Hi all,
sorry for reading this too quickly: my option 3) would only work for
estimating the decay rate of a contineously decaying signal. Not for
calculating direcly how fast a HRF responses amplitude decays, as was
the original question. Taking the Log would here also affect the other
aspects of the modelled signal.
So indeed, as suggested, one would either have to (iteratively I guess)
estimate the exp(-kx) pattern of the HRF amplitudes with a nonlinear
approximation (option 1), or expand the model 1+exp(-kx) in a series of
polynomial or Taylor (http://mathworld.wolfram.com/TaylorSeries.html)
basis functions, estimate them, and derive k from the estimated
regression coefficients of the basis functions (option 2). 2) is
probably most feasible within the context of SPM.
Have a nice weekend,
Bas
Bas Neggers wrote:
>> Very interesting point. Given that SPM uses the GLM such an analysis
>> is not
>> possible because your are trying to estimate parameter k in an
>> equation like
>> f(x) = exp(1-k*x) where exp is the exponential function. This is
>> non-linear.
>> There are however two options:
>> 1) Take voxel timeseries that you are interested in and and use a
>> nonlinear
>> estinmation technique to estimate k
>> 2) Linearize the problem around a point of interest by using the idea
>> of a
>> Taylor expansion: Practically you create two exponential decay curves
>> using
>> two different values of k that are in the range of k that you are
>> interested
>> in. The difference between the two curves can then be seen as the
>> partial
>> derivative with respect to k i.e. shape. This is exactly the same
>> idea as
>> used within SPM when trying to estimate latency differences of the
>> hrf by
>> using the hrf with its time derivative.
>>
>>
> What about option 3):
>
> take a timeseries (per voxel or ROI) Y, calculate the vector log(Y),
> then the equation is linearized:
>
> log(Y)=log(exp(1-kx))=1-kx and one can use a simple GLM to estimate k ...
> Linearization and exactly solving a non-linear problem is usually
> preferred over approximation.
>
> just my 2 cts...
>
> cheers,
>
> Bas
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