## 4 Experimental Results and Analysis
### Normality Test and Analysis of Prediction Errors  
Figure 12 shows the normality test results of the forecast errors of the LightGBM, Prophet, and SARIMA models in 2023, including error histograms and normal probability  
Figure 11: Box plots of PM\({}_{2.5}\) Data (Original & Predicted) in 2023.  
plots. In order to better illustrate the normal distribution, we also marked the relevant information of the Kolmogorov-Smirnov (K-S) test in the chart (when performing a normality test, 0.05 is usually used as the limit of the significance level). From the error histogram analysis, the error distribution of the LightGBM model shows an obvious multi-peak shape, and the intervals between the peaks are relatively scattered, indicating that its error values are relatively complex, with a certain frequency of occurrence in different intervals, and no obvious central trend. The error distribution of the Prophet model fluctuates more disorderly and lacks regularity. The error frequency of each interval fluctuates, and it is difficult to find a unified distribution pattern. Although the SARIMA model shows a certain unimodal trend, the peak is not located in the center of the distribution, and the left and right sides are asymmetric, indicating that its error distribution deviates from the symmetric characteristics of the normal distribution. Overall, the error histograms of the three models do not conform to the typical characteristics of the normal distribution. In the normal probability graph, the sample quantile curves of the three models deviate significantly from the theoretical quantile straight line. The curve of the LightGBM model deviates greatly from the straight line, and fluctuates significantly in different quantile intervals, indicating that its error distribution is quite different from the normal distribution. The curve of the Prophet model is separated from the straight line in most intervals, and only a few points are close to the straight line, showing the inconsistency between its error distribution and the normal distribution. Although the curve of the SARIMA model overlaps with the straight line to a certain extent in some intervals, there is still a significant deviation overall, especially at the quantiles at both ends. At the same time, the K-S test  
Figure 12: Visualizing 2023 PM\({}_{2.5}\) Prediction Errors: Histograms and Normal Probability Plots of 3 Models. results all rejected the null hypothesis that the errors follow the normal distribution, further confirming that the errors of the three models do not follow the normal distribution. Although the errors of the three models do not follow the normal distribution, this study is still of great value for PM\({}_{2.5}\) concentration prediction. From the perspective of model improvement, the multi-peak and dispersed error distribution of the LightGBM model suggests that in subsequent optimization, we can focus on analyzing the influencing factors corresponding to different peaks, such as whether they are related to specific time periods and data characteristics, and then adjust the feature selection and parameter settings of the model in a targeted manner to reduce the discrete degree of the error. For the irregular error distribution of the Prophet model, it is possible to consider strengthening the ability to handle outliers and sudden changes in time series, such as introducing a more flexible anomaly detection mechanism, or improving the modeling method for special factors such as holidays, so that the model can better adapt to the dynamic changes of data. For the asymmetric single-peak error distribution of the SARIMA model, we can try to adjust the model's stabilization method, or combine it with other model components that can capture asymmetric characteristics to improve the model's ability to fit the data distribution. In terms of new model development or model fusion strategy formulation, the data and experience accumulated in this study provide rich references.