Apologies. I'll try to be more clear.

The 3dDeconvolve documentation clearly outlines the matrix algebra that is employed by 3dDeconvolve for estimating beta and F values for the -nosvd option; I haven't found any documentation for what 3dDeconvolve does differently with the -svd option.

I am interested in looking at the similarity structure between variables in a rapid event-related design. Every timepoint corresponds to one and only one condition, so the unwavered X matrix looks something like this:

x =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0

where the rows are timepoints and the columns are 4 regressors. Clearly that matrix is singular. But if I am only interested in the differences between conditions, I can multiply this X matrix by a contrast matrix, such as:

c' =
1 1 -1 -1
1 -1 0 0
0 0 1 -1

This will retain all of the variance between conditions, but remove the collinearity. If I ran the product of (X*c) through 3dDeconvolve, then I would get 3 betas that reflect the differences between the 4 conditions. The first beta would reflect the difference between regressors [1, 2] and regressors [3, 4]. If I wanted estimates of each condition, I would need to reconstruct the individual betas by dividing by the contrast matrix, c.

I always thought that AFNI was doing something similar with the SVD step in 3dDeconvolve: by running an SVD on your X matrix, you end up with a matrix of the same size, but with orthogonal regressors; all of the variance in your original X matrix is still there, but its distributed differently across the N regressors.

If you pass the SVDed X matrix through the 3dDeconvolve GLM, you wouldn't suffer from collinearity, but just like in the contrast matrix case above, your betas no longer reflect conditions, they reflect principle components. To extract the betas for each condition, you would have to use the inverse of the SVD transformation.

This seemed to jive with the documentation that said that if two regressors are identical, 3dDeconvolve will run, and it will split the variance explained in Y equally between two. Unless my intuition is misleading me, I would expect this to be the outcome of the process of (1) using a transformation matrix (SVD) to create principle components from your X matrix; (2) estimating betas for each of the principle components in a GLM; and (3) use the inverse of the transformation matrix from step (1) and the PC betas to estimate betas for each condition.

Before continuing, is this account generally correct? I'd imagine that you wouldn't actually need to go through each of these steps explicitly, but is this in principle what the SVD is achieving?

Thanks,
-cdm