Current Status¶
\[\begin{split}\mathbf{Y}_{i,t,g} & = \mathbf{B}_g x_{i,t} + d_{i,g} \mathbf{1} + \mathbf{E}_{i,t,g} \\
e_{i,t,g,\ldots} & \in \mathbf{E}_{i,t,g} \\
e_{i,t,g,\ldots} & \sim \text{N}(0,\sigma_y^2) \\
\sigma_y^2 & \sim \text{Inverse Gamma}(a_\sigma,b_\sigma) \\
\mathbf{B}_g & = \sum_{r=1}^{R} \boldsymbol{\beta}_{1,r,g} \circ \boldsymbol{\beta}_{2,r,g} \circ \cdots \circ \boldsymbol{\beta}_{D,r,g} \\
\boldsymbol{\beta}_{j,r,g} & \sim \text{N}\left(\mathbf{0},\phi_{r,g}\tau_g \mathbf{W}_{j,r,g}\right) \\
\mathbf{W}_{j,r,g} & = \text{diag}\left( \omega_{j,r,g,1}, \ldots, \omega_{j,r,g,p_j} \right) \\
\omega_{j,r,g,\ell} & \sim \text{Exp}\left( \frac{\left(\lambda_{j,r,g}\right)^2}{2} \right) \\
\lambda_{j,r,g} & \sim \text{Gamma} (a_\lambda,b_\lambda) \\
\text{for } r = 1,\ldots,R-1, \, \phi_{r,g} & = \xi_{r,g}\prod_{k=1}^{r-1} (1 - \xi_{k,g}) \\
\text{for } r = 1,\ldots,R-1, \, \xi_{r,g} & \sim \text{Beta}(1,\alpha) \\
\phi_{R,g} & = 1 - \sum_{r=1}^{R-1} \phi_{r,g} \\
\tau_g & \sim \text{Gamma}(a_\tau,b_\tau) \\
\begin{pmatrix} d_{i,1} \\ \vdots \\ d_{i,G} \end{pmatrix} & \sim \text{N}(\mathbf{0},\boldsymbol{\Sigma}^{-1}) \\
\sigma_{ii} \in \boldsymbol{\Sigma} & \sim \text{Exponential} \left(\frac{\zeta}{2}\right) \\
\sigma_{ij: i < j} \in \boldsymbol{\Sigma} & \sim \text{Double Exponential}(\zeta) \\
\zeta & \sim \text{Gamma} (a_\zeta,b_\zeta)\end{split}\]
Data Structure¶
The tensor-valued fMRI output for a given region is taken as the response in a task-based fMRI study. The covariate is the difference between the modulated, centered number of pumps given to treatment balloons and the modulated, centered number of pumps given to control balloons. More information on the experiment design can be found here. The indices map as follows: \(i\) corresponds to the subject, \(t\) corresponds to the time, \(g\) corresponds to the region, and \(r\) corresponds to the rank. Capitalized bold letters signify tensor-valued data structures, and non-bold letters signify scalars.
