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# Appendix

The general form of the Sherman-Morrison-Woodbury formula [8] is

 (18)

Also, the inverse for a matrix in block form is given by

Theorem A:

Any model of the form

can be rewritten as

where whilst being completely equivalent in terms of the estimated parameters of interest ( ) and the modelled signal space: in the pre-whitened space. Note that for both models.

That is, the signals of interest can be made orthogonal to the confounds without affecting the estimation of the parameters or the residuals.

Proof:

The proof is by construction, where we show that orthogonalising with respect to gives the desired results. Let

where is the projection matrix for in the pre-whitened space.

These equations give . Also, the combined span of and is clearly the same as that of and .

Now consider the covariances

Using the block matrix inverse, this gives

while, since the off-diagonal blocks are zero in the second case, the calculation simply gives

For the first model, the parameter estimates, given by equation 5, can be written using the matrix block inversion formula, giving

 (19)

while for the second model, the block diagonal form yields the familiar form

Applying the Sherman-Morrison-Woodbury formula to the second term in equation 12 gives

Substituting this into equation 12 gives

Theorem B:

Given the standard GLM, , and a set of linearly independent contrasts specified by such that , then an equivalent model without contrasts, but with confounds, exists in the form

That is, , and the modelled signal space: in the pre-whitened space. Note that for both models.

Proof:

The proof is, again, by construction. Firstly, let be a set of contrasts that when combined with form a complete linearly independent set of contrasts. That is, the matrix will be full rank (and hence invertible). Then let

where

From these definitions it is easy to see that , which represents an orthogonality condition. As before, it is straightforward to verify that the combined span of and is equal to the span of . Consequently,

Therefore, and are orthogonal as well.

The estimation equations for the model become

Thus

and

Next: Bibliography Up: tr01cb1 Previous: Acknowledgements
Christian Beckmann 2003-07-16