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Determining Basis Set Constraints

In section 2.5 we determined a basis set to use. In this section we describe how we can apply constraints on the linear combinations of the basis set. In previous work Friman et al. (2003) also looked to constrain the possible linear combinations of the basis set, but within the canonical correlation analysis framework. However, they only looked to constrain the linear combination coefficients to be positive. In this work we look to apply a more complete constraint by fitting a multivariate Normal distribution to describe the desired constrained space probabilistically within the GLM framework. To date, basis sets are used by convolving the constituent basis functions with the known stimulus to give the same number of regressors as there are basis functions stimuli. The resulting regressors are then used in the linear model. This approach corresponds to the model we have described earlier but with $ m=0$ and $ C=I$ in equation 13. Figure 4(a) shows example HRF shapes obtained randomly drawn from this basis set when the linear combinations are unconstrained (i.e. with $ m=0$ and $ C=I$). If we compare these HRFs with those in figure 2, it is clear that unconstrained linear combinations of our basis set allows for nonsensical HRF shapes. To provide constraints on the possible linear combinations, we regress the HRF samples we used to obtain our basis set back onto the basis set:
$\displaystyle W=GR+e$     (15)

where $ W$ is the $ N_T\times N_H$ matrix containing our $ N_H$ HRF samples, $ G$ is the $ N_T\times N_b$ matrix of our $ N_b$ basis functions and $ e\sim N(0,\sigma^2I)$. We can perform standard Ordinary Least Squares (OLS) to obtain an estimate of the $ N_b
\times N_H$ matrix, $ R$:
$\displaystyle \hat{R}=(G^TG)^{-1}G^TW$     (16)

We then fit a $ N_b$-dimensional multivariate Normal distribution, $ MVN(\tilde{m},\tilde{C})$, to the matrix $ \hat{R}$. The parameters of this multivariate Normal distribution are used to set the parameters in equation 13 (i.e. $ m=\tilde{m}$ and $ C=\tilde{C}$). Using the HRF parameter value probabilities in equation 14 for our half-cosine HRF parameterisation, $ 1000$ resulting HRF samples, and the resulting top three eigenHRFs, we obtain the multivariate Normal distribution parameters:
$\displaystyle \tilde{m}$ $\displaystyle =$ $\displaystyle [0.86, 0.09, 0.01]^T$  
$\displaystyle \tilde{C}$ $\displaystyle =$ $\displaystyle \left[\! \begin{array}{ccc} 0.018 & 0.028 & -0.015 \\
0.028 & 0.185 & -0.009\\
-0.015 & -0.009 & 0.030
\end{array}\! \right]$ (17)

Figure 4(b) shows example HRF shapes obtained randomly drawn from this basis set when the linear combinations are constrained with these $ m=\tilde{m}$ and $ C=\tilde{C}$. We can see that compared with figure 4(a) this is a much more faithful representation of the sensible HRF shapes in figure 2. It is important to point out that constraining the linear combinations by using a multivariate Normal distribution is an approximation. This is because a multivariate Normal will not capture all of the detail of the distribution in the $ N_b$ parameter space. As figure 4(b) shows, whilst we are producing much more sensible shapes than the unconstrained case, there are still some undesirable HRF shapes. The alternative is a fully parametric HRF approach (Woolrich et al., 2004b), which is very slow to infer upon, or a different distribution to a multivariate Normal which can capture the required detail in the $ N_b$ parameter space. However, we choose to use a multivariate Normal as it will make inference much more easy to handle within the Variational Bayesian framework, and it does capture the required structure.
Figure 4: Samples from the basis set: (a) unconstrained with $ m=0$ and $ C=I$, (b) constrained with $ m=\tilde{m}$ and $ C=\tilde{C}$.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\psfig{file=priorhrf_fpo.ep...
...a) Unconstrained & (b) Constrained
\end{tabular}
\end{center}
\end{figure}

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Next: Inference Up: Model Previous: Choosing a Basis Set