[Eeglablist] Should Average Reference Include Eye Channels?

[Scott Makeig]

Unlike Alejandro, I regard decently recorded EEG data as having rather
little noise, e.g., Gaussian noise, though it does have some, of course --
plus many (non-Gaussian) non-brain 'artifact' sources contributing to it
(some, including eye movement artifacts, large).

Biophysically, referencing to the average field at the whole surface of a
spherical head would be ideal in the sense that the resulting potentials
would not be spatially biased by the reference. But an EEG montage cannot
cover the whole head (when neck is attached...), and the head is not quite
spherical, etc. ...

Simply including biopolar channels in the average reference calculation
does not make sense, because this will introduce additional bias in the
reference:

For example, if scalp channels (s_1 - r_1) and (s_2 - r_1) are both ref'd
to r_1, and eye channel e_1 to e_2, then average reference for the
common-reference (s) channels is,  for s_1, ...

(s_1-r_1) - [(s_1-r_1) + (s_2-r_1)]/2
  = s_1-r_1-s_1/2+r_1/2 - s_2/2+r_1/2
  = s_1 - (s_1+s_2)/2          <== this average reference channel contains
no r_1...

But including bipolar channel (e1-e2) and then taking average reference,

(e_1-e_2) - [(s_1-r_1) + (s_2-r_1) + (e_1-e_2)]/3
  =  e_1 - e_2 - s_1/3 +r_1/3 - s_2/3 +r_1/3 - e_1/3 + e_2/3
  =  2/3(e_1-e_2) + 2/3r_1 - (s_1 + s_2)/3  <== this has no simple
interpretation

However, performing average reference on the common reference channels, and
then combining these with the bipolar channels into the ICA training matrix
is possible -- since all the signals are linearly related to the
independent source activations.

However, in this case the IC scalp maps may have a discontinuity between
the  eye and scalp channels, since they do not share a reference.  Removing
the bipolar eye channels from these maps involves performing a
pseudo-inverse on the IC map matrix, however, which may introduce some
degree of inaccuracy ...

In practice, ICA typically picks up eye activities very well from their
frontal scalp channel projections -- so I'd recommend using the extra
channels for any other common-reference channel positions of interest, and
not for bipolar eye channels.

Scott Makeig


On Fri, Feb 17, 2012 at 2:40 PM, Matthew Stief <ms2272 at cornell.edu> wrote:

> Thank you very much for this Alejandro.
>
> Just to confirm I understand you correctly, you are saying that it does
> not matter that the eye movement artifacts in the EOG monopolar channels
> are larger than those present in scalp electrodes, because the assumption
> is that the overall average of the noise will still be gaussian and
> therefore sum to zero, regardless of whether it includes the higher
> magnitude activity in EOG channels?
>
>
>
>
> On Wed, Feb 15, 2012 at 4:50 AM, Alejandro <alejo.ojeda83 at gmail.com>wrote:
>
>>
>> Here I will  address the question: should I include EOG monopolar
>> channels in the average reference? After some years now trying this and
>> that for pre-processing my EEG data, doing some reading, and asking to
>> people more knowledgeable than me, I'll explain why I would answer YES to
>> that question.
>>
>> The problem of referencing the data comes from the fact that
>> physiological signals measured on the scalp are always contaminated by
>> noise, which is several orders of magnitude bigger. For instance, let's
>> suppose we recorded the voltage from three different places:
>>
>> **x1 = x~1 + n1
>> x2 = x~2 + n2
>> x3 = x~3 + n3
>>
>>
>> Where the signal we measure *X* is assumed to be a mixture of two
>> components: 1) the electrical activity generated by physiological processes
>> inside the brain, transmitted across several layers of tissue that finally
>> reach the scalp , x~, and 2) the noise at that point, *n*. Because the
>> noise itself is a signal formed by a contribution of many non-physiological
>> processes from a different nature (instrumental, electromagnetic, thermal,
>> etc.) is usually represented as a random variable assumed gaussian with
>> zero mean.
>>
>> The purpose of the average reference is to minimize the contribution of
>> the noise keeping the tiny variations inside it that represent the
>> physiological activity. Let's do the math for one electrode:
>>
>> x1r = x~1 + n1 - ( x~1 + n1 + x~2 + n2 + x~3 + n3 ) / 3
>>
>> where x1r represents the signal in the electrode 1 after removing the
>> average reference. Re-grouping terms we have:
>>
>> x1r = x~1  - ( x~1  + x~2  + x~3  ) / 3  -  ( *n*1 - n1 + n2 + n3 ) / 3
>>
>> But because the sum of several gaussian processes ( as the noise is
>> considered ) is also a gaussian process, then the contribution of the noise
>> goes to zero, ending up with a signal x1r that is still a mixture of
>> things but at least those things represent more closely the physiological
>> precesses:
>>
>> x1r = x~1  - ( x~1  + x~2 + x~3  ) / 3
>>
>> Generalizing to n electrodes we could say that:
>> *X* r = ( *I* n -*1*⋅*1* T /n) *X*
>> *
>> *
>> the average reference data *X* r is equal to the so called "average
>> reference operator" *H*=( *I* n -1⋅1 T ) times the raw data *X*. To
>> construct this operator in Matlab just type:
>>
>> >> H = eye(n) - ones(n)/n;
>>
>> where *n* represents the number of channels we include in the average.
>>
>> One interesting thing about *H* is that it is what the mathematicians
>> call an idempotent matrix, this is a matrix where *H*H*...H=H. *This has
>> the practical implication that removing the average reference of your data
>> several times, without doing anything else in between, doesn't introduce
>> additional modifications.
>>
>> However it's true that multiplying by *H* could be seen as a mixing
>> process of "hopefully" physiological signals, and there is when ICA comes
>> to play an important roll. ICA will find (as far as it can and its
>> assumptions are close to the reality) those sources x~1 , x~2 , x~3 ... associated
>> to the local space and temporal coherent electrical activity generated
>> inside the tissue (I say "tissue" to be general because usually we have
>> components representing brain signals, muscle,  EKG, etc). Just for the
>> record, I'm not saying that *H* is the mixing matrix per se, I'm just
>> saying that it contributes to the mixing process that takes place in the
>> scalp and the other layers of tissue.
>>
>> I hope these thoughts can give you some clues about your pre-processing.
>> They represent part of my practical experience and reading, but this still
>> remains as an open field so don't take my words as pure dogma.
>>
>> Regards,
>> Alejandro
>>
>> --
> _________________________________________________________________
> Matthew Stief
> Human Development | Sex & Gender Lab | Cornell University
> http://www.human.cornell.edu/HD/sexgender
>
>
> Heterosexuality isn't normal, it's just common.
> -Dorothy Parker
>
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-- 
Scott Makeig, Research Scientist and Director, Swartz Center for
Computational Neuroscience, Institute for Neural Computation; Prof. of
Neurosciences (Adj.), University of California San Diego, La Jolla CA
92093-0559, http://sccn.ucsd.edu/~scott
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