## 2 Materials and Methods
### Method
#### 2.2.3 Details of Spatial-Temporal Co-Clustering Algorithm  
The spatial-temporal co-clustering algorithm views the SST data as a joint matrix of timestamps and locations, denoted by the symbol \(O(S,T)\), where \(S\) takes a value from the site set \(\{s_{1},s_{2},\ldots,s_{m}\}\), \(m\) indicates the total number of sites, and \(T\) takes its values from the timestamps set \(\{t_{1},t_{2},\ldots,t_{n}\}\), where \(t\) stands for the number of timestamps. The output of the algorithm is given by \(O(\hat{S},\hat{T})\), where \(\hat{S}\) is derived from the site clustering outcomes \(\{\hat{s}_{1},\hat{s}_{2},\ldots,\hat{s}_{k}\}\), \(k\) represents the number of spatial clusters (\(k<m\)), \(\hat{T}\) is derived from the timestamp clustering outcomes \(\bigl{\{}\hat{t}_{1},\hat{t}_{2},\ldots,\hat{t}_{1}\bigr{\}}\), and \(l\) represents the number of time clusters (\(l<n\)). The goal of the algorithm is to minimize the information divergence between \(O(S,T)\) and \(O(\hat{S},\hat{T})\). The algorithm flowchart is shown in Figure 3. Step 3 is the most crucial part of the spatial-temporal co-clustering algorithm, illustrating how we consider time when performing spatial clustering, and how we consider space when conducting temporal clustering. **Step 1:** Initialize at random the station-to-station mapping and timestamp-to-timestamp mapping. This is the subsequent step in calculating the loss of information divergence. The matrix after random initialization is \(A(S,T)\), as shown in Equation (2):  
\[A(S,T)=RO(\hat{S},\hat{T})C^{T} \tag{2}\]  
where \(R\) and \(C\) are the matrices of \(m*k,n*l\), respectively, representing the cluster members in space and time. **Step 2:** For the two matrices before and after mapping, compute the loss of information divergence. The algorithm determines the loss of information by calculating the information divergence between the original data matrix \(O(S,T)\) and the approximated matrix \(O(\hat{S},\hat{T})\), as shown in Equation (3):  
\[I(S,T)-I\big{(}\hat{S},\hat{T}\big{)}=D_{I}(O(S,T)\|A(S,T)) \tag{3}\]  
\(D_{I}(\bullet\|\bullet)\) is representative of the information divergence between the two matrices. \(O(S,T)\) is the original data matrix, whereas \(A(S,T)\) is the approximate matrix of the original matrix. According to the definition of information divergence, Equation (3) in step 2 can be written as Equation (4), wherein \(o(s,t)\) originates from \(O(S,T)\), and \(a(s,t)\) originates from \(A(S,T)\). \[D_{I}(O(S,T)||A(S,T))=\sum_{\hat{s}}\sum_{\hat{t}}\sum_{s\leq\hat{s}}\sum_{t \in\hat{T}}o(s,t)log\frac{o(s,t)}{a(s,t)} \tag{4}\]  
**Step 3:** This step is the key to spatial-temporal co-clustering, where we consider the spatial dimension when mapping new temporal clusters, and the temporal dimension when mapping new spatial clusters. Equation (4) can be decomposed into the information divergence of the mapping from stations to station groups, as shown in Equation (5). Equation (5) considers the temporal dimension when mapping new station clusters, which  
Figure 3: Flowchart of spatial–temporal co-clustering algorithm for discovering spatial–temporal patterns of SST. “**” indicates the specific SST value. helps us understand how the spatial patterns of SST change over time, as shown in step 3a of Figure 3.  
\[D_{I}(O(T\mid s)|A(T\mid s))=\sum_{\hat{s}}\sum_{S\in\hat{s}}o(T\mid s)\text{log} \frac{o(T\mid s)}{a(T\mid\hat{s})} \tag{5}\]  
The information divergence of the mapping from timestamps to clusters of timestamps is shown in Equation (6). Similarly, Equation (6) considers the spatial dimension when mapping new temporal clusters, which helps us understand the impact of the spatial dimension on the temporal patterns of SST, as shown in step 3b of Figure 3.  
\[D_{I}\big{(}O(S\mid t)\big{|}A\big{(}S\mid\hat{t}\big{)}\big{)}=\sum_{\hat{t} }\sum_{t\in\hat{t}}o(S\mid t)\text{log}\frac{o(S\mid t)}{a\big{(}S\mid\hat{t} \big{)}} \tag{6}\]  
**Step 4:** As stated previously, optimal co-clustering minimizes mutual information loss.