:AANLkTikXcO_O9Oh1q3sB9F=SZn+sdj9CL=bF9SNhdctk@mail.gmail.com"
type="cite">
<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 moz-do-not-send="true"
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>
<br>
<br>
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<blockquote style="border-left: 2px solid
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padding-top: 5px;">
<hr>
<b>Von:</b> "Yuval Harpaz" <<a
moz-do-not-send="true"
href="mailto:yuvharpaz@gmail.com"
target="_blank">yuvharpaz@gmail.com</a>><br>
<b>Gesendet:</b> Mar 24, 2011 10:41:02 AM
<div>
<div class="h5"><br>
<b>An:</b> "Email discussion list for
the FieldTrip project" <<a
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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
moz-do-not-send="true"
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
moz-do-not-send="true"
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
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target="_blank">fieldtrip@donders.ru.nl</a>><br>
Betreff: Re: [FieldTrip] Different
way of calculating the covariance
for LCM<br>
<div>
<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 moz-do-not-send="true"
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
moz-do-not-send="true"
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
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Université 33 (0)4 91 38 55 62
[<a moz-do-not-send="true"
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33 (0)4 91 38 55
62<br>
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