# A Hybrid Autoformer Network for Air Pollution Forecasting Based on External Factor Optimization

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## Abstract

**:**

## 1. Introduction

- (1)
- A Hybrid Autoformer Network with a Genetic Algorithm Model was proposed to predict the air pollution variation, where the genetic algorithm was used to optimize the external variable weighting problem for different variables that have different effects on the target pollution.
- (2)
- The Elite Variable Operator was proposed to vote at fixed intervals of generations to find out the variables that have a greater impact on the target prediction to be selected as elite variables, which are explored to perform a more refined search.
- (3)
- The proposed Archive Storage Operator, using genetic algorithms, led to deviations in the final results due to the influence of the initialization of individual models, where the individuals with better value may be less effective due to initialization and vice versa. The archive mechanism was used to store the individuals with good results and to filter them to get the really good ones.
- (4)
- We conducted comprehensive experiments on the Ma’anshan air pollution dataset to verify the proposed model, where the prediction accuracy was greatly improved, and the selection of model influencing factors was more interpretable.

## 2. Study Area and Data Requirement

#### 2.1. Study Area

#### 2.2. Data Requirement

- (1)
- Air pollution gas detection content, including carbon monoxide (CO), nitrogen dioxide (NO${}_{2}$), and sulfur dioxide (SO${}_{2}$).
- (2)
- Air pollution index, including particulate matter with a particle size below 10 microns (PM
_{10}) and total suspended particulates (TSPs). - (3)
- Environmental factors of the target AQI station, including wind direction, wind speed, precipitation, vapor pressure, humidity, visibility, atmospheric pressure, and temperature.

## 3. Preliminary

#### 3.1. Transformer for Time Series Forecasting

#### 3.2. Genetic Algorithm

- (1)
- Selection

- (2)
- Crossover

- (3)
- Mutation

#### 3.3. Problem Formulation

## 4. Methodology

Algorithm 1 Frame process of the GA-autoformer model |

Require: D: Multivariate time series dataset on air pollution used in this iteration, |

t: Number of iterations, |

p: Population number, |

k: Interval iteration of executive operator, |

M: Neural network model |

Ensure: ${W}_{b}$: Best weight |

Randomly generate W = $[{W}_{1},{W}_{2},{W}_{3},\dots ,{W}_{p-1},{W}_{p}]$ |

$archive$_$weight$ = ∅ |

$archive$_$fitness$ = ∅ |

$elite$ = ∅ |

for $i=1$ to t do |

$Fitnes{s}_{W}\leftarrow M(D,W)$ |

$Fitnes{s}_{a}$$\leftarrow M(D,archive)\cup Fitnes{s}_{a}$ |

W = tournament_selection(W,$Fitnes{s}_{W}$) |

if i%k==0&&i!=0 then |

$elite$← elite_voting(${W}^{\prime}$,$Fitnes{s}_{W}$) |

$archive$← archive_storage($archive$_$weight$, $archive$_$fitness$,${W}^{\prime}$,$Fitnes{s}_{a}$,$Fitnes{s}_{W}$) |

