Spatio-Temporal Heterogeneous Ensemble Learning Method for Predicting Land Subsidence

Abstract

The prediction of land subsidence is of significant value for the early warning and prevention of geological disasters. Although numerous land subsidence prediction methods are currently available, two obstacles still exist: (i) spatio-temporal heterogeneity of land subsidence is not well considered, and (ii) the prediction performance of individual models is unsatisfactory when the data do not meet their assumptions. To address these issues, we developed a spatio-temporal heterogeneous ensemble learning method for predicting land subsidence. Firstly, a two-stage hybrid spatio-temporal clustering method was proposed to divide the dataset into internally homogeneous spatio-temporal clusters. Secondly, within each spatio-temporal cluster, an ensemble learning strategy was employed to combine one time series prediction model and three spatio-temporal prediction models to reduce the prediction uncertainty of an individual model. Experiments on a land subsidence dataset from Cangzhou, China, show that the prediction accuracy of the proposed method is significantly higher than that of four individual prediction models.

1. Introduction

Land subsidence is a geological phenomenon characterized by a slow loss of ground elevation due to natural processes or human activities [1,2,3]. Land subsidence commonly affects large and medium-sized cities, with significant impacts on human activities (e.g., casualties), life, and ecology (e.g., damage to historical heritage, drainage system failures). Over 100 cities worldwide, including China [4,5], Indonesia [6], and Italy [7,8], suffer from tens of centimeters of land subsidence annually [3]. The primary challenge in studying land subsidence is accurately characterizing it and implementing effective prevention and control measures. The accurate prediction of land subsidence is crucial for the early monitoring of geological hazards.

Existing methods for predicting land subsidence can be categorized into three types: physical mechanisms-based methods [4,9,10], statistics-based methods, and machine learning-based methods. Physical mechanisms-based methods utilize field surveys and experiments to characterize complex geological and hydrophysical parameters for predicting land subsidence [11,12]. Bajni, Apuani, and Beretta [9] developed a hydro-geotechnical numerical model with good interpretability for predicting land subsidence. However, these models rely on complex assumptions about data and parameters, which often fail to align with actual land subsidence data, leading to poor fitting results. Statistical methods predict land subsidence by establishing mathematical functions related to the spatial and temporal characteristics of the study area, based on historical observation data. For example, Tang, et al. [13] applied a grey model to predict land subsidence in Shanghai’s Lujiazui area. Deng et al. [14] used time series deformation to build a prediction model based on an improved Grey–Markov model (IGMM) for predicting future subsidence. Statistical methods are simple, easy to use, and low-cost. However, these methods require data to meet certain ideal assumptions, which often conflict with the complex nonlinear characteristics of land subsidence data, resulting in poor prediction results. Machine learning algorithms have been widely used in land subsidence prediction tasks due to their powerful ability to extract high-dimensional, nonlinear features from data, including backpropagation (BP) algorithms, CART, support vector regression (SVR), convolutional neural networks (CNNs), Bayesian models, fuzzy set theory, and long short-term memory (LSTM) networks [15,16,17,18,19,20,21,22]. However, the underlying assumptions of these algorithms may be not satisfied in various scenarios, leading to limited generalization performance. Consequently, ensemble learning methods, which combine multiple individual base learners to improve prediction accuracy, have been applied to land subsidence prediction [23,24,25,26]. Shi et al. [27] proposed a land subsidence prediction model based on XGBoost for the Beijing Plain. Li, Liu, Tao, and Zhai [24] combined the Gradient Boosting Decision Tree (GBDT) and Random Forest (RF) methods to predict land subsidence. Both boosting and bagging strategies combine the same base model multiple times, thus categorizing them as homogeneous ensemble learning methods.

Despite the development of various prediction models, a critical gap remains in addressing the spatio-temporal heterogeneity of subsidence data and the inconsistent performance of individual models. Specifically, two obstacles still exist:

(i)

The spatio-temporal heterogeneity of land subsidence is not adequately considered in existing prediction models. Spatio-temporal heterogeneity refers to the variation in observed values of a spatio-temporal object across time, space, or both. Land subsidence is a spatio-temporally dynamic process, and its spatio-temporal heterogeneity is difficult to characterize. Existing land subsidence prediction methods primarily focus on time series analysis of historical data, often neglecting the spatial dependence of land subsidence data [27]. Recently, some methods have incorporated spatial clustering or time series clustering to consider spatial or temporal heterogeneity in land subsidence prediction [12,21]. However, these methods still struggle to effectively model the spatio-temporal heterogeneity of land subsidence, directly impacting the accuracy and reliability of predictions.

(ii)

The prediction performance of individual models is unsatisfactory when the data deviate from the models’ assumptions. Although some studies have attempted to solve this problem by using homogeneous ensemble learning methods, when there is an inherent bias in the base model, the ensemble model generated through homogeneous ensemble learning methods will inevitably carry this bias [28]. Consequently, homogeneous ensemble models cannot overcome the inherent biases of individual models, resulting in limited improvements in prediction accuracy.

To address these issues, a heterogeneous ensemble learning-based method that considers spatio-temporal heterogeneity was proposed for land subsidence prediction. The main contributions of this study are twofold:

(i)

To address the challenge of adequately considering the spatio-temporal heterogeneity of land subsidence in existing prediction models, we propose a two-stage hybrid clustering method based on co-clustering. In the first stage, the Bregman Block Average Co-clustering with I-divergence (BBAC_I) algorithm [29] is used to compress the original spatio-temporal data, dividing the large-scale land subsidence dataset into a series of small, internally homogeneous spatio-temporal clusters. In the second stage, the REgionalization with Dynamically Constrained Agglomerative clustering and Partitioning (REDCAP) method [30] is employed to further cluster the small spatio-temporal clusters, revealing complex cluster structures. Prediction models are then constructed within each spatio-temporal cluster, allowing for accurate modeling of spatio-temporal heterogeneity of land subsidence.

(ii)

To improve prediction performance when data do not meet the assumptions of individual models, this study adopts a heterogeneous ensemble learning method to combine the results of multi-view and multi-type machine learning prediction models. We developed a heterogeneous ensemble method based on the Blending strategy. For time series prediction, we chose the Sequence-to-Sequence model. For spatio-temporal prediction, we chose the GCN-Sequence-to-Sequence model, which captures the complex spatio-temporal dependencies in the graph structure; the Diffusion Convolutional Recurrent Neural Network model, which can capture the dynamic changes in spatio-temporal relationships; and the Graph Multi-Attention Network model, which incorporates the attention mechanism. The Blending strategy is then used to combine the prediction results of these four individual models, improving the accuracy of land subsidence predictions.

