## 3 Coupling Structures  
In Equation (9), the hyperparameter \(\widetilde{\gamma}_{I,n}\) is defined as a combination of different elements of \(\gamma\), organized based on specific structural dependencies. It can be expressed as:  
\[\widetilde{\gamma}_{I,n}=\gamma_{I,n}+\gamma_{I,n}^{s}+\gamma_{I,n}^{t}, \tag{15}\]  
where \(\gamma_{I,n}^{s}\) represents the spatial coupling term and \(\gamma_{I,n}^{t}\) represents the temporal coupling term. According to the coupling structure proposed in Ref. [32], these components are further detailed as:  
\[\gamma_{I,n}^{s}=\kappa_{I,n}^{s}\sum\limits_{i\in\mathbf{i}_{I}}\gamma_{i,n}, \tag{16}\]  
\[\gamma_{I,n}^{t}=\kappa_{I,n}^{t}\sum\limits_{j\in\mathbf{j}_{n}}\gamma_{I,j}. \tag{17}\]  
Here, \(\kappa_{I,n}^{s}\) and \(\kappa_{I,n}^{t}\) are coupling parameters within the range [0,1], reflecting the degree of correlation between the coefficient \(w_{I,n}\) and its neighboring coefficients \(w_{i,n}\) (\(i\in\mathbf{i}_{I}\)) and \(w_{I,j}\) (\(j\in\mathbf{j}_{n}\)) in the spatial and temporal domains, respectively. The set \(\mathbf{i}_{I}\) and \(\mathbf{j}_{n}\) index the spatial and temporal neighbors of \(w_{I,n}\). The Gaussian prior for \(w_{I,n}\) in Equation (9) accounts not only for its own hyperparameter \(\gamma_{I,n}\), but also incorporates contributions from neighboring hyperparameters \(\gamma_{i,n}\) and \(\gamma_{I,j}\). This coupling design enables the model to capture both spatial and temporal dependencies, providing a more structured and correlated representation of the coefficients. In the temporal dimension, indexed by \(n\), the neighboring coefficients considered are the immediate forward and backward natural neighbors [25,29], represented as \(\mathbf{j}_{n}=[n-1,n+1]\). Similarly, in the spatial dimension, indexed by \(I\), the virtual point sources corresponding to the coefficient \(w_{I,n}\) provide the neighboring coefficients \(\mathbf{i}_{I}\), as determined by the partitioning method [32,36]. In the coupling structure under consideration, the coupling parameters \(\kappa_{I,n}^{s}\) and \(\kappa_{I,n}^{t}\) are placed outside the summation, resulting in a configuration referred to as Centered Pattern-Coupled with Spatio-Temporal Coupling (CPC-ST). This structure simultaneously accounts for both temporal and spatial patterns. While the advantage of CPC-ST lies in its comprehensive consideration of dependencies, its complexity is a drawback, as it involves a higher number of parameters to determine. To simplify the model, alternative configurations can be used. For instance, by only considering temporal coupling term and setting \(\gamma_{I,n}^{t}=0\), the structure becomes Centered Pattern-Coupled with Temporal Coupling (CPC-T). Conversely, if only spatial coupling term is considered, with \(\gamma_{I,n}^{t}=0\) as defined in Equation (17), the structure is referred to as Centered Pattern-Coupled with Spatial Coupling (CPC-S). These simplified configurations of coupling terms provide flexibility depending on the modeling needs. Another simplification approach to reduce model complexity and mitigate the risk of overfitting, particularly when data availability is limited, is to simplify the coupling parameters. In Equations (16) and (17), the coupling parameters are linked to both spatial and temporal indices (\(I\) and \(n\)). Following the work of [PERSON]. [29,34], it has been shown that coupling parameters in the spatial dimension may not always be necessary for certain applications. Hence, simplifying the structure by retaining only the temporal coupling dependency results in a reduced representation of the coupling structure. The simplified structure is expressed as:  
\[\gamma_{I,n}^{s}=\kappa_{n}^{s}\sum\limits_{i\in\mathbf{i}_{I}}\gamma_{i,n}, \tag{18}\]  
\[\gamma_{I,n}^{t}=\kappa_{n}^{t}\sum\limits_{j\in\mathbf{j}_{n}}\gamma_{I,j}. \tag{19}\]This structure is referred to as Reduced Pattern-Coupled with Spatio-Temporal Coupling (RPC-ST) when considering both spatial and temporal patterns.