Constraining Basis Function Linear Combinations

Here we are going to describe how we constrain the basis function linear combinations. To do this we need to reparameterise the regression parameters, $\beta_{ieb}$, 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 $\beta_{ie}$ as the $N_b\times 1$ vector of the regression parameters for the $N_b$ basis functions for the $e^{th}$ underlying condition at voxel $i$. Then, we reparameterise $\beta_{ie}$ as being:

$$\beta_{ie}=\bar{D}_{ie}\frac{D_{ie}}{\sqrt(\sum_b D_{ieb}^2/N_b)}$$ (11)

where $D_{ie}$ is an $N_b\times 1$ vector of parameters describing the HRF for underlying condition $e$, and $\bar{D}_{ie}$ is the scalar value representing the scaling of that HRF. We want the scalar $\bar{D}_{ie}$ to contain all of our size information. However, left unchecked there is an arbitrary scale factor on vector $D_{ie}$. We have removed this arbitrary scale factor by normalising the vector using its root mean square. Hence, we now have a normalised vector, $D_{ie}/{\sqrt(\sum_b D_{ieb}^2/N_b)}$, representing the shape of the HRF, and a scalar, $\bar{D}_{ie}$, representing the size of the HRF. For the scaling parameters we assume a noninformative prior:

$$p(\bar{D}) = \prod_{ie} p(\bar{D}_{ie})$$

$$\bar{D}_{ie} \sim N(0, \phi_{\bar{D}_0}^{-1})$$ (12)

where the precision, $\phi_{\bar{D}_0}$, is fixed to be very small (1e-6) for all voxels. It is via the prior on $D_{ie}$ that we can constrain the possible linear combinations of basis functions to represent the HRF for an underlying condition. We specify the prior on $D_{ie}$ as:

$$p(D) = \prod_{ie} p(D_{ie})$$

$$D_{ie} \sim MVN(m, C^{-1})$$ (13)

where $m$ and $C$ will contain the information constraining the possible linear combinations of the basis functions (see section 2.6 for how we set $m$ and $C$).