Experiments on a land subsidence dataset from Cangzhou, one of the most affected areas in China, demonstrate that the proposed method outperforms four state-of-the-art individual prediction models in terms of prediction accuracy. The proposed method considers spatio-temporal heterogeneity and combines multiple individual models, providing a novel approach for spatio-temporal data prediction that may aid in the early monitoring of ground geological hazards.

2. Methods

An overall research flowchart of this study is presented in Figure 1. First, a two-stage hybrid spatio-temporal clustering method based on co-clustering was proposed to divide the dataset into internally homogeneous spatio-temporal clusters. Then, within each cluster, a heterogeneous ensemble learning approach was developed to combine four heterogeneous prediction models for land subsidence prediction. In practice, some land subsidence outliers are useful for the early monitoring of ground geological disasters. Indeed, errors and meaningful outliers cannot be well distinguished. Therefore, in this study, we did not remove outliers from the dataset.

2.1. Two-Stage Hybrid Spatio-Temporal Clustering Method

In this study, a two-stage hybrid clustering method based on co-clustering was proposed to consider the spatio-temporal heterogeneity of the land subsidence dataset. In the first stage, the BBAC_I algorithm was applied to compress the large-scale spatio-temporal dataset, dividing the land subsidence dataset into a series of small, internally homogeneous spatio-temporal clusters. However, the limitation of BBAC_I is that it has difficulty detecting spatio-temporal clusters with complex shapes. Therefore, the REDCAP algorithm was employed in the second-stage to further cluster the small spatio-temporal clusters obtained by the BBAC_I, revealing complex cluster structures.

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. Banerjee et al. [29] 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 [31]. 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 [31]. The BBAC_I algorithm optimizes this loss function to minimize information loss during spatio-temporal cluster partitioning. Assuming the spatio-temporal data matrix is $O_{\mathit{ST}}$, 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 $O_{\widehat{S}\widehat{T}}$ and $O_{\mathit{ST}}$:

$$S:\left\{s_1,s_2,\cdots s_m\right\}\to \widehat{S}:\left\{{\widehat{s}}_1,{\widehat{s}}_2,\cdots {\widehat{s}}_k\right\}$$

(1)

$$T:\left\{t_1,t_2,\cdots t_n\right\}\to \widehat{T}:\left\{{\widehat{t}}_1,{\widehat{t}}_2,\cdots {\widehat{t}}_l\right\}$$

(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 $O_{\widehat{S}\widehat{T}}$.

(2)

Calculate the mutual information loss between the co-clustered matrix and the original data matrix:

$$L_I=D_I\left(O_{\mathit{ST}}\parallel O_{\widehat{S}\widehat{T}}\right)=\sum _{\widehat{s}} \sum _{\widehat{t}} \sum _{s\in \widehat{s}} \sum _{t\in \widehat{t}} o\left(s,t\right)\mathit{log}{\displaystyle \frac{o\left(s,t\right)}{a\left(s,t\right)}}$$

(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.

i.

Update the spatial cluster partition i to find the partition that minimizes mutual information loss:

$$i =\mathit{arg}\underset{i\in \{1,\dots ,k\}}{\mathit{min}} L_I$$

(4)

ii.

Update the temporal cluster partition j to find the partition that minimizes mutual information loss:

$$j=\mathit{arg}\underset{i\in \{1,\dots ,l\}}{\mathit{min}} L_I$$

(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-temporal resolutions and extensive coverage, with the number of spatio-temporal data matrix feature points exceeding millions. This scale poses convergence and computational challenges for BBAC_I, limiting its use to the first stage for dividing the dataset into intermediate internally homogeneous clusters.

2.1.2. Discovery of Spatio-Temporal Patterns Based on REDCAP

The REDCAP algorithm, which considers both attribute similarity and adjacency relationships of spatial points, is more efficient in large data sets compared to other regionalization methods. Therefore, to identify spatio-temporal clusters of complex morphology, the REDCAP algorithm was employed in the second-stage to cluster the small spatio-temporal clusters obtained from BBAC_I. The REDCAP algorithm adopts a hierarchical agglomerative–partitioning strategy to discover complex cluster structures.

The REDCAP algorithm first constructs a minimum spanning tree (MST) connecting all points based on adjacency relationships, using first-order (direct connection between points) or full-order (indirect connection between points) strategies. In this study, the adjacency relationships among spatial points in different spatio-temporal clusters are computed based on the union of their K-Nearest Neighbor (KNN) adjacency relations. If two spatio-temporal clusters have adjacent spatial points, they are considered adjacent; otherwise, they are considered non-adjacent. This rule defines the adjacency relationships among the intermediate spatio-temporal clustering results.

To ensure strong spatio-temporal consistency, the full-order strategy is used to construct adjacency relationships among small spatio-temporal clusters. The REDCAP algorithm can be classified into six strategies based on the chosen adjacency relationship determination and inter-cluster distance metric: first-order-SLK, first-order-ALK, first-order-CLK, full-order-SLK, full-order-ALK, and full-order-CLK. Among these, full-order-CLK generally provides high clustering quality and is therefore chosen for agglomerative clustering in this study, generating a spatial connectivity tree using a bottom-up approach [30].

The REDCAP algorithm segments the edges of the spatial connectivity tree in a top-down manner to obtain the final clustering results. This study selects the optimal cut-edge result based on the principle of regional homogenization optimization, using the sum of squared errors (SSE) as the heterogeneity measure:

$$H\left(R\right)=\sum _{j=1}^d \sum _{i=1}^{n_r} {\left(x_{\mathit{ij}}-\overline{x_j}\right)}^2$$

(6)

where R is the subtree containing the spatial objects, $H\left(R\right)$ represents its heterogeneity, d is the attribute dimension of each spatial object, $n_r$ is the number of objects in R, $x_{\mathit{ij}}$ is the j-th attribute value of the i-th object, and $\overline{x_j}$ is the mean value of the j-th attribute value of all objects in R.

