Classifying SPMs in FMRI

The mixture models could be used as a second stage to analyze the SPMs resulting from a first stage of a univariate temporal FMRI analysis (Woolrich et al., 2001). This is the same two-stage approach used when using the commonly used random field theory of Worsley et al. (1992) and is also the same approach that Everitt and Bullmore (1999); Hartvig (2000) take with the mixture models they consider.

The result of a univariate temporal analysis is effectively the parameter estimate, $a_i$, of the correlation with the assumed response at each voxel $i$. We might consider using $a_i$ as the observations, $y_i$, for our mixture models. However, this carries over none of the uncertainty in the estimation of $a_i$ from the univariate temporal analysis. Hence, we use instead the normalised version:

$$y_i = \frac{a_i}{std(a_i)}$$ (20)

See Woolrich et al. (2001) for one way of estimating $a_i$ and $std(a_i)$ using a univariate temporal analysis.

We now need to specify $p(y_i\vert x_i=k,\theta_k)$ for all classes. For this we propose to use a similar approach to Hartvig and Jensen (2000), who uses three classes: activation, deactivation and non-activation. The non-activating class is modelled as a Normal distribution:

$$y_i\vert x_i=k_n,\theta_{k_n} \sim N(y_i;\mu_{k_n},\sigma_{k_n}^2)$$ (21)

where $k_n$ represents the label for the non-activating class. To reflect the assumption that the activating class can only have positive values of $y_i$, we use a Gamma function for the activation component:

$$y_i\vert x_i=k_a,\theta_{k_a} \sim Ga(y_i;\tilde{a}_{k_a},\tilde{b}_{k_a})$$ (22)

where $k_a$ represents the label for the activating class. For the deactivating class, we also use a Gamma function:

$$y_i\vert x_i=k_d,\theta_{k_d} \sim Ga((-y_i);\tilde{a}_{k_d},\tilde{b}_{k_d})$$ (23)

where $k_d$ represents the label for the deactivating class.

Note that we sometimes find it useful for the interpretation of parameters to reparameterise a Gamma distribution in terms of its mean, $\mu_k$, and variance, $\sigma_{k}^2$. See the appendix for this transformation.

The hyperpriors $p(\vec{\theta})$ to be used on the component parameters are non-informative, disperse priors. However, we do place a restriction on the mode of the activation and deactivation Gamma classes. The mode of the activation class is constrained to be greater than the mean of the non-activation class, and the mode of the deactivation class is constrained to be less than the mean of the non-activation class. This encodes a sensible prior belief about the expected shape of the activation and deactivation distributions with respect to the non-activation. The mode of a Gamma distribution is given in equation 27.