## 3 Experimental Results and Analysis
### Experimental Results  
To evaluate the effectiveness of the MCRTF-CM model in calibrating low-cost air quality sensors, we compared its calibration errors with six other models: random forest (RF), support vector regression (SVR), multi-layer perceptron (MLP), multiple linear regression (MLR), extreme gradient boosting (XGBoost), and Long Short-Term Memory (LSTM). These models were implemented using open-source libraries, with necessary adjustments to suit the specific characteristics of the dataset. The implementation was optimized and adjusted according to the specific requirements of the experiments, and the hyperparameters for each model were fine-tuned to ensure optimal performance. To ensure fairness, all models were tested on the same dataset with identical preprocessing methods. Each model was trained 20 times with optimal hyperparameters to mitigate result variability, and the average calibration value was used. Uncalibration results, calibration results, and reference results are visualized in Figure 7, where points closer to the diagonal reference line indicate smaller errors and better calibration performance. Due to the large and complex test set data, we selected a subset from 24 May 2019 12:00 to 24 May 2019 20:00 to illustrate the calibration values and actual values for the seven models, as shown in Figure 8. Figure 9 illustrates the variation of calibrated values from different models compared to the reference values using error bars. The error bars represent the standard deviation of the residuals, where residuals are the differences between the calibrated values and the true values. The standard deviation reflects the distribution of the residuals, indicating the degree of dispersion of the calibrated values relative to the true values. Additionally, the coverage ratio for each model was calculated, representing the proportion of reference values falling within the error range of the calibrated values. Specifically, the coverage ratio is determined by checking whether the reference values lie within the range of the calibrated value \(\pm\) error bar (i.e., standard deviation). To further verify the reasonableness of the calibration values of these seven models, both error bars and coverage ratios are utilized. If the error bars include the reference value, the calibration values are considered within a reasonable range. The coverage ratio indicates the proportion of error bars that include the reference value. As shown in Figure 9, the error bars are reasonable. This approach provides a more comprehensive assessment of the calibration performance of each model by considering not only the absolute calibration error but also the consistency of calibration results over different periods. The incorporation of error bars and coverage ratios offers a deeper evaluation of model performance and an understanding of calibration uncertainty.  
Table 3 presents the test results for each model on the data from 22 May 2019 12:00:00 to 11 June 2019 15:00:00. The MCRTF-CM model achieved superior calibration performance, with reductions in the MAE, RMSE, MSE, and MAPE compared to the best-performing models. Specifically, MCRTF-CM reduced the MAE, RMSE, MSE, and MAPE by 14.04%, 13.59%, 25.33%, and 8.22%, respectively, compared to LSTM, and by 18.24%, 14.89%, 27.57%, and 24.10%, respectively, compared to MLP. This indicates that the MCRTF-CM model has better feature extraction and nonlinear mapping capabilities.  
Figure 7: Scatter plots of the calibrated values and reference values for the seven models, as well as scatter plots of the uncalibrated values and reference values.