When the number of clusters k is set for the REDCAP algorithm, it should include k subtrees obtained from the segmentation. The overall heterogeneity $H_k$ of the partitioning result is the sum of the heterogeneity of the k subtrees:

$$H_k=\sum _{j=1}^k H\left(R_j\right)$$

(7)

During each split, the REDCAP algorithm divides the region R into two sub-regions, $R_a$ and $R_b$, identifying the edge with the maximum homogeneity gain $h_g^{*}\left(R\right)$ (i.e., the reduction in heterogeneity), calculated as:

$$h_g^{*}\left(R\right)=\mathit{max}\left(H\left(R\right)-H\left(R_a\right)-H\left(R_b\right)\right)$$

(8)

By iteratively finding the optimal cut-edge based on the principle of maximum homogeneity gain, the MST is segmented until the number of subtrees reaches k. At this point, the REDCAP algorithm is complete, and the spatial objects in each subtree form a spatial cluster. The REDCAP algorithm can produce clustering results with varying numbers of clusters. The optimal number of clusters is determined using the silhouette coefficient [32]. The average silhouette coefficient of all points in the clustering result is taken as the silhouette coefficient. Typically, the k value corresponding to the maximum silhouette coefficient is chosen.

2.2. Heterogeneous Ensemble Learning-Based Prediction Method

2.2.1. Individual Models for Land Subsidence Prediction

To ensure the diversity of surface subsidence prediction, four models were selected from different perspectives to predict land subsidence. For time series prediction, the Sequence-to-Sequence model, which makes minimal assumptions on the sequence structure, was chosen to model the time dependencies [33]. For spatio-temporal series prediction, the Graph Convolutional Sequence-to-Sequence model (GCN-Seq2Seq) [34] was chosen to capture the local spatial structure representation of land subsidence; the Diffusion Convolutional Recurrent Neural Network model (DCRNN) [35] was chosen to model highly nonlinear spatio-temporal dependencies; and the Graph Multi-Attention Network model (GMAN) [36], which integrates an attention mechanism, was used to model the impact of the spatio-temporal factors on land subsidence.

(i)

Sequence-to-Sequence model

The Seq2Seq model is an encoder–decoder model that performs well in natural language processing and time series analysis. This study uses the Seq2Seq model to predict sedimentation from a temporal perspective. The Seq2Seq model consists of three main components: the encoder, intermediate vector, and decoder, all of which are built using RNN units [33].

The encoder transforms each element of the input sequence into a low-dimensional embedding vector to generate a hidden state, represented as follows:

$$h_t=f_R\left(x_t,h_{t-1}\right)$$

(9)

where $f_R\left(\ast \right)$ represents the function of the RNN unit.

The intermediate vector connects the encoder and the decoder and can be obtained by the hidden state of the last RNN unit of the encoder or the weighted average of all hidden states.

The decoder gradually builds the output sequence based on the intermediate vector. Assuming that the predicted output at time step t is $y_t$, the calculation is as follows:

$$y_t=f_a\left(W{h}_t+b\right)$$

(10)

The basic mechanism behind Seq2Seq model prediction is to construct hidden states and calculate intermediate vectors through the encoder, and then initialize and predict future moments using the decoder. To alleviate the error accumulation problem in predictions, a scheduled sampling strategy is adopted, where the decoder uses a certain probability of using real label data or the prediction results of the previous time step as input during the training phase [33]. This study uses the gated recurrent unit (GRU) method as the core component of the Seq2Seq model to enhance both performance and stability.

(ii)

Graph Convolutional Sequence-to-Sequence model

The GCN-Seq2Seq model combines the Graph Convolutional Network (GCN) [34] with a sequence-to-sequence architecture, focusing on processing spatio-temporal sequences with graph structures. It simultaneously considers the spatial interdependence and temporal continuity, capturing the spatial propagation patterns and temporal evolution trends of land subsidence to enhance predictions’ accuracy and reliability. In the GCN-Seq2Seq model, GCN is responsible for capturing the structural characteristics of graph data. By performing convolution operations on graph nodes and identifying and utilizing the dependencies between nodes, the features of graph data can be effectively extracted and represented.

The main steps of GCN-Seq2Seq model prediction include graph feature extraction, sequence encoding, and sequence decoding [34]. During graph feature extraction, GCN updates the feature representation of each node, as follows:

$$h_v^{\left(l+1\right)}=\sigma \left(\sum _{u\in N\left(v\right)\bigcup \left\{v\right\}}{\displaystyle \frac{1}{c_{\mathit{vu}}}}{W}^{\left(l\right)}{h}_u^{\left(l\right)}\right)$$

(11)

where $\sigma$ is a nonlinear activation function (such as ReLU), N(v) represents the set of neighboring nodes of node $v$, and $c_{\mathit{vu}}$ is a normalization constant. The commonly used symmetric normalization $\sqrt{d_v{d}_u}$ is deployed, where $d_v$ and $d_u$ are the degrees of nodes v and u, respectively, $W^{\left(l\right)}$ is the weight matrix of the l-th layer, and $h_u^{\left(l\right)}$ is the feature representation of the neighboring node u in the l-th layer.

After graph feature extraction, the model proceeds to the sequence encoding, similar to the Seq2Seq encoding process. In the sequence decoding stage, the model gradually generates a predicted sequence based on the encoded information. By combining graph structure and sequence information, the GCN-Seq2Seq model fully integrates spatial and temporal dependencies and is suitable for complex graph data and sequence prediction tasks.

(iii)

Diffusion Convolutional Recurrent Neural Network model

The DCRNN model [35] is a deep learning framework for spatio-temporal series prediction that combines Graph Convolutional Networks (GCNs) and recurrent neural networks (RNNs). DCRNN captures land subsidence propagation characteristics through diffusion convolutional layers and uses RNNs to handle time series data, adapting to nonlinear and dynamic changes [35]. This study selects the DCRNN model as the primary prediction model to account for the dynamic changes in spatio-temporal relationships. The DCRNN model prediction includes two steps. First, in the spatial dependency modeling stage, DCRNN models spatial dependencies through random walks on the graph and the state transition matrix $D_O^{-1}$, as follows:

$$P_{t|1}=\alpha \times D_O^{-1}\times P_t+\left(1-\alpha \right)\times P_0$$

(12)

where $P_t$ is the flow distribution at time step $t$ and $P_0$ is the initial distribution. In the diffusion convolution process, DCRNN applies a defined diffusion convolution operation to the graph signal and filter, combining the forward and backward diffusion processes to capture the upstream and downstream impacts, as follows:

$$H^{\left(l+1\right)}=\left(W^l\times H^l+b^l\right)⨀M^l$$

(13)

where $H^{\left(l+1\right)}$ is the node representation of the (l + 1)-th layer, $W^l$ is the weight matrix, $b^l$ is the bias term, $M^l$ is the diffusion matrix of the l-th layer, and $⨀$ represents element-by-element multiplication. Then, in the temporal dynamic modeling stage, DCRNN uses gated recurrent units (GRUs) to model temporal dependencies and forms DCGRUs by replacing matrix multiplications in GRUs with diffusion convolutions. DCGRUs process time series data by resetting gates, updating gates, and utilizing candidate hidden state mechanisms, all combined with spatial diffusion information. Finally, the DCRNN uses a Seq2Seq structure for multi-step prediction, with the encoder processing historical time series and the decoder generating predictions based on the final encoder state.

