[Eeglablist] dipfit uses average reference?

[Robert Oostenveld]

On 23 Nov 2006, at 2:18, arno wrote:
> David Groppe wrote:
>> I'm using Dipfit 2 to localize independent components using the  
>> spherical head model.  Apparently the software requires the data  
>> to use the average reference.  Why is this?
> Source localization assumes that the data is average reference (I  
> think it is because no current should get in or out). I do not  
> think it is really an option not to use average reference. Robert  
> might have more insight about that.

Although the "no current leaving the body/head" argument is valid, we  
also have to account for the limited sampling of the head: only the  
upper half is sampled sparsely. In principle you could use an  
arbitrary reference in your source reconstruction. The practical  
reason to use an average reference over the sampled electrodes in  
source estimation is that this prevents the solution to be biassed  
due to forward modelling errors at the reference electrode. Let me  
give a partially intuitive, partially mathematical explanation. This  
follows on the idea of Joseph.

Assume that you would use left mastoid as reference. That would mean  
that the measured value "V" at each electrode "x" is V_x, so the list  
of all measured values in the N channels is
  V_C3-V_M1
  V_Cz-V_M1
  V_C4-V_M1
  ...
  V_M1-V_M1  (this is zero)
  V_M2-V_M1

Those values can be modeled using the source model and the volume  
condution model. Now, lets assume a spherical volume conduction  
model. That is especially inaccurate for low electrodes, and the bony  
structure of the mastoid is definitely not modelled appropriately in  
a spherical model. So for the model potential "P" we would have the  
value at each of the N electrode also referenced to the model mastoid  
electrode:
  P_C3-P_M1
  P_Cz-P_M1
  P_C4-P_M1
  ...
  P_M1-P_M1  (this is zero)
  P_M2-P_M1

The source estimation algorithm tries to minimize the quadratic error  
between model potential distribution and the measurement, so the  
error term to be minimized is
  Total_Error
  = sum of quadratic error over all channels
  = [(V_C3-V_M1)-(P_C3-P_M1)]^2 + ....
  = [(V_C3-P_C3)-(V_M1-P_M1)]^2 + ....   (here the terms are re-ordered)

So for each channel the error term consists of a part that  
corresponds to the potential at the electrode of interest, plus a  
part that corresponds to the reference electrode. The error term  
corresponding to the reference electrode is identical over all  
channels (i.e. repeats in each channel), hence for each channel you  
are adding some error term for the reference electrode. Therefore,  
the minimum error ("minimum norm") solution will be one that  
especially tries to minimize the model error at the reference  
electrode (since that is included N times). In the case of a mastoid  
reference we know that there is a large volume conductor model error  
at M1, hence the source solution would mainly try to minimize that  
error term. The result would be that the source solution would be  
biassed, because it tries to reduce the (systematic) error at the  
reference.

The solution is to use an average reference (average over all  
measured electrodes). That implicitely assumes that the model error  
over all electrodes is on average zero, hence the minimum norm  
solution is not biassed towards a specific reference electrode.

I hope that this clarifies it.

best regards,
Robert

PS the maths in my explanation above are rather sloppy, but the  
argument still holds for a more elaborate mathematical derivation  
which would assume the forward model inaccuracies are uncorrelated  
over electrode sites.