# A Hybrid Deep Learning Model for Air Quality Prediction Based on the Time–Frequency Domain Relationship

^{1}

^{2}

^{3}

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## Abstract

**:**

_{2.5}and PM

_{10}) had an obvious correlation in the low-frequency band and a low correlation in the high-frequency band. PM

_{2.5}and temperature had a negative correlation in the high-frequency band and an obvious positive correlation in the low-frequency band. PM

_{2.5}and wind speed had a low correlation in the high-frequency band and an obvious negative correlation in the low-frequency band. These results showed that the laws of variables in the time–frequency domain could be found by the model, which made it possible to explain the model. The experimental results show that the prediction performance of the established model was better than that of multilayer perceptron (MLP), one-dimensional convolutional neural network (1D-CNN), gate recurrent unit (GRU), long short-term memory (LSTM) and Transformer, in all time steps (1, 4, 8, 24 and 48 h).

## 1. Introduction

_{2.5}, CO, SO

_{2}, NO

_{2}, etc., which can cause many diseases, such as asthma, heart disease, chronic obstructive pulmonary disease and cancer [4,5]. According to the World Health Organization (WHO), simple breathing behavior causes 7 million deaths each year due to air pollution, which seriously endangers human health. To reduce the harm caused by air pollution, researchers have introduced various models to predict changes in air pollution to take necessary measures at the corresponding time [6,7]. Among these models, the deep learning model has the best prediction effect [8]. However, the deep learning models have the “black box” problem, and the prediction behavior of the models is difficult to explain. In addition, the time-series data of the atmospheric environment integrates signals of different frequencies and is mixed with incorrect noise signals, the correlation between atmospheric and pollutant variables is masked by these entangled signals, so it is difficult to be reliably found. Therefore, it is very important to improve the interpretability and accuracy of the prediction by separating the entangled different frequency signals from the original data to obtain clearer signals and designing an interpretable network to extract these correlation rules.

_{3}using a CMAQ modeling system to help air pollution control in China [18]. However, detailed and accurate external environmental parameters were required as inputs to the mechanism model. Owing to the complexity of real environment, these parameters were difficult to obtain reliably, which makes prediction of the mechanism models have great limitations. Statistical models were used to predict future changes in variables by discovering the evolution of data from historical data, including linear regression models [19,20], perceptron [21,22], support vector machine (SVM) [23,24], tree models [25,26,27], deep neural networks (DNN) [28,29,30], etc. Linear regression models include univariate linear regression and multivariate linear regression; between them, multivariate linear regression had better nonlinear fitting ability, but it may be insufficient compared with perceptron, tree models, DNN, etc.

## 2. Problem Scenario

_{2.5}was used as the target parameter, other pollutants and meteorological parameters as the impact parameters, including PM

_{10}, CO, wind speed, temperature, humidity, etc., and there is a correlation between them [42]. Meteorological and pollutant data are typical time-series data. Because of the complexity of the real environment, the time-series data were affected by factors of different frequencies, and the time-series data were mixed with signals of different frequencies.

_{2.5}concentrations.

#### 2.1. Wavelet Transform Used for Time-Series Decomposition

#### 2.2. Encoder

_{2.5}, which are greater after enhancement by the correlation matrix. Self-attention can be described in mathematical language as follows:

#### 2.3. Decoder

_{2.5}concentration.

#### 2.3.1. LSTM

#### 2.3.2. Attention

#### 2.4. Data Sources

_{2.5}, PM

_{10}, NO

_{2}, CO, SO

_{2}and O

_{3}. PM

_{2.5}in the sample area was used as the prediction target for the model.

## 3. Methods

#### 3.1. Framework

_{2.5}concentration. Fourth, root mean square error (RMSE), mean absolute error (MAE) and symmetric mean absolute percentage error (SMAPE) were used as evaluation parameters. The WTformer was compared with the established baseline model and ablation model to verify the prediction performance. Finally, the correlation matrix learned by the model was analyzed to obtain the deeper time–frequency law of the meteorological and environmental variables.

#### 3.2. Data Processing

#### 3.3. Construction of Frequency Separator

_{2.5}data.

_{1}and high-frequency component CD

_{1}by SWT. Because the decomposition process of SWT did not extract coefficients at each transform level, CA

_{1}and CD

_{1}had the same dimension as the original data. Then, CA

_{1}was decomposed in the same way to obtain CA

_{2}and CD

_{2}. The low-frequency component CA

_{n}and the high-frequency component CD

_{n}were obtained by recursive calculation, where n represents the decomposition scale. The larger the number of decomposition layers, the more detailed information was lost in the low-frequency signal. The number of decomposition layers used in this study was four.