(iv)

Graph Multi-Attention Network model

The GMAN model is an encoder–decoder architecture that integrates an attention mechanism, including the ST-Attention module (STAtt) with residual connections [37]. GMAN incorporates graph structure and temporal information into a multi-attention mechanism through spatio-temporal embeddings, enriching contextual information and enhancing the expression ability of spatio-temporal data [36].

The basic mechanism of the GMAN model prediction is as follows. First, the spatio-temporal embedding module learns vertex representations through the node2vec method and inputs them into a two-layer, fully connected neural network to obtain spatial embedding. Since spatial embeddings provide only static representation, GMAN also introduces temporal embedding, encoding each time step into a vector, which is then processed through a fully connected neural network. The spatial embedding and temporal embedding are fused into the spatio-temporal embedding module (STE), which is defined as follows:

$$e_{v_i,t_j}=e_{v_i}^S+e_{t_j}^T$$

(14)

The temporal attention module includes the spatial attention module, the temporal attention module, and the gated fusion. The model represents the input of the l-th block as $H^{(l-1)}$ and the outputs of the spatial and temporal attention mechanisms as $H_S^{\left(l\right)}$ and $H_T^{\left(l\right)}$, respectively. After gated fusion, the output of the l-th block is obtained as $H^{\left(l\right)}$.

GMAN employs an encoder–decoder architecture with a residual-connected ST-Attention block to effectively capture and integrate features in both spatial and temporal dimensions, making it suitable for complex spatio-temporal data and multi-dimensional sequence prediction tasks.

2.2.2. Blending-Based Ensemble Learning Strategy

Blending [38] is a heterogeneous ensemble learning algorithm and a variant of Stacking [39]. Similar to Stacking, Blending improves overall prediction performance by combining the prediction results of multiple independent models of different types. However, Blending uses a hold-out strategy for dataset splitting during training, in contrast to the cross-validation strategy used in Stacking. The core mechanism of Blending is that by combining individual models, the advantages of each model can be fully utilized to achieve a prediction effect that exceeds that of an individual model [40]. The Blending algorithm contains two important components: base learners and a meta-learner. The base learners are a diverse set of pre-selected models. The base learners are as diverse as possible to capture the advantages of different aspects of the data. First, the base learners are trained on the training set. The meta-learner uses the prediction results of the base learner on the test set as input to learn how to best combine the predictions of the base learners, so as to combine different base learners to obtain more accurate prediction results. The four individual models for land subsidence prediction introduced in Section 2.2.1 were combined using the Blending algorithm. The flowchart of the Blending ensemble algorithm is illustrated in Figure 2.

For the selection of meta-learners, existing ensemble learning research has shown that simple linear models can achieve good performance [41], so linear regression was selected in this study as the meta-learner. The basic mechanism involves using the prediction results of the base learners as independent variables and generating the final prediction value through linear combination, represented as follows:

$$Y={\beta }_1{X}_1+{\beta }_2{X}_2+\dots +{\beta }_i{X}_I+\epsilon$$

(15)

where ${\beta }_i$ represents the contribution of an individual model, i.e., the weight; $X_i$ represents the predicted value of an individual model. During the training process, the weight of each base learner is determined by minimizing the prediction error. In this way, the Blending algorithm effectively combines the prediction results of different models achieving more accurate land subsidence prediction results. The four individual models are implemented in the pytorch library and the TensorFlow python library. The pseudo-code of the proposed method (Algorithm 1) is presented below.

Algorithm 1: Spatio-temporal ensemble predicting (LS_Mat, k, l, k_final, split_ratio)

// Two-stage hybrid spatio-temporal clustering Co_clusters= BBAC_I (LS_Mat, k, l); //first stage co_clustering using BBAC_I algorithm Aggregated_clusters = Aggregate (LS_Mat, Co_clusters) //Aggregate each co-cluster into single object ST_clusters = REDCAP(Aggregated_clusters, k_final); // second stage clustering using REDCAP algorithm // Blending-based ensemble learning for prediction Base_models = [S2S, GCN_S2S,DCRNN, GMAN] Testing_results_all = [] for cluster in ST_clusters do    Train_set, Test_set, Valiadate_set = Split(cluster, split_ratio)    for model in Base_models do       Trained_model = Train(model)       Testing_result = Trained_model(Test_set)       Testing_results_all.add(Testing_result)    end for    // Train Linear regression model for Blending ensemble      Trained_LR = LR.fit(Testing_results_all)    Valiadating_result = Trained_LR(Valiadate_set) end for

2.3. Model Evaluation

To comprehensively evaluate the prediction performance of each model, this study selected root mean square error (RMSE), mean absolute error (MAE), and the coefficient of determination ($R^2$) as the main evaluation indicators. RMSE measures prediction error by calculating the square root of the average squared deviations between the predicted value and actual observed value. RMSE is highly sensitive to large errors, making the performance of the model in extreme cases particularly important. MAE provides a more intuitive error assessment by calculating the average of the absolute deviations between the predicted value and actual value. The advantage of MAE is its simple calculation and ease of understanding, making it a direct indicator for evaluating the prediction accuracy. The $R^2$ indicator measures the model’s ability to explain variability in the data. The $R^2$ value ranges from 0 to 1, with values closer to 1 indicating better model fit and prediction accuracy. The specific calculation formulas for each metric (RMSE: Equation (16); MAE: Equation (17); $R^2$: Equation (18)) are as follows:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N_S{N}_E}}\sum _{s=1}^{N_S}\sum _{t=1}^{N_E}{\left(y_s^t-{\widehat{y}}_s^t\right)}^2}$$

(16)

$$\mathrm{MAE}={\displaystyle \frac{1}{N_S{N}_E}}\sum _{s=1}^{N_S}\sum _{t=1}^{N_E}\left|y_s^t-{\widehat{y}}_s^t\right|$$

(17)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{s=1}^{N_S}}{\displaystyle \sum _{t=1}^{N_E}}(y_s^t-{\widehat{y}}_s^t{)}^2}{{\displaystyle \sum _{s=1}^{N_S}}{\displaystyle \sum _{t=1}^{N_E}}(y_s^t-\bar{Y}{)}^2}}$$

(18)

where $y_s^t$ and ${\widehat{y}}_s^t$ represent the observed and predicted subsidence values at monitoring station j at time step t, respectively; $N_E$ is the total number of subsidence monitoring point samples in the test set; $N_S$ is the total number of subsidence monitoring points; and $\bar{Y}$ is the overall average of the observed values.

