## 4 Methodology
### Hybrid Predictor  
In this section, we detail our proposed structure for prediction. The main purpose of the prediction layer in Figure 1c is to obtain the predicted AQI values from the final embedding vector. First, the final embedding vector is obtained by concatenating the output of the inversed distance attention (\(e_{S}^{3}\)) and the embedding vector of unlabeled grids (\(e_{U}^{2}\)). Although our predicted target can be taken as a regression problem, we can use MLP to process embedding vectors and predict the results. Compared to the common regression problem, directly using traditional MLP does not yield good accuracy, due to the lack of historical AQI data from unlabeled grids. Therefore, we propose a hybrid predictor to solve the problem and achieve a better long-term prediction by splitting the predicted values into two data flows. We assume that predicted values can be separated into two independent fractions: time-independent and time-dependent. We design two divided predictor structures that have two independent data flows to predict the AQI, respectively. In the end of the predictor, we sum the predicted values up with dynamic weights to combine these two data flows on each prediction timestamp. We apply the first dense layer behind the inversed distance attention layer to learn the cross features of unlabeled grid embedding and attention output. In Equation (8), we can have a cross embedding vector (\(e_{Z}\)) for the prediction of two data flows after the dense layer. \[e_{Z}=ReLU(W_{Z}(e_{U}^{2}\oplus e_{S}^{3})+b_{Z}) \tag{8}\]We then directly use MLP, a common method for a regression problem, to predict one data flow. As Equation (9), we can have time-independent predicted values for future time t by using an additional dense layer. \[X_{m}=W_{x}e_{Z}+b_{x},\text{for }m=0,1,\ldots,t \tag{9}\]  
Another predicted data flow, time-dependent flow, means the prediction for the next timestamp is based on the current prediction and so on for further timestamp prediction. To achieve this goal, we propose an LSTM predictor that is based on the structure of LSTM AutoEncoder [25]. AutoEncoder is a common unsupervised learning method that usually trains a hidden embedding vector for the image, or compresses the dimension of vectors. AutoEncoder is a structure that minimizes the error between input and output vectors. We use the concept of the decoder part for time-dependent data flow in our model. LSTM Enc. refers to the structure of LSTM Encoder in Figure 2. Because the input of LSTM should be a sequence, we repeat the cross embedding vector (\(e_{Z}\)) several times as a sequence input. Through the processing of the encoder, we can have output (\(e_{Enc. }\)) and hidden state (\(h_{t}^{AE},c_{t}^{AE}\)) of the encoder for the next step in Equations (10) and (11). \[e_{LSTM},(h_{t}^{AE},c_{t}^{AE})=LSTMEnc.(e_{Z}) \tag{10}\]  
\[e_{Enc.}=ReLU(e_{LSTM}) \tag{11}\]  
LSTM Dec. in Figure 2 is composed of one LSTM and one dense layer. We have a total of t LSTM decoders which have the same hidden sizes as the t number of the encoder's LSTM. In Equation (12), we take the output of the encoder that usually represents the last layer of \(h_{t}^{AE}\) as the initial input of the decoder. The hidden state of the encoder is also the initial hidden state for all decoder units. In other words, we set the same hidden state for both the encoder and all decoders in back-propagation. Meanwhile, in Equation (13), the output of the first decoder would be the input of the second decoder, and so on until t time periods. With our proposed LSTM predictor, the data flow of time-dependent prediction can be dealt with. \[Y_{0}=LSTMDec.(e_{Enc},(h_{t}^{AE},c_{t}^{AE})) \tag{12}\]  
\[Y_{m}=LSTMDec.(Y_{m-1},(h_{t}^{AE},c_{t}^{AE})),\text{for }m=1,2,\ldots,t \tag{13}\]  
From two separate predicting data flows, we acquire two complementary predicted values that are time-independent, as well as time-dependent values for the future \(t\) time periods. Then we aggregate the results of the two data flows.