#### 3.4. Construction of Encoder

_{2.5}, the greater its value would be after feature enhancement, thereby enhancing the feature information for the main influencing factors and reducing the impact of interference signals.

#### 3.5. Construction of the Decoder

_{2.5}concentrations. Its structure is shown in Figure 8.

_{2.5}and meteorological and pollutant parameters in different frequency bands. Third, the SoftMax layer was input to obtain the correlation matrix of each frequency domain for the prediction results. Fourth, the correlation matrix was multiplied by the data after time decoding to enhance the feature information for the main frequency bands. Finally, the PM

_{2.5}concentration in the future was predicted by fusing the feature information with the linear layer.

#### 3.6. Evaluation Criterion

## 4. Experimental Results and Analysis

#### 4.1. Network Parameters

#### 4.2. Prediction Performance

_{2.5}concentration of the two periods was used as the test data to evaluate the performance of WTformer in different situations. During the period from 7 to 23 May 2021, the change in PM

_{2.5}concentration showed a normal trend. From 1 July to 16 July 2021, PM

_{2.5}concentration changes frequently. Figure 9 and Figure 10 show the prediction structure of WTformer in two time periods. In different situations, the model showed good prediction performance.

#### 4.3. Ablation Experiment

_{2.5}in the next 48 h was predicted by WTformer and the three ablation models. The predicted and observed values are shown in Figure 11. By analyzing the performance of each model at markers 1, 2, 3, 4, it can be found that the WTformer model performs best, whereas the SA-LA and LA models, which lacked the wavelet decomposition module, were less sensitive to the mutation. The WT-LA and LA models, which lacked the feature enhancement encoder, had a prediction lag problem at markers 1 and 3.

#### 4.4. Correlation Analysis between PM_{2.5} and Other Variables

_{2.5}at a deeper level to show the interpretability of the model. To analyze the factors affecting the variation in PM

_{2.5}for different frequency bands, Figure 12 shows the attention matrix learned by self-attention in the encoder. Among the pollutant factors, the correlation between PM

_{2.5}and PM

_{10}/NO

_{2}/CO/SO

_{2}was reflected mainly in the low-frequency band and slower frequency band in the high-frequency band, and the correlation was lower in the high-frequency band. The correlation between PM

_{2.5}and O

_{3}was reflected mainly in the slower high-frequency band, and the correlation was lower in the low-frequency band and the faster high-frequency band. Among meteorological factors, the correlation between PM

_{2.5}and temperature/wind speed/pressure/precipitation were reflected mainly in the low-frequency band, and the correlation was weak in the high-frequency band. The correlation between PM

_{2.5}and humidity was reflected mainly in the slower high-frequency band, and the correlation was weak in the low-frequency band.

_{10}/NO

_{2}/CO/SO

_{2}/temperature/wind speed/pressure/precipitation on PM

_{2.5}was reflected mainly in a wider time scale, and the influence was long term, whereas the influence of O

_{3}and humidity on PM

_{2.5}was reflected mainly in the high-frequency band, and the influence was short term. This shows that the time–frequency law between variables was found, and the prediction behavior of the model could be explained by analyzing the attention matrix.

#### 4.5. Comparison of WTformer with Other Methods

_{2.5}concentration predicted by WTformer with five baseline models and three ablation models at a time step of 4h. The prediction curves for each model were basically consistent with the observation curves, and there was a linear correlation. Compared with the time-series prediction models LSTM and GRU, there was a greater difference between the predicted and observed values of the non-time-series prediction models MLP and CNN1D, which indicates that capturing time dependence in time-series prediction helps to improve the prediction ability of the models. Transformer captures time dependence by embedding position coding, which had similar performance to LSTM and GRU when the predicted time step is 4. The ablation models could not predict some mutation values and extreme values, and the prediction performance was not as good as WTformer. Compared with the previous models, the WTformer model had the best prediction effect in each stage. The main reasons are that the WTformer model obtains richer time–frequency domain information, exhibits more sensitivity to local changes and improves the prediction accuracy by reducing the influence of noise signals.

## 5. Discussion

_{2.5}can also be found. For example, the meteorological and environmental variables from 1 to 30 May 2021 were selected as the input to the model, and PM

_{2.5}was used as the prediction target. The analysis of the attention matrix shows that the correlation between PM

_{2.5}and PM

_{10}/NO

_{2}/CO/SO

_{2}were reflected in the low-frequency region and in the high-frequency region, which were reflected in the slower frequency band. The correlation between PM

_{2.5}and O

_{3}/humidity were reflected in the slower high-frequency band. The correlation between PM

_{2.5}and temperature/wind speed/pressure/precipitation were reflected in the low-frequency band. It shows that the developed WTformer model as described in the paper has strong explanatory power and effectively provides a data basis for pollution control.