end if |

${W}^{\prime}$← crossover($Fitnes{s}_{W}$,${W}^{\prime}$,$elite$) |

${W}^{\prime}$← mutation($Fitnes{s}_{W}$,${W}^{\prime}$,$elite$) |

W← generate_population($Fitnes{s}_{W}$,${W}^{\prime}$) |

end for |

${W}_{b}$← select_best($archive$) |

return ${W}_{b}$ |

#### 4.1. Generate Random Individuals

#### 4.2. Calculation of Fitness Values

#### 4.3. Selection

#### 4.4. Elite Variable Voting Operator/Archive Storage Operator

#### 4.5. Crossover

#### 4.6. Mutation

#### 4.7. Iteration

#### 4.8. Elite Voting Operator

- (1)
- Candidates are selected from the top 30% of the population based on population fitness, and these individuals are the best candidates in the population; they represent the evolutionary direction of the population.
- (2)
- Among the candidates, we want to have some particles that can lead the candidates to the optimization direction more effectively; we call these particles “chairman”. We appoint two particles with highest fitness value among the candidates as “chairman”. In order to maintain diversity, the candidates that differ most from the “chairman” are added in “chairman”. We use the Euclidean distance as Equation (5) to measure the distance between two particles.
- (3)
- The elite variables chosen by vote are able to give special treatment in the process of mutation and crossover: there is a higher probability of becoming larger in the process of mutation and crossover.

Algorithm 2 Elite Voting Operator |

Require: W: Population, |

$Fitnes{s}_{W}$: Population fitness value |

Ensure: $elite$: Elite Variables |

$chairman$←∅ |

$voters$←∅ |

$chairman$← Take the top 2 fitness values in $voters$ as $chairman$ |

$chairman$←$chairman$+Take the particle in W that is most different from the particle in $chairman$$voters$←W-$chairman$ |

$elite$← voting($chairman$,$voters$) |

return $elite$ |

#### 4.9. Archive Storage Operator

Algorithm 3 Archive Storage Operator |

Require: $archive$_$weight$, |

$archive$_$fitness$, |

W: population, |

$Fitnes{s}_{W}$: Population fitness value, |

$Fitnes{s}_{a}$: archive fitness value |

Ensure: W: population |

d← size($archive$) |

$\overline{Fitnes{s}_{W}}$ = $Fitnes{s}_{W}$/size($Fitnes{s}_{W}$) |

for $i=1$ to d do |

if $Fitnes{s}_{a}^{i}$ >$\overline{Fitnes{s}_{W}}$ then |

W← rejoing_population(${W}_{i}$, W) |

else |

Discard(${W}_{i}$, $archive$_$weight$,$archive$_$fitness$) |

end if |

end for |

return W |

- (1)
- Every k generations, the best particle in all k generations is copied into the archive.
- (2)
- In the next k iterations, individuals in archive will not be involved in the process of the genetic algorithm, but only in the calculation of the fitness.
- (3)
- Every k generations, we perform an examination. If the fitness value of the individual is greater than the average fitness value of the current population, this individual will replace the individual with the lowest fitness in the population; otherwise, it will be discarded.

#### 4.10. Prediction and Optimization

## 5. Experiments and Results

#### 5.1. Dataset Descriptions

#### 5.2. Parameter Setting

#### 5.3. Evaluation Metrics

#### 5.4. Baselines

- (1)
- RNN: RNN is a classical time series prediction model that is capable of extracting time series features. Unlike feed-forward neural networks, which use the output of the previous neuron as input to the next neuron, RNN involves a structure that gives the network the ability to remember information about trends and cycles [15].
- (2)
- LSTM: LSTM belongs to the class of RNNs and also belongs to the recurrent network model. LSTM solves the problem that RNNs cannot extract long-term time dependence and uses multiple gate mechanisms to alleviate the gradient explosion and gradient disappearance problems that exist in RNNs [16].
- (3)
- EA-LSTM: EA-LSTM is based on the attention LSTM and uses the genetic-algorithm-based competitive random search (CRS) instead of gradient-based approach to explore the attention layer weights; thus, it better assigns the weights of features within the time window [43].
- (4)
- Transformer: Recently, the Transformer model has made a big breakthrough in time series prediction. Unlike the RNN and LSTM, Transformer is not a cyclic sequence model. Its prediction efficiency and its ability to predict long-term time series are greatly improved [32].
- (5)
- Informer: The authors designed an efficient transformer-based long-time prediction model, named Informer, by proposing a ProbSparse self-attentive mechanism, which utilizes self-attentive distillation to highlight dominant attention by halving the cascade layer input with a generative decoder for the one-time prediction of long-time sequence sequences. A new solution to the long-time sequence prediction problem is provided [34].
- (6)
- Autoformer: The authors used a deep decomposition architecture. The authors designed sequence decomposition units to embed deep models, implement progressively predictive, auto-correlation mechanisms, discard point-wise connected self-attention mechanisms, and implement series-wise connected autocorrelation mechanisms to break information utilization bottlenecks [35].

