## 3 Coupling Structures
In a manner similar to CPC-S and CPC-T, Reduced Pattern-Coupled with Spatial Coupling (RPC-S) and Reduced Pattern-Coupled with Temporal Coupling (RPC-T) are derived from RPC-ST by omitting \(\gamma_{l,n}^{t}\) and \(\gamma_{l,n}^{s}\), respectively. Building upon the CPC-ST structure, another coupling configuration is proposed by modifying Equations (16) and (17), where coupling parameters are moved from outside the summation to within it, allowing it to vary with every element in \(\mathbf{i}_{l}\) and \(\mathbf{j}_{n}\). This adjustment results in a structure referred to as Adjacent Pattern-Coupled with Spatio-Temporal Coupling (APC-ST), and the equations are reformulated as follows:  
\[\gamma_{l,n}^{s}=\sum\limits_{i\in\mathfrak{I}_{l}}\kappa_{i,n}^{s}\cdot\gamma _{i,n}, \tag{20}\]  
\[\gamma_{l,n}^{t}=\sum\limits_{j\in\mathfrak{I}_{n}}\kappa_{i,j}^{t}\cdot\gamma _{l,j}. \tag{21}\]  
In the APC-ST configuration, the spatial and temporal coupling parameters \(\kappa_{i,n}^{s}\) and \(\kappa_{l,j}^{t}\) are element-specific, allowing the model to capture localized dependencies more flexibly. In a manner similar to CPC-S and CPC-T, Adjacent Pattern-Coupled with Spatial Coupling (APC-S) and Adjacent Pattern-Coupled with Temporal Coupling (APC-T) are derived from APC-ST by omitting \(\gamma_{l,n}^{t}\) and \(\gamma_{l,n}^{s}\), respectively. The coupling parameters in the APC-ST, APC-S, and APC-T are tailored to specific indices, allowing the model to effectively capture intricate patterns in spatially or temporally correlated data. This adaptability comes at the cost of increased complexity in determining the coupling parameters, which must be carefully optimized to avoid overfitting. The parameter configurations for different coupling structures can be found in Table 1.  
With the introduction of pattern coupling, a new set of coupling parameters \(\kappa\) is incorporated into the model, which contains elements \(\kappa_{i,n}^{s}\) and \(\kappa_{l,n}^{t}\). To derive the Bayesian model that includes \(\kappa\), this study assumes a uniform distribution as the hyperprior for \begin{table}
\begin{tabular}{c c} \hline \hline Coupling Structures & \(\widehat{\gamma}_{l,n}\) \\ \hline CPC-ST & \(\gamma_{l,n}+\kappa_{l,n}^{s}\sum\limits_{i\in\mathfrak{I}_{l}}\gamma_{i,n}+ \kappa_{l,n}^{t}\sum\limits_{j\in\mathfrak{I}_{n}}\gamma_{l,j}\) \\ CPC-T & \(\gamma_{l,n}+\kappa_{l,n}^{t}\sum\limits_{j\in\mathfrak{I}_{n}}\gamma_{l,j}\) \\ CPC-S & \(\gamma_{l,n}+\kappa_{l,n}^{s}\sum\limits_{i\in\mathfrak{I}_{l}}\gamma_{i,n}\) \\ \hline RPC-ST & \(\gamma_{l,n}+\kappa_{n}^{s}\sum\limits_{i\in\mathfrak{I}_{l}}\gamma_{i,n}+ \kappa_{n}^{t}\sum\limits_{j\in\mathfrak{I}_{n}}\gamma_{l,j}\) \\ RPC-T & \(\gamma_{l,n}+\kappa_{n}^{t}\sum\limits_{j\in\mathfrak{I}_{n}}\gamma_{l,j}\) \\ RPC-S & \(\gamma_{l,n}+\kappa_{n}^{s}\sum\limits_{i\in\mathfrak{I}_{l}}\gamma_{i,n}\) \\ \hline APC-ST & \(\gamma_{l,n}+\sum\limits_{i\in\mathfrak{I}_{l}}\kappa_{i,n}^{s}\cdot\gamma_{i, n}+\sum\limits_{j\in\mathfrak{I}_{n}}\kappa_{l,j}^{t}\cdot\gamma_{l,j}\) \\ APC-T & \(\gamma_{l,n}+\sum\limits_{j\in\mathfrak{I}_{n}}\kappa_{l,j}^{t}\cdot\gamma_{l,j}\) \\ APC-S & \(\gamma_{l,n}+\sum\limits_{i\in\mathfrak{I}_{l}}\kappa_{i,n}^{s}\cdot\gamma_{i,n}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameter configurations for different coupling structures. \(\kappa\).