Dear Markus,
> Having just returned from the excellent SPM course and thinking about
> all the stuff I have heard there, I remember somebody saying that if
> you use temporal derivatives then you do not have an orthogonal set of
> basis functions.
>
> This is indeed the case when I check the design matrix for
> orthogonality, but I don't understand it:
>
> Given a ("well-behaved") function f and its derivative f', then
>
> integral(f*f') = 0.5 * integral( (f^2)' ) = 0.5 * ( f(inf)^2
> - f(-inf)^2 )
>
> since f is zero for -infinity and +infinity, the integral should be
> zero. Hence the function and its derivative should be orthogonal!
In general any process and its derivative are orthogonal, so you are
absolutely right. However, if you then filter the two processes (i.e.
with high or low pass filtering) a small amount of correlation is
usually re-introduced. If you omitted filtering then the regressors
would indeed be exactly orthogonal.
With very best wishes - Karl
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