# [FieldTrip] SVD and projectmom

RICHARDS, JOHN RICHARDS at mailbox.sc.edu
Wed Oct 14 15:49:17 CEST 2020

```Eelke

Thanks for your response. It's a good answer and helps to elucidate my question, but part of the question remains. Your description of a) b) and c) were implicit in my question, so maybe this is the crux:

The 1) and 2) below are probably identical.  Doing this in the leadfield results in the 1 x nvoxel LF coefficients; then applying a filter (eLORETA, sLORETA) to the 1 x nv LF coefficients, is probably the same as first doing the filter and then the max orientation SVD (e.g., 2).

I guess the critical question is what 3) is doing?  What does it mean "the data covariance is projected through the filters"?

I have not tried 1) (leadfield step), but have done 2) and 3) and they give different results.  I presume this is because the orientation of the filters in 2) is not the same as the orientation of the filters modified by the data in 3).

Also, re your note below...   If I have X and Y datasets, and do 2), then the orientation of the projection will be the same for X and Y.  But doing 3), the projection is based on the data,
e.g.    svd(filter) * data and svd(filter*covX*filter')*data give different results; so that the orientation of svd(filter*covX*filter') is a different orientation than svd(filter*covY*filter).  I worry that the difference in orientation between the two datasets might confound experimental conditions effect differences in X and Y. Are the "data sources" different, or is it just a chance occurrence due to the orientation differences.

One reason this came up is that we have two related sets of data, like X and transform(X), where the transform is a cwt operation. It turns out that
Correlation (X and transformX in the channel*time) = 1
svd(filter)*X and svd(filter)*transform(x) also result in an identity matrix across nvoxels; (correlation over time, for each nvoxel)
But  svd(filter*covX*filter')*X, and svd(filter*CovY*filter')*Y, show weird patterns of correlations-some nvoxels are 1, some nvoxeks are -1, and then some are in the middle range; or they are strong positive but not 1 (like .8 to 1).
The use of the cov in the svd definitely changes the results

This might not make a difference in the pattern of responses, but the orientation of some of the analyses flips from 1 to -1, and the intermediate correlations mean that the results from the svd(filter*COV*filter') is changing the results.

John

Hi John,

I'm not sure the following is a full answer to your questions, but
it's definitely relevant to both of them. As you say, there are by
default 3 dipole orientations along which activity is considered. We
can combine these three into one (using SVD or similar) at (at least)
three possible levels:

(a) the level of the leadfield;
(b) the level of the filters;
(c) the level of the source-reconstructed data.

What you are suggesting in question 2 is approach b. Your phrasing
a, but unless I'm misunderstanding you really are proposing approach
b.

What by default is implemented in FieldTrip (cfg.projectmom = 'yes')
is approach c. This I think answers your question 1: the data
covariance is projected through the filters in order to determine the
maximum orientation of the source-reconstructed data.

Also of relevance: you can use the non-downprojected (3xN) filters, as
computed based on some data X, and apply them to some data Y, and only
downproject after that. This might result in a different orientation
being the "maximal" one, as the maximum (in approach c) depends not
only on the filters (and hence the leadfield), but also on the data.

Hope that helps.

Best,
Eelke

***********************************************
John E. Richards
Carolina Distinguished Professor
Department of Psychology
University of South Carolina
Columbia, SC  29208
Dept Phone: 803 777 2079
Fax: 803 777 9558
Email: richards-john at sc.edu<mailto:richards-john at sc.edu>
https://jerlab.sc.edu
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