# A Gaussian Process Method with Uncertainty Quantification for Air Quality Monitoring
## 4 Experiments and Analysis
### Impacts of Measurement Noise Level and Hyperparameters  
To demonstrate how noise level \(\sigma_{n}^{2}\) and hyperparameters affect prediction accuracy and uncertainty, three sets of experiments are conducted. This paper adopts the squared exponential (SE) kernel, with hyperparameters \(s_{f}\) and \(l\). The analytical derivation can be found in Appendix C. The prediction accuracy is identified by the root mean square error (RMSE), as shown in Equation (34), while the uncertainty is identified by \(\frac{1}{2}\sigma\) confidence bound. Configurations of the experiments are as follows.  
**Experiment 1:** Impacts of \(\sigma_{n}^{2}\) on prediction accuracy and uncertainty. Both \(s_{f}\) and \(l\) are fixed to be the optimised values. \(\sigma_{n}^{2}\) varies from 0.1 through to 20.0. NO, NO\({}_{2}\), SO\({}_{2}\), and PM\({}_{2.5}\) data from both cities are processed. Six inducing points are applied to both FITC and VFE.  
**Experiment 2:** Impacts of \(s_{f}\) on prediction accuracy and uncertainty. \(l\) is set to the optimised value. \(s_{f}\) varies from 0.1 through to 30.0. \(\sigma_{n}^{2}\) is set to 0.5 and 1.5, respectively. NO data from both cities are processed. Six inducing points are applied to both FITC and VFE.  
**Experiment 3:** Impacts of \(l\) on prediction accuracy and uncertainty. \(s_{f}\) is set to the optimised value. \(l\) varies from 0.1 through to 30.0. \(\sigma_{n}^{2}\) is set to 0.5 and 1.5, respectively. NO data from both cities are processed. Six inducing points are applied to both FITC and VFE.  
\[\text{RMSE}=\sqrt{\frac{\sum_{i=1}^{Num}(y_{i}-\widehat{y_{i}})^{2}}{Num}}, \tag{34}\]  
where \(y_{i}\) is the ground truth value and \(\widehat{y_{i}}\) represents predicted meant. \(Num\) is the sample number in testing set.  
Figures 5 and 6 show the results from **Experiment 1**. To make the results more distinguishable, the horizontal axes of the figures are set to \(\log(\sigma_{n}^{2})\). We can see from Figure 5 that when \(\sigma_{n}^{2}\) is small, GPs perform the best in general, while the performance of FITC and VFE varies. We can also observe that as \(\sigma_{n}^{2}\) keeps increasing, the RMSE becomes very significant for all methods/pollutants. Similar results can be observed from Figure 6 as well. Both comply with our theoretical conclusions, despite the fact that the Neumann series is used to approximate the matrix inverse. We also notice that \(\sigma_{n}^{2}\) has a more significant impact on Sheffield data as RMSE increases ealier after \(\log(\sigma_{n}^{2})\) reaches  
Figure 4: Prediction and absolute error of pollutants in Peshawar: (**a**) NO, (**b**) NO\({}_{2}\), (**c**) SO\({}_{2}\), (**d**) PM\({}_{2.5}\).  
zero. From Figure 6b,c, we also see that the uncertainty bounds of Sheffield data are greater after \(\log(\sigma_{n}^{2})\) reaches zero. We think the reason is that Sheffield data are generally less periodical than Pershawar data (see Figure 2), which influences the performance of the models.  
Figure 5: Relationship of \(\sigma_{n}^{2}\) with four pollutants prediction RMSE: (**a**) NO, (**b**) NO\({}_{2}\), (**c**) SO\({}_{2}\), (**d**) PM\({}_{2.5}\).  
Figure 6: Relationship of \(\sigma_{n}^{2}\) with pollutants prediction uncertainty bound: (**a**) NO, (**b**) NO\({}_{2}\), (**c**) SO\({}_{2}\), (**d**) PM\({}_{2.5}\).