# Prediction of PM2.5 Concentration on the Basis of Multitemporal Spatial Scale Fusion
## 2 Materials and Methods
### ConvLSTM  
The ConvLSTM network enhances the traditional LSTM model by incorporating convolution operations, allowing it to capture both temporal and spatial correlations among air quality monitoring stations, thereby extracting intricate spatiotemporal correlation features related to air pollution. The structure is shown in Figure 4.  
The classic LSTM model is limited to capturing the temporal relationship of just one air quality monitoring station. The ConvLSTM network enhances traditional LSTM models by incorporating convolutional operations, similar to LSTM models, to capture temporal correlations and CNN to capture spatial correlations among air quality monitoring stations, ultimately extracting intricate spatiotemporal correlation features of air pollution (Figure 3) [35]. Furthermore, ConvLSTM networks are able to address the issues of gradient explosion and vanishing during error backpropagation, allowing for improved capture of long-term, short-term, and neighboring information compared to conventional LSTM networks. Similar to LSTM networks, ConvLSTM networks also provide information transmission through forget gates f\({}_{t}\), input gates it, and output gates ot. X\({}_{t}\), C\({}_{t}\), and h\({}_{t}\) represent the input, internal, and external conditions at a specific time. The forget gate f\({}_{t}\), input gate it, and output gate ot are determined using convolution and batch normalization, along with internal C\({}_{t-1}\) or C\({}_{t}\) based on states X\({}_{t}\) and h\({}_{t-1}\). ConvLSTM networks outperform traditional LSTM models in capturing long-term information dependencies. The information transmission formula in the ConvLSTM network is as follows:  
\[N_{t}=BN(W_{x}x_{t})+BN(W_{h}h_{t-1})+b_{N} \tag{1}\]  
\[i_{t}=\sigma(N_{t}+W_{ci}\circ c_{t-1}+b_{i}) \tag{2}\]  
\[f_{t}=\sigma\Big{(}N_{t}+W_{cf}\circ c_{t-1}+b_{f}\Big{)} \tag{3}\]  
\[C_{t}=f_{t}\circ c_{t-1}+i_{t}\circ\tanh(N_{t}+b_{c}) \tag{4}\]  
\[o_{t}=\sigma(N_{t}+W_{co}\circ C_{t}+b_{o}) \tag{5}\]  
N\({}_{t}\) is the middle variable of the input feature at the present time t computed by convolution; W\({}_{x}\) and W\({}_{h}\) are the weight matrices of N\({}_{t}\); BN is for batch normalization; Wci and Wcm are the weight matrices for variables connecting C\({}_{t-1}\) and input gate it and C\({}_{t-1}\)  
Figure 4: ConvLSTM structure diagram.  
and forget gate \(\rm{f_{f}}\); \(\rm{W_{co}}\) is the weight matrix for variables connecting internal \(\rm{C_{t}}\) and output gate it; \(\rm{B_{N}}\), \(\rm{b_{i}}\), \(\rm{b_{f}}\), \(\rm{b_{c}}\), and \(\rm{b_{o}}\) are the respective biases; and \(\circ\) represents matrix operations.  
The model is developed using multi-grain features to forecast PM2.5. This model consists of various branches, and the final prediction is generated by combining the results from these branches. The process of fusion is described as follows:  
\[P_{fusion}=\sum\limits_{i=1}^{m}a_{i}P_{i} \tag{6}\]  
M signifies the number of branches. The forecasted outcome for the i-th division can be denoted as \(\rm{P_{i}}\). The final prediction is expressed in Pfusion.