# Spatio-Temporal Heterogeneous Ensemble Learning Method for Predicting Land Subsidence
## 2 Methods
### Two-Stage Hybrid Spatio-Temporal Clustering Method
#### 2.1.1 Co-Clustering of Large-Scale Land Subsidence Data  
In this study, the original subsidence spatio-temporal data were treated as a data matrix, with rows representing different spatial points and columns representing different time observations. This study adopted a co-clustering strategy, treating spatial and temporal dimensions equally and dividing them into \(k\) spatial clusters and \(l\) temporal clusters to obtain a co-clustering spatio-temporal matrix. [PERSON] et al. (2019) proposed the Bregman co-clustering algorithm, which uses information divergence combined with multiple intra-cluster summary statistics. The Bregman co-clustering algorithm (BBAC_I) was used in the first stage because it preserves the mean of co-clusters, effectively identifying co-clusters and revealing the spatio-temporal pattern and its dynamic evolution in the dataset ([PERSON] et al., 2019). The BBAC_I algorithm is highly effective and has been successfully applied in document analysis, temperature studies, and phenological spatio-temporal pattern analysis. Hence, in this study, the BBAC_I algorithm was used to compress the original subsidence spatio-temporal dataset, dividing it into a series of small, internally homogeneous spatio-temporal clusters. BBAC_I is suitable for co-clustering a data matrix with real and positive elements, representing a co-occurrence matrix of joint probabilities between two random variables in the rows and columns. It is an information-theoretic clustering method that treats co-clustering as an optimization problem, calculating the difference between the co-clustered and original data matrices as a loss function ([PERSON] et al., 2019). The BBAC_I algorithm optimizes this loss function to minimize information loss during spatio-temporal cluster partitioning. Assuming the spatio-temporal data matrix is \(O_{ST}\), where \(S\) is the number of spatial sampling points and \(T\) is the number of time clusters, the objective of BBAC_I is to partition rows and columns (Formulas (1) and (2)) to minimize the mutual information loss between \(O_{\hat{\mathcal{S}}\hat{\mathcal{T}}}\) and \(O_{ST}\):  
\[S:\{s_{1},s_{2},\cdots s_{m}\}\rightarrow\hat{\mathcal{S}}:\{\hat{s}_{1},\hat {s}_{2},\cdots\hat{s}_{k}\} \tag{1}\]  
\[T:\{t_{1},t_{2},\cdots t_{n}\}\rightarrow\hat{T}:\{\hat{t}_{1},\hat{t}_{2}, \cdots\hat{t}_{l}\} \tag{2}\]  
The main steps of the BBAC_I algorithm are as follows:  
1. Randomly assign spatial sampling points and time slices to \(k\) spatial clusters and \(l\) temporal clusters, generating the initial co-clustered matrix \(O_{\hat{\mathcal{S}}\hat{\mathcal{T}}}\). 2. Calculate the mutual information loss between the co-clustered matrix and the original data matrix: \[L_{I}=D_{I}\big{(}O_{ST}\parallel O_{\hat{\mathcal{S}}\hat{\mathcal{T}}}\big{)} =\sum_{\hat{s}}\sum_{\hat{t}}\sum_{s\in\hat{t}}\sum_{t\in\hat{t}}o(s,t)log \frac{o(s,t)}{a(s,t)}\] (3)
3. Iteratively update spatial and temporal cluster partitions for each spatial point and time cluster to minimize the mutual information loss between the co-clustered matrix and the original data matrix. 1. Update the spatial cluster partition \(i\) to find the partition that minimizes mutual information loss: \[i=arg\min_{i\in\{1,\ldots,k\}}L_{I}\] (4)
2. Update the temporal cluster partition \(j\) to find the partition that minimizes mutual information loss: \[j=arg\min_{i\in\{1,\ldots,l\}}L_{I}\] (5)  
The algorithm concludes when the mutual information loss between iterations falls below a predefined threshold, indicating stability. However, BBAC_I cannot guarantee convergence to a global optimum for complex data, so multiple executions of the algorithm are often performed, with the best result used as the final clustering outcome. For medium-sized and smaller datasets, the BBAC_I algorithm efficiently generates co-clustering results. However, land subsidence data often have high spatio-temporalresolutions and extensive coverage, with the number of spatio-temporal data matrix feature points exceeding millions.