## 3 Experimental Results  
### Settings  
The dataset used in this paper is from the weather and air pollution index collected by the US embassy in Beijing, China from 2013 to 2017 (see [[http://archive.ics.uci.edu/ml/datasets/Beijing+Multi-Site+Air-Quality+Data](http://archive.ics.uci.edu/ml/datasets/Beijing+Multi-Site+Air-Quality+Data)]([http://archive.ics.uci.edu/ml/datasets/Beijing+Multi-Site+Air-Quality+Data](http://archive.ics.uci.edu/ml/datasets/Beijing+Multi-Site+Air-Quality+Data)), accessed on 20 September 2019) [27]. In the experiment of this paper, only one year of data is selected randomly from the dataset, which is from 1 April 2014 to 1 February 2015 (a total of 8760 data). It is big enough for the PM\({}_{2.5}\) concentration prediction. Each air quality record contains six pollutants: PM\({}_{2.5}\), PM\({}_{10}\), SO\({}_{2}\), NO\({}_{2}\), CO and O\({}_{3}\). Each meteorological record contains seven items--time (the interval is one hour), temperature, pressure, dew point temperature, precipitation, wind direction and wind speed. The average, median and standard deviation of PM\({}_{2.5}\) concentration at the central site are 90.07 \(\mu\)g/m\({}^{3}\), 68.72 \(\mu\)g/m\({}^{3}\) and 83.30 \(\mu\)g/m\({}^{3}\), respectively. The standard deviation is relatively large, indicating that the data are widely distributed.  
The first 75% of the dataset is selected as the training data and the remaining 25% as the test data. If the continuous missing values are greater than one row, IDW interpolation [25] is used to fill the missing values according to the neighboring sites PM\({}_{2.5}\) concentration. It is because PM\({}_{2.5}\) concentration data from various sites are highly correlated. If the continuous missing values are less than two rows, the linear interpolation method is used.  
To prove that the proposed method is more suitable for long-term prediction, the 1-24 h are divided into three time lags (1-6, 7-12 and 13-24 h) and a separate model is trained to predict the average PM\({}_{2.5}\) concentration of each time lag. Each model is set with appropriate hyper parameters to produce the best performance. The predicted time lag of 1-6 h is 8, 7-12 h is 16, and 13-24 h is 28.  
The back propagation algorithm is used to train all models. DIstortion Loss including shApe and TimE (DILATE) loss function [28] is used in the experiment. In the training process, the small-batch stochastic gradient descent (SGD) is combined with Adam optimizer, the size of the random small batch is set to 128, the upper limit of the training period is 200, the learning rate is set to 0.001 and the number of neighboring sites is set to 4. A layer of an LSTM network is used for each attention module, and the hidden state of the LSTM network is set to the same, that is, \(m=p=q=128\).  
In order to evaluate the effectiveness of the method, two indicators are used in the experiment, including root mean square error (RMSE) and mean absolute error (MAE). These indicators are defined as follows:  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-y_{i^{\prime}}\right)^{2}} \tag{31}\]  
\[MAE=\frac{1}{n}\sum_{i=1}^{n}\left|y_{i}-y^{\prime}_{i}\right|, \tag{32}\]  
where \(y_{i}\) is the true value and \(y_{i^{\prime}}\) is the predicted value.