#### 5.5. Analysis of Prediction Result

#### 5.6. Model Interpretability

#### 5.7. Ablation Experiment

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The daily trend of O${}_{3}$ with PM${}_{2.5}$ and precursor NO${}_{x}$ on a certain day at a monitoring station:(

**a**) O${}_{3}$-PM${}_{2.5}$; (

**b**) O${}_{3}$-NO${}_{x}$.

**Figure 2.**The map on the right identifies the location of the urban area of Ma’anshan, and the red dots represent the location of the monitoring stations where the data are concentrated. Location A is located in Ma’anshan Hudong Road Fourth Primary School (Resident area), Location B is located in Poutang National Resort Park (Rurial area), and C is located in Maanshan Economic and Technological Development Zone (industrial area), which correspond to Location A, Location B, and Location C used in the experiment, respectively.

**Figure 5.**Architecture of GA-autoformer. The whole neural architecture consists of two parts. The right side is the neural network architecture, and the autoformer model is used as back-bone network in this paper. The left side is the genetic algorithm, and the lower left side is two operators that we proposed: Elite Voting Operator and Archive Storage Operator.

**Figure 6.**Description of crossover. We separated the elite and non-elite variables for shuffle crossover and then combined them.

**Figure 7.**Distribution of the AQI, PM${}_{2.5}$, and O${}_{3}$ data by seasons. (

**a**) AQI-PM${}_{2.5}$. (

**b**) AQI-O${}_{3}$.

**Figure 8.**Actual normalization value and average predicted value for the test experimental run for each monitor station with different target pollutants. The y-axis represents the predicted values and ground truth, and the x-axis is the predicted time-step.

**Figure 12.**We took the weights of the external variables for six random individuals in the final population; the color is more red, which indicates that the variable was more important.

**Figure 13.**The box plot represents the distribution of the final prediction results of different models. The experimental result is the average fitness value of the population for the last iteration using Location A, with ozone as the prediction target.

Dataset | Location A Location B Location C | |
---|---|---|

Time_interval | 1 h | |

Time span | 1 January 2020–6 October 2020 | |

Prediction target | O${}_{3}$, PM${}_{2.5}$, AQI | |

External factors | Relevant pollutants | CO, NO${}_{2}$, SO${}_{2}$ |

Air pollution index | PM${}_{10}$, TSP | |

Weather factors | wind direction, wind speed, precipitation, vapor pressure, humidity, visibility, atmospheric pressure, temperature. |

Variables | Measurement |
---|---|

O${}_{3}$ | [2, 300] |

PM${}_{2.5}$ | [30, 320] |

AQI | [30, 320] |

CO | [0, 5] |

NO${}_{2}$ | [0, 150] |

SO${}_{2}$ | [5, 210] |

PM${}_{10}$ | [0, 330] |

TSP | [0, 330] |

WindDirection | [0, 360] |

WindSpeed | [0, 15] |

Precipitation | [0, 82.1] |

vapor pressure | [0, 54.9] |

humidity | [0, 99] |

visibility | [0, 31,565] |

atmospheric pressure | [0, 1019.8] |

temperature | [−0.8, 37.1] |

Model | Location A | ||||||||
---|---|---|---|---|---|---|---|---|---|

Target | O${}_{\mathbf{3}}$ | PM${}_{\mathbf{2.5}}$ | AQI | ||||||

MAE | RMSE | MAPE | MAE | RMSE | MAPE | MAE | RMSE | MAPE | |

RNN | 0.634 | 1.341 | 1.469 | 0.033 | 0.052 | 0.423 | 0.253 | 0.481 | 1.151 |

LSTM | 0.612 | 1.313 | 1.419 | 0.034 | 0.047 | 0.346 | 0.251 | 0.477 | 1.072 |

EA-LSTM | 0.573 | 1.226 | 1.297 | 0.029 | 0.041 | 0.335 | 0.249 | 0.473 | 1.081 |

transformer | 0.514 | 1.175 | 1.069 | 0.030 | 0.037 | 0.311 | 0.241 | 0.468 | 1.011 |

