PCA fix scale power

[rcassani]

• Scales the power of the 1stPC to be the same as the average power as done in MNE-Python: https://github.com/mne-tools/mne-python/blob/31a83063557fbd54d898f00f9527ffc547888395/mne/source_estimate.py#L2818

[rcassani]

Hi @Moo-Marc, this is the PR for the power scaling

[Moo-Marc]

I'm not sure it's very useful, other than looking more similar to mean. But if we want to keep a meaningful scale comparable to mean, here's what I'd suggest. For mean, the sum of weights = 1, whereas for a (PCA) component/projection, the sum of squared component values = 1. To compare mean to PCA, we can multiply the mean weights by sqrt(N), and we get sum(w^2) = 1, a valid component. To go the other way, we can divide PCA by sqrt(N). This is simple and maintains a meaningful relative power between the two options.

[ftadel]

If you need additional inputs on this topic, put it on the agenda of the next Wednesday meeting. Dimitrios, Richard or John might be able to help.

[rcassani]

we can divide PCA by sqrt(N).

Yeah, this makes more sense than scaling the 1PC to match the same power of all N signals. Can you add it in #559?