3. Experiments

3.1. Study Area

The Cangzhou area is located in the southeast of Hebei Province, China. It is 165 km wide from north to south and 187 km long from east to west, with a total area of about 14,304.26 km². Since the 1960s, to meet the needs of economic development, the region has conducted large-scale groundwater resource exploitation, resulting in a sharp drop in groundwater levels, which in turn has caused the most severe land subsidence problem in North China. Therefore, Cangzhou was selected as the study area for this research.

3.2. Dataset

Eighty Sentinel-1A images acquired in the Cangzhou area between January 2017 and December 2019 were used as the dataset. The multiple Master-image Coherent Target Small-baseline Interferometric SAR method was used [42] to generate the InSAR deformation time series. Figure 3 shows the spatio-temporal distribution of spatio-temporal land subsidence data in the Cangzhou area (for part of the timestamps). It indicates that high subsidence values are mainly distributed in the western part of Cangzhou, whereas the eastern part of the Cangzhou area mainly consists of points with low subsidence values and uplift points. In terms of temporal distribution, high subsidence points are mainly found in the early stages, while low subsidence points are predominantly found in the later stages. This study organizes the subsidence data into a data matrix of (69,852, 80) for spatio-temporal clustering. The three representative spatio-temporal clusters obtained by clustering are selected as input data for subsequent predictions.

3.3. Data Preprocessing

Since the BBAC_I algorithm requires each entry in the data matrix to be positive, the original data matrix is transformed into a positive matrix by adding the opposite number of the minimum value. After spatio-temporal clustering, all subsidence values are transformed to [0, 1] and input into the ensemble model for prediction.

3.4. Parameter Setting

In this study, the number of clusters in the BBAC_I algorithm and the REDCAP algorithm of the two-stage hybrid clustering needs to be manually specified. To determine the number of spatial and temporal clusters in the BBAC_I algorithm, this study tested spatial clusters ranging from 200 to 400 with an interval of 50 and temporal clusters ranging from 3 to 8 with an interval of 1. It was found that when the number of spatial clusters was 200 and the number of temporal clusters was 3, the BBAC_I algorithm could reach a convergence state, and the clustering results have the minimum mutual information loss with the original data matrix. To determine the number of clusters k in the REDCAP algorithm, the silhouette coefficient was used in this study to find the optimal clustering. As shown in Figure 4, when k is 4, the silhouette coefficient reaches an inflection point; thus, the final number of clusters is set to 4.

To achieve optimal results, the BBAC_I algorithm was set with a maximum of 3000 iterations, a convergence threshold of 1 × 10⁻⁵, and 200 repetitions. For the adjacency relationship and difference measurement method of the REDCAP algorithm, Guo [30] demonstrated that using the full-order configuration for constructing the adjacency relationship and CLK for the difference measurement method achieves the best clustering performance compared with other methods. Therefore, this study adopts the REDCAP algorithm under the full-order-CLK configuration.

This study sets a unified input sequence length (i.e., 10 steps) and sliding window step length (i.e., 1 step) for the selected prediction model. The prediction output sequence length is set to 1 step (each step corresponds to 1 monitoring interval). These settings ensure that the model effectively captures short-term dynamic changes in land subsidence, providing accurate data support for prediction analysis. This study uses the KNN algorithm to find the 10 nearest neighbors around each point to establish spatial connections and uses the spatial distance between points as the weight of the edges to construct a spatial graph.

The dataset was split into training and testing sets with an 80-20 ratio, meaning 80% of the data was used for training the model, while the remaining 20% was reserved for testing its performance. This 80-20 split is a common practice in machine learning, as it provides a balanced approach: the majority of the data is used to train the model, allowing it to learn effectively, while a sufficient portion is held back to evaluate how well the model generalizes to unseen data. Within the training set, 20% was further utilized to train the ensemble learner. This step ensures that the ensemble learner, which typically combines predictions from multiple models, is trained on diverse subsets of the data, enhancing its robustness. A potential bias of the 80-20 split is that the model may not encounter all data variations during training, particularly in small or unevenly distributed datasets. To mitigate this, techniques such as cross-validation can be employed, where the training set is split into multiple subsets, and the model is trained and validated on these different subsets iteratively. This approach ensures the model is trained on the maximum data variation, reducing the risk of overfitting or underfitting. Overall, the 80-20 split was chosen to balance model training efficiency and evaluation reliability, while potential biases were addressed through careful validation strategies.

To comprehensively evaluate and compare the performance of different individual models in predicting spatio-temporal land subsidence, this study selected three clusters with representative subsidence patterns based on clustering results and conducted a detailed evaluation of the prediction accuracy of four individual models. In this study, a trial-and-error approach was employed for parameter tuning. Specifically, different parameter combinations were tried, and their performance was evaluated until the highest-performing set was determined. While this process was time-consuming, it ensured that the model was in its optimal state for the experiment. The specific parameter configurations of the basic models are listed in

Table 1. Parameters of the base models.

Model	Hyper Parameters
Seq2Seq	batch size = 64, epochs = 100, hidden units = 16, learning rate = 0.001, teacher forcing ratio = 0.2
GCN-Seq2Seq	batch size = 64, epochs = 100, hidden units = 16, learning rate = 0.001, teacher forcing ratio = 0.2
DCRNN	batch size = 64, epochs = 100, learning rate = 2.0 × 10−6
GMAN	batch size = 64, epochs = 100, learning rate = 0.005

.

