Jason,
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> I am looking at the paper: Analysis of fMRI Time-Series
> Revisited-Again and I have some questions. I understand and accept
> that SPM uses the pseudoinverse when performing the least sqaure
> analysis to avoid any singularites that may arise due to
> multicollinearity in the design matrix. However on page 3 of the above
> manuscript are the equations for Var{b} and T which both include:
> (GT*G)-1*GT which is evaluated with the pseudoinverse. However
> what about:
> G*(GT*G)-1 which exist in both of these equations?
> Is this inversion possible? If the psudoinverse is used to
> avoid any multicollinearity issues how does this second equation get
> evaluated if there is multicollinearity?
As far as I can see, the pseudo-inverse (or some generalised inverse)
needs to be used throughout this paper. So the answer is that if
'multicollinearity' is present (i.e. G has less than maximum rank) then
the evaluation is via a generalised inverse e.g. the pseudo-inverse.
There is no arbitrariness in this. The 'multicollinearity' restricts
the domains of contrasts, or linear compounds of the parameters, which
are 'estimable', and is to just these estimable combinations of the
parameters that the theory applies. If we reparameterise so that
multicollinearity is removed, though all compounds of the new
parameters are now estimable the domain of such compounds is exactly
equal to the estimable domain in the old parameters, and the estimators
and their variances are equivalent.
Ian
--
Ian Nimmo-Smith
MRC Cognition and Brain Sciences Unit
15 Chaucer Road, Cambridge CB2 2EF
Tel: +44 (0) 1223 355294 x 710 Fax: 359062
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