[Mne_analysis] equivalence of averaged single trial STCs and evoked STCs

[Alexandre Gramfort]

hi Hari,

>    Thanks for the gist! Just ran it.. The results actually see to suggest
> that the two stc (one with the original nave of 55, and the new nave of
> 10000) are pretty much scaled versions of each other with the scale
> factor being exactly what you would expect, namely the ratio of the
> sqrt(nave) in the two cases:
>  See: http://nmr.mgh.harvard.edu/~hari/MNEnotes/dSPM_nave_scaling.pdf

indeed

> I agree with you that the regularization would change the numerical values
> very slightly and hence they won't be *exactly* equal.

agreed

> Also, I do think dSPM is pretty much z-scoring when you use the baseline
> periods to estimate the noise covariance. If w_i is the row of the inverse
> operator corresponding to the weights needed to estimate the MNE of source
> #i, and C is the noise covariance, then the normalization factor used in
> dSPM for that source is:
> sqrt(w_i*C*w_i.T) (where .T denotes a transpose), which is indeed the
> estimated standard deviation of the baseline. The reason that the dSPM
> standard deviation is not *exactly* equal to 1 in the baseline is that the
> noise-cov is estimated from raw-data samples and scaled by 'nave' =
> #trials_in_evoked, rather than being estimated from the baseline of the
> evoked data.
>   On the other hand, if you had many datasets/sources and calculated the
> baseline standard deviation, then across the datasets/sources you would
> have the baseline sample standard deviation be distributed narrowly
> around 1. In other words, the dSPM has a "null distribution" that is
> mean zero and variance 1, which is what makes it an "SPM".
>
> What am I missing?

nothing. This is correct. This makes me think that one should have
a quality insurance routine that checks how much dSPM is an
"SPM" map as you explained. I you have a good suggestion please share.

Best,
Alex