3.5. Results

3.5.1. Spatio-Temporal Clustering Results

Following the two-stage hybrid spatio-temporal clustering, the spatio-temporal data matrix of land subsidence was divided into four clusters. Based on the mean value of subsidence in each spatio-temporal cluster, the four homogeneous spatio-temporal clusters are defined as slight uplift, slight subsidence, moderate subsidence, and severe subsidence. According to these definitions, the spatio-temporal clustering results are shown in Figure 5. The time clusters are arranged in chronological order, and the spatial clusters are arranged by the mean value of the rows within the cluster. The clustering result matrix shows that most spatio-temporal points fall into the slight uplift and slight subsidence categories. Statistical information of the spatio-temporal clustering results is presented in

Table 2. Statistics of spatio-temporal clustering matrix of land subsidence data.

Cluster ID	Pattern	Mean (mm)	Proportion
1	slight uplift	15	43.28%
2	slight subsidence	−17	42.42%
3	moderate subsidence	−65	11.19%
4	severe subsidence	−124	3.11%

. The mean values for the four subsidence patterns are as follows: slight uplift (15 mm), slight subsidence (−17 mm), moderate subsidence (−65 mm), and severe subsidence (−124 mm). Judging from the proportion of the four spatio-temporal subsidence patterns, the spatio-temporal points of land subsidence in the entire study area are mainly categorized as slight uplift and slight subsidence, accounting for 85.7%, indicating that the overall subsidence in the study area is slight. The spatio-temporal points of land subsidence with moderate and severe subsidence patterns account for a small proportion, totaling 14.3%.

(i)

Spatial distribution pattern

The spatio-temporal distribution pattern of subsidence was initially analyzed from a spatial perspective. Columns corresponding to the same time cluster in the spatio-temporal clustering matrix were merged to derive three spatial patterns, as illustrated in Figure 6. The proportion of each type of spatio-temporal subsidence pattern points in each spatial pattern and the number of occupied time periods are presented in

Table 3. The proportion of each type of spatio-temporal subsidence pattern and the number of time periods they occupy in each spatial pattern.

Spatial Pattern	Slight Uplift	Slight Subsidence	Moderate Subsidence	Severe Subsidence	Time Period
Spatial pattern 1	33.78%	66.10%	0.1%	0	27
Spatial pattern 2	41.20%	37.75%	18.33%	2.72%	29
Spatial pattern 3	56.47%	21.40%	15.04%	7.08%	24

.

In spatial pattern 1, the slight subsidence and slight uplift patterns account for nearly all spatio-temporal points, with the slight subsidence pattern representing the largest proportion. Moderate subsidence is represented minimally, while severe subsidence is absent in spatial pattern 1. In spatial pattern 3, the slight subsidence and slight uplift patterns still account for most spatio-temporal points, with the slight uplift pattern representing the largest proportion.

(ii)

Temporal distribution pattern

Additionally, the subsidence pattern was analyzed from a temporal perspective. Spatial points exhibiting the same changes in the temporal dimension in the clustering result matrix were merged and classified into six temporal patterns. As illustrated in Figure 7, six temporal patterns are identified based on the temporal and spatial patterns of spatial points in the time series: stable slight uplift (temporal pattern 1, Figure 7a), stable slight subsidence (temporal pattern 2, Figure 7b), slight subsidence–slight uplift (temporal pattern 3, Figure 7c), slight subsidence–moderate subsidence (temporal pattern 4, Figure 7d), slight subsidence–moderate subsidence–severe subsidence (temporal pattern 5, Figure 7e), and moderate subsidence–severe subsidence (temporal pattern 6, Figure 7f). The spatial distribution of each temporal pattern is presented in Figure 8, with different color markings distinguishing the spatial points of various temporal patterns. The proportion of spatial points in various temporal patterns within the spatial extent of Cangzhou area is detailed in

Table 4. The proportion of each time pattern in the study area.

Time Pattern	Trend	Proportion
Time pattern 1	Stable slight uplift	33.78%
Time pattern 2	Stable slight subsidence	21.40%
Time pattern 3	Stable slight subsidence–uplift	22.69%
Time pattern 4	Slight subsidence–moderate subsidence	15.03%
Time pattern 5	Slight subsidence–moderate subsidence–severe subsidence	2.61%
Time pattern 6	Moderate subsidence–severe subsidence	4.46%

.

The overall temporal pattern of land subsidence in Cangzhou area falls into three categories: stable (stable slight uplift, stable slight subsidence), fluctuating (slight subsidence–slight uplift, slight subsidence–moderate subsidence, and moderate subsidence–severe subsidence), and violent fluctuation (slight subsidence–moderate subsidence–severe subsidence). Of these, the stable temporal pattern accounts for the largest proportion (more than 50%), followed by the fluctuating temporal pattern (accounting for more than 40%), while the violent fluctuation temporal pattern accounts for the lowest proportion (less than 5%). The spatial distribution of the six temporal pattern points in the Cangzhou area is illustrated in Figure 8. Slight subsidence–moderate subsidence is primarily found in the southwest part, while moderate subsidence–severe subsidence is primarily found in the westernmost part. This indicates that the land subsidence conditions in these areas have been continuously intensifying over the 80-period timeframe. Slight subsidence–moderate subsidence–severe subsidence is primarily found in the southwest part; moderate subsidence–severe subsidence is mainly distributed in the northwest part. This suggests that the land subsidence in these areas is worsening and necessitates intervention and control by relevant authorities.

This study employs a two-stage hybrid spatio-temporal clustering method to conduct co-clustering on the land subsidence data of the Cangzhou area. Although each spatio-temporal cluster is roughly continuous in space, there are still a considerable number of points mixed with other clusters. To facilitate subsequent subsidence value prediction of land subsidence spatio-temporal data, the spatio-temporal clustering results were post-processed to obtain spatio-temporally continuous and homogeneous subclusters. First, KNN is employed to identify the spatio-temporal neighbors of each spatio-temporal object. Then, the spatio-temporal cluster IDs of all neighbors in its spatio-temporal neighborhood are voted on, and the cluster ID with the highest vote is assigned as the final ID of the object. Finally, the isolated points from the KNN spatio-temporal neighbor voting results are manually assigned the IDs of the surrounding points to obtain the final clustering results. For subsequent land subsidence prediction, this study selected three representative subsidence pattern clusters: the slight subsidence pattern (cluster 1), the moderate subsidence pattern (cluster 2), and the severe subsidence pattern (cluster 3). The statistical information of these three spatio-temporal clusters is presented in

Table 5. Spatio-temporal cluster information of three representative types of subsidence patterns.

