Dear Ying,
> Thank you very much for your reply. I can understand the difference
> (or indifference) of the two design matrices now.
> However just on the last point about the parameter estimation, I
> thought if a design matrix has 5 columns, but suffers colinearity and
> therefore has rank of 4, isn't it true that there should be only 4
> non-zero singular values associated with the design matrix? Hence even
> with pseudoinverse, one can only obtain 4 parameters instead of 5, is
> that right? I tested this with SPM, entering 4 box-car columns which
> add up to a constant, exactly equal to column 5. SPM seems to be able
> to produce 5 parameter files: beta_001 upto beta_005, although in the
> 'parameter estimability' section, the first four boxes are grey,
> indicating they are not estimable. I don't understand how SPM obtained
> the 5 beta files even though not all of them are estimable........?
You are absolutely right in that the dimensionality of the parameter
estimates is only 4. This is why some contrasts are not estimable.
The peudoinverse effectively applies an extra constraint in
rank-deficient problems that emulates a minimum norm solution (i.e. the
variance of the parameter estimates is itself minimal). This is a
standard approach to rank deficient design matrices and means there are
no restrictions on its form as specified by the user.
I hope this helps - Karl
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