# Spatio-Temporal Heterogeneous Ensemble Learning Method for Predicting Land Subsidence
## 2 Methods
### Heterogeneous Ensemble Learning-Based Prediction Method  
#### 2.2.1 Individual Models for Land Subsidence Prediction  
Sequence-to-Sequence model  
The Seq2 Seq model is an encoder-decoder model that performs well in natural language processing and time series analysis. This study uses the Seq2 Seq model to predict sedimentation from a temporal perspective. The Seq2 Seq 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:  
\[h_{t}=f_{R}(x_{t},h_{t-1}) \tag{9}\]  
where \(f_{R}(*)\) 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:  
\[y_{t}=f_{a}(Wh_{t}+b) \tag{10}\]  
The basic mechanism behind Seq2 Seq 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 Seq2 Seq model to enhance both performance and stability. The GCN-Seq2 Seq 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-Seq2 Seq 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-Seq2 Seq 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:  
\[h_{v}^{(l+1)}=\sigma\left(\sum_{u\in N(v)\bigcup\{v\}}\frac{1}{c_{vu}}W^{(l)}h_ {u}^{(l)}\right) \tag{11}\]  
where \(\sigma\) is a nonlinear activation function (such as ReLU), \(N(v)\) represents the set of neighboring nodes of node \(v\), and \(c_{vu}\) 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, \(W^{(l)}\) is the weight matrix of the \(l\)-th layer, and \(h_{u}^{(l)}\) 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 Seq2 Seq 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-Seq2 Seq model fully integrates spatial and temporal dependencies and is suitable for complex graph data and sequence prediction tasks. 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).