Short version:

Could someone point me in the direction of some documentation for precisely what the SVD is doing in 3dDeconvolve?

Long version:

I have a rapid event-related design with N conditions, and I am trying to look at the similarity structure of those N conditions across local populations of voxels.

If I use an X matrix with N regressors, and every TR has one and only one condition that it belongs to, then my X matrix is singular. According to the (limited) documentation, if you have two identical regressors, the SVD approach in 3dDeconvolve would equally distribute variance attributable to this pattern between the two regressors. Now I have N variables, each predicted by the other N-1 variables. If I were to simply pass this to 3dDeconvolve, my understanding is that the SVD would "clump" my N regressors the orthogonal N-1 regressors that explain all of the variance in the original N.

My belief was that the SVD is multiplying the data by a contrast matrix of sorts, running the GLM for the product of the X matrix and the contrast matrix, and then dividing the beta vector by the inverse of the contrast matrix. The problem is that if I do this manually, I get very different results:

I should be able to multiply my X matrix by a contrast matrix along the lines of:

3 -1 -1 -1
0 2 -1 -1
0 0 1 -1

This would turn my NxTime matrix into an (N-1)xTime matrix with orthogonal columns, completely eliminating the singularity. But critically, the resulting (N-1)xTime matrix retains the variance between every condition and every other condition, so even though I lose a column of data, I retain all of relationships between variables. I just need to divide the vector of betas by the contrast matrix, to turn my (N-1)x1 vector of betas into an Nx1 vector of betas.

Thus, although the grand mean should be removed, the similarity structure of my regressors should remain the same. That is, for Voxel X, if I were to find that regressors 1 and 2 are similar (e.g., positive), and 3 and 4 are similar (e.g., negative), but 1/2 differ from 3/4, I should find that pattern regardless of whether I let the SVD reduce collinearity or if I do it myself with a contrast matrix.

However this is not true; these methods produce very different results. Why?

Thanks in advance for the help,
-cdm