Cluster ID	Pattern	Mean	Timestamp Range	Data Matrix Size
1	Slight subsidence	−21	Z27–Z55	26,508 × 29
2	Moderate subsidence	−64	Z27–Z55	6499 × 29
3	Moderate subsidence	−144	Z56–Z79	1602 × 24

. As the subsidence pattern becomes more severe, the size of the spatio-temporal cluster matrix increases correspondingly.

3.5.2. Spatio-Temporal Prediction Results

(i)

Comparison of overall prediction accuracy

The overall prediction accuracy of the ensemble model and the four base models when predicting subsidence values for cluster 1, cluster 2, and cluster 3 is displayed in

Table 6. Prediction accuracy of ensemble model and base models.

Model	Metric	Cluster 1	Cluster 2	Cluster 3
Seq2Seq	RMSE	3.5680	4.6509	6.9436
	MAE	2.8608	3.7389	5.1175
	R2	0.9520	0.9591	0.9676
GCN-Seq2Seq	RMSE	2.8094	4.7664	7.3231
	MAE	2.1040	3.6568	6.2494
	R2	0.9742	0.9570	0.9640
DCRNN	RMSE	2.9939	4.5349	5.7840
	MAE	2.1669	3.3007	3.6702
	R2	0.9662	0.9610	0.9775
GMAN	RMSE	3.1274	4.2175	4.6243
	MAE	2.4888	3.3909	3.8729
	R2	0.9627	0.9662	0.9857
Ensemble	RMSE	1.3988	1.4696	0.9597
	MAE	1.0355	1.2248	0.6788
	R2	0.9925	0.9959	0.9994

. The experimental results demonstrate that the ensemble prediction model developed in this study exhibits the best prediction performance among the five models across various subsidence clusters. It is evident that the prediction accuracy of subsidence values significantly improves after the Blending ensemble is implemented in the model. Compared with the Seq2Seq model, the ensemble model reduces the RMSE by up to 86.2% and the MAE by up to 86.7% for cluster 3; compared with the GCN-Seq2Seq model, the RMSE is reduced by up to 86.9% and the MAE by up to 89.1% for cluster 3; compared with the DCRNN model, the RMSE is reduced by up to 83.4% and the MAE by up to 81.2% for cluster 3; and compared with the GMAN model, the RMSE is reduced by up to 79.2% and the MAE by up to 82.5% for cluster 3. Methods based on spatio-temporal graph neural networks (such as GCN-Seq2Seq, DCRNN, and GMAN) generally outperform Seq2Seq on the three clusters because these models consider the spatio-temporal correlation between different monitoring points. Among the methods based on spatio-temporal graph neural networks, GCN-Seq2Seq has the best prediction performance for cluster 1 (slight subsidence area), while GMAN performs best for cluster 2 (moderate subsidence area) and cluster 3 (severe subsidence area). Compared to the basic models, the prediction accuracy of the ensemble model is significantly improved for cluster 3, as the basic models achieve their best performance on cluster 3, leading to optimal ensembling results.

To intuitively compare the prediction effects of different models, this study visualized the prediction results of the five models across clusters. Given the short time series of the data, the time series in the prediction results are averaged, and the prediction results for each monitoring point are visualized, as shown in Figure 9, Figure 10 and Figure 11. The results indicate that the prediction outcomes of the ensemble model are highly consistent with the actual subsidence values and significantly outperform the other base prediction models.

This study further utilized a paired t-test to evaluate whether the prediction accuracy of the ensemble model is statistically significantly improved compared to the base model [43]. This test helps assess whether the observed differences in prediction performance are statistically significant or likely due to random variation. For each model $M_i$, the error sequence is calculated:

$$E_i=\left\{\left|y_A^1-{\widehat{y}}_A^1\right|,\left|y_A^2-{\widehat{y}}_A^2\right|,\dots ,\left|y_A^t-{\widehat{y}}_A^t\right|,\dots \left|y_A^{N_E}-{\widehat{y}}_A^{N_E}\right|\right\}$$

(19)

where $y_A^t$ and ${\widehat{y}}_A^t$ represent the observed and true values of the subsidence at each monitoring point under the average time step.

For two different models $M_i$ and $M_j$, the null hypothesis (no significant difference in the prediction accuracy of the two models) and the alternative hypothesis (the prediction accuracy of the $M_i$ model is significantly better than the $M_j$ model) can be defined as follows:

$$H0:\mu \left(E_i\right)=\mu \left(E_j\right)$$

(20)

$$H1: \mu \left(E_i\right)<\mu \left(E_j\right)$$

(21)

In this study, the significance level was set to 0.05. The statistical comparison of the prediction accuracy of different models is shown in

Table 7. Statistical comparison of prediction accuracy of different models (cluster 1).

Model	Seq2Seq	GCN-Seq2Seq	DCRNN	GMAN
Ensemble	Yes	Yes	Yes	Yes
Seq2Seq		No	No	No
GCN-Seq2Seq			Yes	Yes
DCRNN				Yes

,

Table 8. Statistical comparison of prediction accuracy of different models (cluster 2).

Model	Seq2Seq	GCN-Seq2Seq	DCRNN	GMAN
Ensemble	Yes	Yes	Yes	Yes
Seq2Seq		No	No	No
GCN-Seq2Seq			No	No
DCRNN				No

and

Table 9. Statistical comparison of prediction accuracy of different models (cluster 3).

Model	Seq2Seq	GCN-Seq2Seq	DCRNN	GMAN
Ensemble	Yes	Yes	Yes	Yes
Seq2Seq		Yes	No	No
GCN-Seq2Seq			No	No
DCRNN				No

. If cell $C_{\mathit{ij}}$ is marked as “Yes”, it means that the prediction accuracy of model $M_i$ is statistically significantly better than model $M_j$; otherwise, there is no significant difference in the prediction accuracy of the two models. It can be observed that the prediction accuracy of the ensemble model is statistically significantly better than that of the base models in all data sets.

(ii)

Comparison of prediction accuracy at extreme monitoring points

This study further compared the prediction performance of different models at peak monitoring points. The sliding window method, with a window size set to 10, was used to identify the peak points of the subsidence value. For each cluster, the maximum and minimum values within the sliding window were identified as peak points. Figure 12 illustrates the peak points of the average time series of all stations for each cluster. As seen in

Table 10. Prediction accuracy of the ensemble model and base models at peak monitoring points.

