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Is there any literature that has addressed electrode pooling? In particular, are there any standards concerning what pools (e.g. frontal, central, posterior vs. electrode pairs) could be used or any particular applications and limitations?

EDIT: By "pooling" I am refering to the procedure in which new channels are created based on pooling the raw data from the original data electrodes. This is a preprocessing step not to be confused with variance pooling for statistical testing.

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    $\begingroup$ What you call "pools" are often called "Regions of Interest" (ROIs). (Statistically, this is indeed of course a form of pooling.) Have you tried searching under that term? Maris discusses in various articles ways of determining ROIs from data (as opposed to a priori). $\endgroup$ – Livius Jun 10 '15 at 14:24
  • $\begingroup$ I am familiar with the pitfalls of a priori definitions of ROI in the context of fMRI (e.g. functional ROI vs anatomical ROIs), but was not aware that this term was used in EEG research. Which Maris paper are you refering to? I could not locate any paper by Maris that addresses ROIs. $\endgroup$ – noumenal Jun 10 '15 at 14:39
  • $\begingroup$ "Randomization tests for ERP topographies and whole spatiotemporal data matrices", but Eric Maris has published several papers on statistics in EEG and MEG, especially on "advanced" (non traditional, e.g. non parametric) techniques as well as closer examinations of fundamentals. $\endgroup$ – Livius Jun 10 '15 at 14:49
  • $\begingroup$ "Topography" might be another search term for you to consider. $\endgroup$ – Livius Jun 10 '15 at 14:53
  • $\begingroup$ @Livius Please refer to my latest edit. I am not refering to statistical testing. There is also nothing about ROIs in that paper. $\endgroup$ – noumenal Jun 10 '15 at 15:53
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This seems to be a rather painful terminology issue.

"Pooling" is used in statistics to describe combining data / different sources of information in a single model. It's of particular interest in hierarchical modelling approaches (e.g. mixed-effects modelling), and partial pooling -- broadly, the sharing of information between groups in a model without discarding the structure of the groups -- is perhaps one of the great innovations of 20th Century regression techniques. For more on this topic, see e.g. Gelman and Hill (2007).

In this sense, the use of regions of interest (ROIs) instead of individual electrodes as a categorical variable can be seen as a form of pooling -- you're combining data sources to gain more information, but losing some structure along the way. Because of the limited spatial resolution of EEG and the high correlation between adjacent electrodes, averaging across a ROI (and this is implicitly what's happening when we use ROI as a factor in ANOVA or regression models -- this is where the term "regression to the mean" comes from), may enhance our ability to see and detect the signal in the noise (i.e. increase statistical power). The coarsening of spatial (more specifically topographic) resolution is perhaps not tragic, because we are currently not able to make precise spatial predictions. Maris (2004) suggests an alternative way of dealing with topography.

This appears to be the sense that BrainVision Analyser uses the word "pooling". You're combining multiple data channels (electrodes) into a single one (ROI) via averaging. This is one of those places where you see that "channel" and "electrode" are not completely synonymous.

Incidentally, there are other ways you can combine information from electrodes. EEG measures voltage, which is per definition potential difference. The question, of course, is "difference to what?" In traditional recording setups, you have a single reference electrode (or perhaps a few linked electrodes as in linked mastoids) and you measure the difference in the electric field between the measurement electrode and the reference -- this is called "common reference". Sometimes, a pair of electrodes is measured relative to each other as "bipolar electrodes". This seems to be more common with the electrooculogram and has the advantage there that it makes eye movements more apparent. It has the disadvantage that you have effectively made one channel from two and cannot distinguish between the two original electrodes! A further disadvantage is that techniques like independent component analysis (ICA) depend on a common reference to do their magic, and as such, bipolar electrodes are problematic for ICA. A more advanced variant of common reference is "average reference", where the voltage is measured to the average voltage of all common reference electrodes. This seems weird at first, but it has its uses and emphasizes how the electrodes differ from each other (topography). Any common reference can be computed from any other via a relatively straight forward linear transformation. For more on this, see Luck (2005, 2014).

All of this stuff with referencing is important because it is possible to compute bipolar channels offline -- they are simply the difference of two common-reference electrodes. (Depending on which electrode you substract from which, you will get a different sign.) ERPLAB allows you to combine channels in fairly arbitrary ways to get new ones. If you're doing linear combinations of channels, it's fairly straightforward to compute what the statistics of the combined channel will look like compared to the statistics of the originals, but for arbitrary combinations, this can get confusing quite fast. Incidentally, the construction of ROIs or "pooling" in the BrainVision Analyser sense can be viewed as a special case of this -- you're adding a number of channels together and dividing by a constant (i.e. calculating the mean), which is a linear transformation.

Finally, you could also use existing channels to simulate missing or bad channels -- this is called interpolation. Good interpolation takes things like the curvature of the scalp and the electric field into account and is thus computationally more complex, often using techniques like splines. A simpler interpolation method is simply taking the average of adjacent electrodes (in some sense constructing a mini-ROI!), but this is not just less accurate numerically but also fails to take into account issues with the way electric fields sum together (see also: equivalent dipoles). Despite Scott Makeig's protests, I suspect that linear interpolation for a single electrode would work quite well in a high density setup. But then again, in a high density setup, you can generally afford to leave one electrode out of your analysis!

In each of the cases above -- construction of ROIs, arbitrary combination of channels, interpolation -- you are in some sense "pooling" the data you have. You are not gaining any information that was not there (e.g. an interpolated electrode does not really add much if any statistical power to your analysis), but you may be using the information that you had more efficiently. And that's what pooling is all about in statistics.

