## 3 Method
### Baseline Models  
To validate the proposed Informer model, we compared the Informer model with four other models, including one linear regression method (least absolute shrinkage and selection operator regression (LASSO)), two machine learning methods (random forest (RF) and extreme gradient boosting (XGBoost)), and one deep learning method (attention-based long short-term memory (AtLSTM)). The following content explains the detailed information about the other four models. LASSO is a regularized regression method and uses L1 regularization [44]. The main parameter of LASSO is alpha, which was tuned using the grid search with cross-validation. RF is an ensemble machine learning algorithm composed of numerous independent decision trees. RF has an accurate prediction ability, thanks to RF avoiding overfitting and injecting correct randomness [45]. XGBoost has parallel and distributed computing, which makes XGBoost have a quicker running speed than other popular machine learning methods [46]. The parameters of RF and XGBoost were tuned using a randomized search with cross-validation. LASSO and RF were realized using the scikit-learn library. XGBoost was trained using the XGBoost library. The LSTM model was proposed by [PERSON] and [PERSON] [21] and has solved the error flow and artificial long-time-lag tasks in existing recurrent neural networks (RNN) [21]. LSTM has been widely applied to real-world sequences processing problems such as natural language processing [47] and crop yield prediction [48; 49]. The AtLSTM model is an attention-based LSTM model that is practical and reliable in many fields, such as the industrial field [50]. In this study, we used a six-layer AtLSTM model to predict rice yield, which consists of one input layer, two LSTM layers, one attention module layer, one fully connected layer, and one output layer (Figure 3). The input is a time series with climate and satellite data at 12 intervals of the rice growth period. The LSTM cell consists of three gates: a forget gate \(g_{t}^{f}\), an input gate \(g_{t}^{i}\), and an output gate \(g_{t}^{o}\), which can control the cell state (\(S_{i}\)) and output (\(O_{i}\)). The calculation formula used for generating hidden features \(O_{i}\) is shown in Equations (2)-(6):  
\[g_{t}^{f}=\sigma(W_{f}\cdot[O_{t-1},I_{t}]+b_{f}) \tag{2}\]  
\[g_{t}^{i}=\sigma(W_{i}\cdot[O_{t-1},I_{t}]+b_{i}) \tag{3}\]  
\[S_{t}=g_{t}^{f}\cdot S_{t-1}+g_{t}^{i}\cdot tanh(W_{c}\cdot[O_{t-1},I_{t}]+b_ {c}) \tag{4}\]  
\[g_{t}^{o}=\sigma(W_{o}\cdot[O_{t-1},I_{t}]+b_{o}) \tag{5}\]  
\[O_{t}=g_{t}^{o}\cdot tanh(S_{t}) \tag{6}\]  
Figure 2: The architecture of the Informer model. where \(g_{t}^{f}\), \(g_{t}^{i}\), and \(g_{t}^{o}\) are the forget gate, input gate, and output gate, respectively. \(W_{f}\), \(W_{i}\), \(W_{c}\), and \(W_{o}\) are the weight matrices of the forget gate, input gate, input candidate element, and output gate, respectively. \(b_{f}\), \(b_{i}\), \(b_{c}\), and \(b_{o}\) are biases of the forget gate, input gate, input candidate element, and output gate, respectively. \(S_{t}\), \(S_{t-1}\), \(O_{t}\), and \(O_{t-1}\) are the current cell state, previous cell state, current cell output, and previous cell output. \(I_{t}\) is the current input. \(\sigma\) is the sigmoid function. To better utilize multi-time-step information for crop yield prediction, the attention module adjusts the contribution of hidden features by normalizing the weight parameters. The attention weight \(\alpha_{t}\) of each hidden feature \(O_{t}^{*}\) obtained by the LSTM layers is generated using a _softmax_ function (Equations (7) and (8)):  
\[\alpha_{t}=softmax(W_{a}\cdot O_{t}+b_{a}) \tag{7}\]  
\[O_{t}^{*}=\alpha_{t}O_{t} \tag{8}\]  
where \(W_{a}\) and \(b_{a}\) are learnable weight matrices in the attention module and shared by all time steps. \(\alpha_{t}\) is the attention weight. \(O_{t}^{*}\) is the adjusted hidden feature. Then, the output of the attention module is aggregated into the fully connected layer.