Separable Models

One approach to simplifying equation 3 is to consider a stationary separable model (28,1). This gives:

$$\vec{q}\sim MVN(0, \vec{\Gamma}\otimes \vec{\Lambda})$$ (4)

where $\otimes$ represents the Kronecker product and where $\vec{\Gamma}$ and $\vec{\Lambda}$ are the spatial ($N\times N$) and temporal ($T\times T$) covariance matrices respectively. This is where the covariance between two observations is decomposed into the temporal covariance at the lag between the observations multiplied by the spatial covariance at the distance between the observations. Such a decomposition would make computation a lot easier, for example when computing the inverse, which is necessary for all viable inference techniques. The use of equation 4 means that the spatial autocovariance is assumed to be the same at all time points. This seems a reasonable assumption. However, it also requires that the temporal autocovariance is the same at all voxels. Previous studies (43,13), clearly demonstrated that this was not the case and would be an incorrect assumption to make. Consequently, a separable model is not considered further.