[Eeglablist] ICA suggests weirdly low rank

[Евгений Машеров]

Hi Makoto,

Thanks for the interesting and useful article.
I encountered the problem of rank deficiency in averaged referenced in connection with dipole localization. It turned out that despite getting rid of artifacts from ear electrodes, localization accuracy decreased. It seems to me that this is due to the fact that the dimension of the signal space is reduced from n to n-1, so that 1/n part of the signal information goes into the orthogonal subspace and is not available for analyze. Trying to eliminate this effect, they tried to replace the average when calculating the averaged electrode with the median, which formally restored the rank, since the median is equal to the average only on the average. However, the desired restoration of accuracy did not happen.
On older devices with analog averaging, differences in resistor values ​​and amplifier characteristics led to the fact that the sum of signals at the averaged lead could differ from zero and the rank could not be deficient. In a digital implementation, the sum of the signals differs from zero by an amount of the order of the error in number representation (10-7 - 10-10).
It seems to me that to use the ICA it is better to work with the ear reference, and after eliminating artifacts, if such a need arises, it can be recalculated to the average.

Eugen Masherov

> Hi Eugen,
> 
> There are three levels of understanding on the issue of average reference.
> 
> Level 1: Why rank -1? I don't understand it.
> Level 2: a1+a2+a3+...+an=0 is a hallmark of linear dependency. Rank -1 is a
> mathematical consequence.
> Level 3: Rank -1 is a result of not including the initial reference
> electrode in calculating the average potential, which is an invalid way to
> re-reference the data.
> 
> If you are curious, here is the recent paper discussing this issue. Note
> that this paper was motivated by discussions on this mailing list.
> https://urldefense.com/v3/__https://www.frontiersin.org/articles/10.3389/frsip.2023.1064138/full__;!!Mih3wA!BGpWl3IPPtL4Xe_3seDjV0fOrNUPhiSC3N4CIZuiSdVRExXxhamOn45qE5-PrqEIbhCMu_W_1An0ORQE7IKL8gFB7og$
> 
> Makoto
>