## 4 Experimental Results and Analysis
### Analysis of Model Performance Metrics  
Figure 9 presents the calculation results of the performance indicators (Mean Squared Error, MSE), (Root-Mean-Squared Error, RMSE), (Mean Absolute Error, MAE), (Coefficient of Determination, R\({}^{2}\)) of the LightGBM, Prophet and SARIMA models for each month in 2023 in the form of a heat map. To present the error magnitudes or goodness-of-fit of the three models more clearly and intuitively, each set of data was normalized independently within itself and sorted by size. They were marked with colors from dark to light to better display the comparative relationships within each group of indicators (The values shown in the figure are still the original data of the indicators). The heat map of the LightGBM model in January, October and December has darker colors, indicating that the evaluation index values in these months are relatively high after normalization, that is, the model prediction value deviates greatly from the actual value. Taking May as an example, the color of this month is relatively light, and the values of various indicators are low, which means that the error index is relatively small and the prediction effect is good. This difference is closely related to the seasonal characteristics of the data in the study area. Although the LightGBM model has a strong nonlinear fitting ability and can handle complex data relationships, in the winter of January and December, the atmospheric diffusion conditions are poor, and the PM\({}_{2.5}\) concentration is affected by a variety of complex factors. Even this model is difficult to accurately capture the concentration change law. In July, the summer meteorological conditions are relatively stable, and the laws in the data are easier to be learned and predicted by the model, so it performs better. Unlike the LightGBM model, the Prophet model has darker colors in May-July, indicating that the MSE, RMSE and MAE values are large during this period, and the prediction effect is poor, while the color is relatively light in other months, and the prediction error is relatively stable and small. The Prophet model is based on the seasonality and trend of time series, and has a certain ability to handle conventional seasonal patterns. However, during the period from May to July, the changes in PM\({}_{2.5}\) concentration in the study area were more complex. There may be some special meteorological conditions or human activities, which led to deviations between the actual situation and the seasonal patterns in the historical data. The model is too dependent on the historical pattern, which increases the prediction error. Looking at the SARIMA model again, the darker colors in January, May, September and November mean that the MSE, RMSE and MAE values are higher and the prediction performance is poor. From the R\({}^{2}\) indicator, the closer the R\({}^{2}\) is to 1, the better the model fits the data. Among the three models, the negative R\({}^{2}\) value of the SARIMA model is smaller than that of the LightGBM and Prophet models, that is, it is closer to 1, indicating that it fits the data relatively well. However, combined with its higher error index in some months, it  
Figure 9: Monthly Heatmap of PM\({}_{2.5}\) Prediction Error Metrics in 2023.  
can be seen that although the model has relatively good fitting performance, it still lacks in prediction accuracy. This is because the SARIMA model has certain requirements for the stability of the data, and the PM\({}_{2.5}\) concentration data in 2023 is affected by a variety of complex factors, and it is difficult to meet its stability assumption in some months, so the overall performance still has room for improvement. Overall, the three models of LightGBM, Prophet and SARIMA have their own advantages and disadvantages in predicting the PM\({}_{2.5}\) concentration in 2023. The LightGBM model has strong fitting ability and can capture complex trends, but it is greatly affected by special environmental factors.