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Here we are going to describe how we constrain the basis function
linear combinations. To do this we need to reparameterise the
regression parameters,
, into parameters which
describe the shape of the HRF, and parameters which scale these
HRF shape parameters, to give the actual fit in the GLM.
Firstly, we specify
as the
vector of
the regression parameters for the basis functions for the
underlying condition at voxel . Then, we
reparameterise
as being:
|
|
|
(11) |
where is an
vector of parameters describing
the HRF for underlying condition , and
is the
scalar value representing the scaling of that HRF. We want the
scalar
to contain all of our size
information. However, left unchecked there is an arbitrary scale
factor on vector . We have removed this arbitrary scale
factor by normalising the vector using its root mean square.
Hence, we now have a normalised vector,
, representing the shape of the HRF, and a scalar,
, representing the size of the HRF.
For the scaling parameters we assume a noninformative prior:
where the precision,
, is fixed to be very small
(1e-6) for all voxels. It is via the prior on that we can
constrain the possible linear combinations of basis functions to
represent the HRF for an underlying condition. We specify the
prior on as:
where and will contain the information constraining the
possible linear combinations of the basis functions (see
section 2.6 for how we set and ).
Next: Choosing a Basis Set
Up: Model
Previous: Basis Functions