EDIT (2015-07-07 05:51 UTC): ICA and common vs. bipolar reference

As stated in the comments, ICA is actually based on relatively general assumptions that are generally indifferent to the choice of reference. In signal terms, the key assumption is that each measured channel is the linear combination (sums and constant multiples) of a number of components, which are typically taken to represent different (neueral) generators or (EEG) sources. In other words, ICA is a type of Blind Source Separation (BSS), which can be used to decompose a set of (directly) measured signals into a set of contributing sources, based on some additional assumptions. There are many different BSS techniques, each with different assumptions / goals, and thus each answering slightly different questions. For example, Principal Component Analysis (PCA) requires that components are orthogonal, while ICA require that components are independent. Somewhat simplified, PCA tries to separate the signal in terms of successive, "maximal" trends, which ICA tries to separate the signal into a series of distinct contributions.

This can be best seen in the following figure:

PCA and ICA decompositions of an X shaped collection of points

(Image reproduced from Jung et al (2001).)

The first component produced by PCA is a larger diagonal one explaining the general trend of increasing $y$ for increasing $x$, while the second component is a measure of "drift" or a correction to the general trend. Importantly, the components are at right angles (orthogonal) to each other. Combining each component in different proportions (i.e. weights), you arrive at the original two measurement series. In ICA, the components are not required to be at right angles to each other, and the result in this example is a component for each of the visually obvious groups.

Applying this to EEG data, we can view the choice of reference as some sort of builtin "offset" for measurement channel. This will change the weights in the recovered components, but ultimately not the components themselves. It's fairly trivial to transform the data from one common reference to another, and it's also fairly trivial to change transform the component weights from one common reference to another.

Now, when the channels have a mix of references, things get a little more interesting. Bipolar (measurement) channels are still linear combinations of EEG sources, but they have unknown offsets.[1,2] This is not a problem for ICA in and of itself -- as @mmh pointed out in the comments -- and thus the decomposition into sources / components works. The bigger issue is that the offset messes with the weights a bit and thus the back-projection; i.e. the generation of scalp maps becomes somewhat more difficult and the resulting scalp maps may be not be interpretable and cannot be source localized.[1] (Moreover, bipolar channels effectively pack the signal change from two discrete spacial locations into one and thus lose some of the topography information, which is also important for things like scalp maps.)

This is ultimately why the official advice from the EEGLAB Wiki is to use common reference for all channels, including the eye channels, because you can always create bipolar channels from unipolar channels, but not vice vera:

We advise recording eye channels (conventionally four channels, two for vertical eye movement detection and two for horizontal eye movement detection) using the same reference as other channels, instead of using bipolar montages. One can always recover the bipolar montage activity by subtracting the activities of the electrode pairs.[2]

For the work I do, ICA is really only interesting if you can generate scalp maps or perform source localization. It is still possible to identify (some types of) problematic components and remove them (e.g. a single noisy electrode, but then again, you can often identify that without ICA), but if the scalp maps are not interpretable, then I would have serious doubts about removing components that depend on electrodes from multiple references -- removing components is effectively substracting the component-associated scalp map from the measurement-associated scalp maps and my intuition is that the resultant difference would not necessarily be the correction you're looking for.

Or more succinctly, Arnaud Delorme, one of the key figures in the development and application of ICA to EEG data, states:

ICA on bipolar montage will not be informative. ICA will attempt to model each channel reference so that common sources may be projected to all channels in a linear fashion. [3]

ICA will work with a mixture of bipolar and common references, but it will be answering a a different question than the one we would like answered.

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  • $\begingroup$ This answer contains factual errors related to ICA. It does not care about the reference. It only cares about that the model x = As holds. Also, most reasonable neuroimaging software use always the average reference. You could also explain why the signals at the sensors are correlated (volume conduction). $\endgroup$ – mmh Jun 12 '15 at 9:02
  • $\begingroup$ The reference affects whether or not $x = As$ holds: ICA doesn't care about which common reference is used, but it does care about whether or not a common reference is used. In particular, bipolar electrooculogram is problematic and must be excluded from the ICA decomposition of the common-reference (monopolar) scalp EEG. There have been several discussions on the EEGLAB listserve to this effect. Intuitively, this is obvious because with bipolar electrodes, you can't which electrode contributed which part of the signal and you effectively collapse two electrodes into a single channel. $\endgroup$ – Livius Jun 12 '15 at 10:01
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    $\begingroup$ While average reference is preferred in some subfields, in others, linked mastoids (e.g. neurolinguistics) or the nasion are the most commonly used references. Average reference has advantages but also disadvantages such as reducing the rank of the data and ICA does not behave well with rank-deficient data. Signals from adjacent electrodes are correlated not just because of the "smearing" from volume conduction but also because of the basic physics of electric fields -- there are no discontinuities in the electric field in typical EEG settings. $\endgroup$ – Livius Jun 12 '15 at 10:16
  • $\begingroup$ Well ICA assumes $A$ is a square matrix. So of course it does not behave well with rank-deficient data. Actually, a more common problem is to extract too many components: most papers just pick the number of extracted components out of the blue. $\endgroup$ – mmh Jun 12 '15 at 11:18
  • $\begingroup$ It does not follow from the assumption of a square matrix that the matrix must be full-rank! A square matrix has full rank if and only if it is invertible, but not all square matrices are invertible -- a trivial example is the $n$x$n$ zero matrix. As mentioned in the link from my last comment, numerical issues can make rank estimation non trivial. But this is getting off topic for this question, which focused on pooling channels and not the linear algebra behind ICA and rereferencing. $\endgroup$ – Livius Jun 12 '15 at 11:41

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