informer | 0.481 | 1.056 | 0.816 | 0.029 | 0.284 | 0.285 | 0.239 | 0.461 | 1.036 |

autoformer | 0.479 | 0.990 | 0.778 | 0.027 | 0.294 | 0.275 | 0.235 | 0.462 | 1.021 |

GA-autoformer | 0.460 | 0.981 | 0.761 | 0.028 | 0.027 | 0.242 | 0.231 | 0.453 | 0.861 |

Location B | |||||||||

Target | O${}_{\mathbf{3}}$ | PM${}_{\mathbf{2.5}}$ | AQI | ||||||

MAE | RMSE | MAPE | MAE | RMSE | MAPE | MAE | RMSE | MAPE | |

RNN | 0.439 | 0.542 | 1.235 | 0.227 | 0.347 | 2.090 | 0.045 | 0.023 | 0.301 |

LSTM | 0.422 | 0.513 | 1.238 | 0.156 | 0.246 | 1.654 | 0.041 | 0.212 | 0.284 |

EA-LSTM | 0.342 | 0.459 | 1.239 | 0.107 | 0.106 | 0.715 | 0.034 | 0.213 | 0.265 |

transformer | 0.333 | 0.454 | 1.234 | 0.031 | 0.044 | 0.225 | 0.036 | 0.189 | 0.235 |

informer | 0.329 | 0.441 | 1.250 | 0.038 | 0.046 | 0.216 | 0.031 | 0.176 | 0.229 |

autoformer | 0.305 | 0.432 | 1.205 | 0.028 | 0.041 | 0.208 | 0.034 | 0.172 | 0.217 |

GA-autoformer | 0.315 | 0.406 | 1.215 | 0.026 | 0.039 | 0.215 | 0.029 | 0.170 | 0.207 |

Location C | |||||||||

Target | O${}_{\mathbf{3}}$ | PM${}_{\mathbf{2.5}}$ | AQI | ||||||

MAE | RMSE | MAPE | MAE | RMSE | MAPE | MAE | RMSE | MAPE | |

RNN | 0.081 | 0.246 | 0.510 | 0.294 | 0.332 | 1.513 | 0.169 | 0.290 | 0.360 |

LSTM | 0.061 | 0.240 | 0.412 | 0.262 | 0.316 | 1.479 | 0.091 | 0.159 | 0.311 |

EA-LSTM | 0.057 | 0.233 | 0.346 | 0.251 | 0.281 | 1.431 | 0.041 | 0.106 | 0.281 |

transformer | 0.056 | 0.231 | 0.348 | 0.241 | 0.279 | 1.419 | 0.033 | 0.063 | 0.245 |

informer | 0.054 | 0.229 | 0.339 | 0.236 | 0.269 | 1.410 | 0.036 | 0.062 | 0.240 |

autoformer | 0.052 | 0.228 | 0.336 | 0.231 | 0.264 | 1.369 | 0.034 | 0.061 | 0.239 |

GA-autoformer | 0.051 | 0.223 | 0.328 | 0.209 | 0.269 | 1.356 | 0.031 | 0.056 | 0.238 |