Model	Metric	Cluster 1	Cluster 2	Cluster 3
Seq2Seq	RMSE	3.7265	4.6453	6.5843
	MAE	3.0159	3.8015	5.0155
	R2	0.9494	0.9575	0.9659
GCN-Seq2Seq	RMSE	2.6863	4.776	7.3002
	MAE	2.0018	3.6522	6.1603
	R2	0.9737	0.9550	0.9624
DCRNN	RMSE	3.0955	4.7104	5.3870
	MAE	2.2744	3.4491	3.7927
	R2	0.9651	0.9563	0.9796
GMAN	RMSE	3.3913	2.7253	4.4028
	MAE	2.6707	2.1196	3.7022
	R2	0.9581	0.9854	0.9864
Ensemble	RMSE	1.5167	1.6109	1.3213
	MAE	1.1697	1.3289	0.9396
	R2	0.9916	0.9949	0.9988

, the performance of the ensemble model at peak monitoring points is also better than that of the base model; the prediction accuracy of models considering spatio-temporal dependence is generally better than that of models considering only time dependence.

This study also conducted a paired t-test on the prediction results of peak monitoring point data to determine whether the prediction accuracy of ensemble model at these points is statistically better than that of the basic model. As shown in

Table 11. Statistical comparison of prediction accuracy of different models at peak monitoring points (cluster 1).

Model	Seq2Seq	GCN-Seq2Seq	DCRNN	GMAN
Ensemble	Yes	Yes	Yes	Yes
Seq2Seq		No	No	No
GCN-Seq2Seq			Yes	Yes
DCRNN				Yes

,

Table 12. Statistical comparison of prediction accuracy of different models at peak monitoring points (cluster 2).

Model	Seq2Seq	GCN-Seq2Seq	DCRNN	GMAN
Ensemble	Yes	Yes	Yes	Yes
Seq2Seq		No	No	No
GCN-Seq2Seq			No	No
DCRNN				No

and

Table 13. Statistical comparison of prediction accuracy of different models at peak monitoring points (cluster 3).

Model	Seq2Seq	GCN-Seq2Seq	DCRNN	GMAN
Ensemble	Yes	Yes	Yes	Yes
Seq2Seq		Yes	No	No
GCN-Seq2Seq			No	No
DCRNN				No

, compared to the basic model, the ensemble model achieves statistically significant improvement at extreme monitoring points.

3.6. Discussion

Experimental results indicate that considering the spatio-temporal heterogeneity of land subsidence can significantly improve the prediction accuracy of the model. Additionally, the Blending ensemble model significantly outperforms individual models in terms of overall prediction accuracy and peak monitoring point prediction performance. The main reasons for this are attributed to the following two aspects:

(i)

The proposed two-stage hybrid clustering method divides the large-scale land subsidence dataset into internally homogeneous spatio-temporal clusters. The heterogeneous ensemble model then shows excellent prediction performance on these clusters, demonstrating that considering the spatio-temporal heterogeneity improves the prediction performance.

(ii)

Ensemble learning enhances prediction accuracy and reduces errors by combining the results of multiple individual models. In this study, the Blending ensemble strategy effectively combines the strengths of different individual models to achieve higher accuracy in complex subsidence prediction tasks. Although the predictions of individual model may not meet the requirements in some cases, satisfactory land subsidence prediction results are obtained after ensembling, greatly improving accuracy.

The experimental results in Section 3.5 highlight the strengths and weaknesses of each individual model. For the slight subsidence pattern (Cluster 1), Seq2Seq, which only considers temporal autocorrelation performs worse than the methods considering spatio-temporal autocorrelation. This indicates that accounting for spatial autocorrelation can improve the prediction accuracy for slight subsidence patterns with relatively uniform spatial distribution. For moderate and severe subsidence patterns (Clusters 2 and 3), spatio-temporal prediction models do not always perform better than the time series prediction model. We found that GCN-Seq2Seq performs worse than Seq2Seq. This may be due to the fact that the user-specified spatial proximity relationship cannot effectively capture the spatial autocorrelation [43] due to the datasets exhibiting significant local spatial heterogeneity. In this situation, the diffusion convolution operation [35] and the attention mechanism [36] are more suitable for modeling spatio-temporal autocorrelation.

Additionally, this study’s results were compared with a previous study on land subsidence prediction in Cangzhou, which accounted for spatial heterogeneity [12]. We found that both of them consider the heterogeneity of land subsidence dataset, so the prediction performance can be improved compared with the global methods. Compared to [12], this study further addresses spatio-temporal heterogeneity in land subsidence prediction. Actually, the model constructed in [12] is similar to the Seq2Seq model tested in this study. According to the experimental results given in Section 3.5, we can infer that the model developed in this study outperforms that developed in [12]. In addition, the proposed method in this study is different from the related study in numerical performance. The reason is that the related study was proposed to predict differential deformation values, while this study predicts cumulative subsidence values.

4. Conclusions

This study proposes a spatio-temporal heterogeneous ensemble learning method for predicting land subsidence. The achievements of this study can be summarized as follows. (i) A hybrid spatio-temporal clustering method is developed to segment the dataset into a series of homogenous spatio-temporal clusters, thereby effectively considering the spatio-temporal heterogeneity of the land subsidence data. (ii) We construct a heterogeneous ensemble prediction model by combining four different individual prediction models, thus leveraging the advantages of multiple models through ensemble learning methods to adapt to the complex land subsidence data distribution and improve prediction accuracy. (iii) We conducted an experiment on a land subsidence dataset collected in Cangzhou, China. The results show that the proposed method performs better than four state-of-the-art models, providing a reference for effectively managing and controlling land subsidence. This study introduces a new approach to spatio-temporal prediction, particularly as a feasible solution for addressing the spatio-temporal heterogeneity of large-scale datasets.

Despite its strong performance in subsidence prediction, the ensemble model has some limitations. Ensemble learning performance is highly dependent on the chosen base models. Additionally, Blending is just one of many ensemble learning strategies, and other methods are worth considering. Their effectiveness in improving model performance and applicability requires further evaluation. Furthermore, the complexity of the ensemble model is typically higher than that of individual models, which may lead to higher computational costs and reduced interpretability. Future work will focus on comparing multiple ensemble learning strategies and developing methods with lower computational requirements.