_{2.5}as a complex indicator, which would be affected by many other factors, such as geographical environment. In future research, we should independently model each point, introduce spatial geographical factors, simulate the correlation of pollutant transmission between cities, discover and explain their laws, and limit the prediction error to a smaller range. This should be our next work direction.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Categories | Variables | Unit |
---|---|---|

Pollutant | PM_{2.5} | ug/m^{3} |

PM_{10} | ug/m^{3} | |

CO | ug/m^{3} | |

NO_{2} | ug/m^{3} | |

SO_{2} | ug/m^{3} | |

O_{3} | ug/m^{3} | |

Climate variables | Wind speed | m/s |

Temperature | °C | |

Humidity | % | |

Rain | mm | |

Pressure | hpa |

MLP | CNN1D | GRU | Transformer | LSTM | LA | WT-LA | SA-LA | WTformer | ||
---|---|---|---|---|---|---|---|---|---|---|

+1 h | RMSE | 7.475 | 7.349 | 6.799 | 8.083 | 6.840 | 6.614 | 6.475 | 6.404 | 6.334 |

MAE | 4.406 | 3.815 | 3.567 | 4.117 | 3.270 | 3.146 | 3.061 | 3.034 | 3.002 | |

SMAPE | 0.119 | 0.106 | 0.086 | 0.117 | 0.084 | 0.081 | 0.080 | 0.077 | 0.076 | |

+4 h | RMSE | 15.554 | 16.364 | 13.099 | 12.607 | 12.172 | 10.703 | 10.287 | 9.681 | 8.162 |

MAE | 10.151 | 10.233 | 8.725 | 8.830 | 8.091 | 6.655 | 6.582 | 6.509 | 5.679 | |

SMAPE | 0.261 | 0.266 | 0.228 | 0.233 | 0.222 | 0.184 | 0.183 | 0.176 | 0.171 | |

+8 h | RMSE | 19.372 | 20.008 | 19.044 | 16.806 | 18.459 | 16.465 | 15.741 | 15.410 | 13.096 |

MAE | 12.695 | 13.647 | 12.665 | 11.492 | 12.451 | 11.069 | 10.814 | 10.468 | 8.604 | |

SMAPE | 0.306 | 0.348 | 0.304 | 0.291 | 0.303 | 0.270 | 0.263 | 0.258 | 0.215 | |

+24 h | RMSE | 27.650 | 29.452 | 27.077 | 24.478 | 26.321 | 22.820 | 21.086 | 20.938 | 17.140 |

MAE | 19.209 | 20.949 | 19.103 | 17.604 | 18.868 | 16.208 | 15.008 | 14.723 | 12.213 | |

SMAPE | 0.445 | 0.491 | 0.439 | 0.401 | 0.432 | 0.361 | 0.336 | 0.332 | 0.271 | |

+48 h | RMSE | 32.492 | 33.115 | 36.878 | 30.027 | 33.649 | 28.905 | 26.794 | 26.419 | 21.379 |

MAE | 23.569 | 23.581 | 25.987 | 21.295 | 24.135 | 20.442 | 18.991 | 18.630 | 14.943 | |

SMAPE | 0.538 | 0.524 | 0.567 | 0.487 | 0.539 | 0.455 | 0.417 | 0.409 | 0.331 |

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**MDPI and ACS Style**

Xu, R.; Wang, D.; Li, J.; Wan, H.; Shen, S.; Guo, X.
A Hybrid Deep Learning Model for Air Quality Prediction Based on the Time–Frequency Domain Relationship. *Atmosphere* **2023**, *14*, 405.
https://doi.org/10.3390/atmos14020405

**AMA Style**

Xu R, Wang D, Li J, Wan H, Shen S, Guo X.
A Hybrid Deep Learning Model for Air Quality Prediction Based on the Time–Frequency Domain Relationship. *Atmosphere*. 2023; 14(2):405.
https://doi.org/10.3390/atmos14020405

**Chicago/Turabian Style**

Xu, Rui, Deke Wang, Jian Li, Hang Wan, Shiming Shen, and Xin Guo.
2023. "A Hybrid Deep Learning Model for Air Quality Prediction Based on the Time–Frequency Domain Relationship" *Atmosphere* 14, no. 2: 405.
https://doi.org/10.3390/atmos14020405