ICA questions

I don’t understand some choices in the ICA calculation process, maybe someone could answer these questions:

1. Why is a so-called pre-whitener (by default channels are divided by their standard deviations) used before PCA/ICA? I think this is not necessary because the next step is PCA with whiten=True.
2. Speaking of PCA and whitening, sphering seems to be more optimal for ICA, and sphering is also used in EEGLAB by default. AFAIK the only change necessary would be another rotation by the eigenvector matrix to convert the PCA whitening transform to a sphering transform. Should we care to implement it?
3. I don’t understand this line - why are the elements of the unmixing matrix divided by the square root of the explained variances?

[cbrnr]

Oh yes, this is definitely nice but hacky (i.e. no one without some experience with MNE will be able to come up with it in less than 5 minutes).

+100 for exposing the remaining parts, this is still my preferred option. I propose the following components:

• ica.unmixing_matrix_ (already exists)
• ica.pca_components_ (already exists)
• ica.standardization_ (removes mean and scales variance to 1) - but we have to think about how the noise_cov argument influences this pre-whitening to return the right thing
• ica.whitening_

Then the docs should say that the final unmixing matrix, which transforms the MEEG data into sources, can be composed as ica.unmixing_matrix_ @ ica.whitening_ @ ica.pca_components_ @ ica.standardization_ (ignoring any necessary transpose operation or dimensionality rectification right now).

[cbrnr]

to do this you need W to be the inverse of a square root of cov(X). Let’s call the svd of this covariance U S^2 U^T then W can be S^{-1} U^T or U S^{-1} U^T. What you call (I think) sphering is U S^{-1} U^T and whitening S^{-1} U^T but they are just 2 choices to whiten the data.

Yes, the terminology seems to be inconsistent. Whitening is a transform that decorrelates the data (i.e. the covariance matrix becomes the identity matrix), e.g. via PCA. There are infinitely many ways to whiten due to rotation. Sphering is a special kind of whitening that rotates the data back to the original space (some call that ZCA). So exactly like you say, but currently we (and scikit-learn) use PCA whitening, but EEGLAB uses sphering (and I remember having read somewhere that ZCA is more optimal for Infomax than PCA whitening, but I can’t find that anymore).

In any case, since I can’t find relevant references, any kind of whitening seems to work just fine, so if I find more on that topic I’ll create a new issue.

So only the last question left - @dengemann?