RFC: Re-applying baseline correction after ICA

Skimming over some Study Template analysis results with @agramfort, we found that some of our data appeared to be not baseline-corrected – some traces clearly didn’t have a zero-mean during the baseline period.

So I started digging and figured that we had indeed applied baseline correction to our data; yet after cleaning the Epochs via ICA, the baseline signals of some channels had started floating away from zero.

Now this may not come as a surprise to some of you, but I have to admit it caught me off-guard. I’d never seen this behavior before with the data I’d worked with so far, and I’ve failed to reproduce a similarly visible effect using our sample data.

However, looking through the ICA tutorial, where we clean Raw data, one can see the essence of this effect too – albeit with Raw and not Epochs (lower panel):

I’d like to propose and discuss the following changes and enhancements to deal with this behavior:

• Clearly state in the docs that baseline correction should be re-applied to Epochs and Evoked data after applying ICA
• viz.plot_ica_overlay() should automatically re-apply baseline correction to the cleaned signal if an Evoked was passed
  • for this, we’d need to add an Evoked.baseline attribute, which gets inherited from the parent Epochs.baseline. This has been on my agenda for a while anyway in order to improve provenance, so I’d might take a shot at this anyway.
• We could even go so far as to auto-reapply baseline correction to Epochs and Evoked data as part of ICA.apply(), so users wouldn’t have to do this themselves. This behavior should be toggle-able using a new kwarg.

[agramfort]

we converge to the same thing.

Do not baseline epochs passed to ICA.

Apply baseline only after doing ICA cleaning.

No need to enable not centering the PCA step.

[larsoner]

SSP is a linear spatial operator, so if all channels have zero mean (during some interval) all channels will have zero mean after the operation (in that same interval). This is because the mean (over that interval) is also a linear operator.

To demonstrate how ICA violates this, let’s say we have two channels of data for one 3-sample epoch that that each already have zero mean over time, for example [[1, 0, -1], [0, 0, 0]]. If the empirical means estimated by ICA during fitting happened to be, say -1 and 0, ICA will:

1. Subtract the empirical estimated means, giving [[2, 1, 0], [0, 0, 0]]
2. Apply the ICA operator, which is just some linear spatial operator. Let’s pick one that just swaps data between channels, [[0, 1], [1, 0]] (ICA in practice will do something different like unmix-exclude-mix, but this gives a simple example to play with). So now our data after the ICA operator is [[0, 0, 0], [2, 1, 0]].
3. Add the means back, giving [[-1, -1, -1], [2, 1, 0]].

Now our previously-zero-mean channels have means -1 and 1, which is not good. Let’s call this operation B = ICA(A). If we apply this operation to identical data, that will (trivially) have B = ICA(A) having means -1 and 1, too.

And this is where you can see that the ICA operation above is not linear. We know that B + B = ICA(A) + ICA(A) but that this means 2B = ICA(A) + ICA(A). We know that since B has means -1 and 1, 2B will have means over time of -2 and 2. For linearity to hold, we need that 2B = ICA(A) + ICA(A) = ICA(A + A) = ICA(2A), but unfortunately ICA(2A) will go from [[2, 0, -2], [0, 0, 0]] to start, through step 1 to [[3, 0, -1], [0, 0, 0]], through step 2 to [[0, 0, 0], [3, 0, -1]], through step 3 to [[-1, -1, -1], [3, 0, -1]], which has means -1 and 2/3, which do not equal -2 and 2. Hence our ICA operation is not a linear operator, which I think it is meant to be…

If we remove the mean subtraction at the beginning and mean re-addition at the end, ICA will be a linear spatial operator (just like SSP) in the end. To me this is starting to seem like the correct option.