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Space-Time Simultaneously specified Auto-Regressive model (STSAR)

The simultaneous auto-regressive approach is analogous to time series auto-regressive modelling (6):

$\displaystyle q_u=\sum^{{NT}}_{v=1}b_{uv} q_v+\epsilon_u$ (5)

where $ u\in \{1\ldots NT\}$ indexes a particular voxel and time point, and $ \vec{\epsilon}\sim MVN(0,\vec{\Lambda})$, giving:

$\displaystyle \vec{q}\sim
 MVN(\vec{0},\vec{(I-B)^{-1}\Lambda(I-B')^{-1}})$ (6)

where $ \vec{B}$ is an $ {NT}\times {NT}$ matrix whose (u,v)th element is $ b_{uv}$ and $ \vec{\Lambda}$ is a diagonal covariance matrix (in this paper the variances along the diagonal of $ \vec{\Lambda}$ vary voxelwise but not through time). Note that for a valid covariance matrix $ \vec{(I-B)}$ must be invertible. In using equation 5, Wikle et al. (42) lagged the spatial correlations by one time point. Then instead of having to consider the entire joint distribution, we can take advantage of conditional independence and factorise as follows:

$\displaystyle p(\vec{q\vert\epsilon}) =
 \prod_{i}^N\prod_{t}^Tp(q_{it}\vert\{q_{j\tau}\},\phi_{\epsilon_i})$ (7)

where $ j\in \{1\ldots N\},\tau \in \{1\ldots t-1\}$. The reason for lagging the spatial correlations by one time point and creating this conditional independence is to introduce directional acyclic dependency between the $ q$ parameters. Avoiding cyclic dependencies will improve the mixing of the MCMC sampling, whilst introducing no loss in the usefulness of the model. This is the approach that we will take in this work.
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Next: STSAR model in FMRI Up: Small Scale Variation Previous: Separable Models