STSAR model in FMRI

We assume a temporally stationary but spatially non-stationary temporal component. It seems sensible that the spatial noise structure would be the same through time, hence we use a temporally stationary spatial autoregressive model. Also, the spatial autocorrelations are assumed to be anisotropic, that is we assume that the spatial autocorrelation is different in the three different spatial directions. Therefore, we use an STSAR which combines a temporally fixed spatial AR of order 1 together with a spatially varying general order temporal AR. It is not clear whether the spatial component should be spatially stationary or non-stationary. Hence, we consider both modelling approaches. In either case we have:

$$q_{it}= \sum_{j\in{\cal N}_{i}} \beta_{ij} q_{j(t-1)} +\sum_{p=1}^P \alpha_{pi}q_{i(t-p)}+\epsilon_{it}$$ (8)

where $\beta _{ij}$ is the spatial autocorrelation between voxel $i$ and $j$ at a time lag of one, $\alpha _{pi}$ is the temporal autocorrelation between time point $t$ and $t-p$ at voxel $i$, and:

$$\epsilon_{it} \sim N(0,\phi_{\epsilon_i}^{-1})$$ (9)

The spatially stationary spatial model sets $\beta_{ij}=\beta_{d}$ where $d$ is the direction of the link $i,j$, giving:

$$q_{it} = \sum_{d=1}^D \left(\beta_d \sum_{j\in{\cal N}_{id}} q_{j(t-1)}\right)$$ (10)

$$+ \sum_{p=1}^P \alpha_{pi}q_{i(t-p)}+\epsilon_{it}$$

where $j\in{\cal N}_{id}$ is the set of neighbouring voxels to voxel $i$ in the $d$ direction, there are $D=3$ directions. The spatially non-stationary spatial model is the same as equation 8, but with the condition $\beta_{ij}=\beta_{ji}$.