Model | Location A | ||||||||
---|---|---|---|---|---|---|---|---|---|

Target | O${}_{\mathbf{3}}$ | PM${}_{\mathbf{2.5}}$ | AQI | ||||||

MAE | RMSE | MAPE | MAE | RMSE | MAPE | MAE | RMSE | MAPE | |

autoformer | 0.479 | 0.995 | 0.778 | 0.027 | 0.029 | 0.275 | 0.235 | 0.462 | 1.021 |

autoformer-GA | 0.478 | 0.991 | 0.782 | 0.036 | 0.029 | 0.269 | 0.234 | 0.461 | 0.979 |

autoformer-GA(elite) | 0.476 | 0.986 | 0.771 | 0.029 | 0.029 | 0.252 | 0.232 | 0.457 | 0.957 |

autoformer-GA(archive) | 0.469 | 0.984 | 0.769 | 0.028 | 0.028 | 0.261 | 0.234 | 0.451 | 0.892 |

our-model | 0.460 | 0.981 | 0.761 | 0.028 | 0.027 | 0.242 | 0.231 | 0.453 | 0.861 |

Location B | |||||||||

Target | O${}_{\mathbf{3}}$ | PM${}_{\mathbf{2.5}}$ | AQI | ||||||

MAE | RMSE | MAPE | MAE | RMSE | MAPE | MAE | RMSE | MAPE | |

autoformer | 0.305 | 0.432 | 1.205 | 0.028 | 0.041 | 0.208 | 0.034 | 0.172 | 0.217 |

autoformer-GA | 0.316 | 0.453 | 1.233 | 0.028 | 0.037 | 0.218 | 0.034 | 0.173 | 0.208 |

autoformer-GA(elite) | 0.311 | 0.443 | 1.234 | 0.029 | 0.038 | 0.195 | 0.033 | 0.171 | 0.205 |

autoformer-GA(archive) | 0.309 | 0.441 | 1.234 | 0.027 | 0.037 | 0.204 | 0.032 | 0.173 | 0.202 |

our-model | 0.315 | 0.406 | 1.215 | 0.026 | 0.039 | 0.215 | 0.029 | 0.170 | 0.207 |

Location C | |||||||||

Target | O${}_{\mathbf{3}}$ | PM${}_{\mathbf{2.5}}$ | AQI | ||||||

MAE | RMSE | MAPE | MAE | RMSE | MAPE | MAE | RMSE | MAPE | |

autoformer | 0.052 | 0.228 | 0.336 | 0.231 | 0.264 | 1.369 | 0.034 | 0.061 | 0.240 |

autoformer-GA | 0.053 | 0.226 | 0.331 | 0.221 | 0.261 | 1.359 | 0.032 | 0.058 | 0.239 |

autoformer-GA(elite) | 0.052 | 0.229 | 0.337 | 0.218 | 0.262 | 1.362 | 0.033 | 0.059 | 0.237 |

autoformer-GA(archive) | 0.051 | 0.224 | 0.329 | 0.212 | 0.261 | 1.358 | 0.034 | 0.059 | 0.237 |

our-model | 0.051 | 0.223 | 0.328 | 0.209 | 0.267 | 1.356 | 0.031 | 0.056 | 0.238 |

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## Share and Cite

**MDPI and ACS Style**

Pan, K.; Lu, J.; Li, J.; Xu, Z.
A Hybrid Autoformer Network for Air Pollution Forecasting Based on External Factor Optimization. *Atmosphere* **2023**, *14*, 869.
https://doi.org/10.3390/atmos14050869

**AMA Style**

Pan K, Lu J, Li J, Xu Z.
A Hybrid Autoformer Network for Air Pollution Forecasting Based on External Factor Optimization. *Atmosphere*. 2023; 14(5):869.
https://doi.org/10.3390/atmos14050869

**Chicago/Turabian Style**

Pan, Kai, Jiang Lu, Jiaren Li, and Zhenyi Xu.
2023. "A Hybrid Autoformer Network for Air Pollution Forecasting Based on External Factor Optimization" *Atmosphere* 14, no. 5: 869.
https://doi.org/10.3390/atmos14050869