# A Gaussian Process Method with Uncertainty Quantification for Air Quality Monitoring
## 5 Conclusions  
This paper proposes a general method to investigate how the performance variation of a Gaussian process model can be attributed to hyperparameters and measurement noises, etc. The method is demonstrated by applying it to process particulate matter (e.g., PM\({}_{2.5}\)) and gaseous pollutants (e.g., NO, NO\({}_{2}\), and SO\({}_{2}\)) from both Sheffield, UK, and Peshawar, Pakistan. Experimental results show that the proposed method provides insights on how measurement noises and hyperparameters, etc. affect the prediction performance of a Gaussian process. The results align with the analytical derivations, which is enabled by adopting Neuman series to approximate matrix inversions in Gaussian process models. The theoretical findings and experimental results combined demonstrate that the proposed method can generate air quality forecasting results. In the meantime, it provides a way to link uncertainties in measurements and hyperparameters, etc. with the forecasting results. This will help with forecasting performance analysis when measurement noise level or model hyperparameters vary, making the method more general.  
Figure 9: Effects of \(\sigma_{n}^{2}\) on ELBO and UBML: (**a**) NO in Sheffield, (**b**) NO in Peshawar.  
**Author Contributions:** Conceptualization, P.W., L.M., M.M., R.C., S.M., K.A. and M.F.K.; methodology, P.W.; software, P.W.; validation, P.W., Z.Z., C.J. and H.F.; formal analysis, P.W., L.M.; investigation, P.W.; data curation, S.M., R.C., K.A. and M.F.K.; writing--original draft preparation, P.W., L.M., R.C., S.M., K.A. and M.F.K.; writing--review and editing, P.W. and L.M.; visualization, P.W., R.C.; supervision, L.M., M.M.; funding acquisition, L.M., P.W., M.M., S.M. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the UK EPSRC through EP/T013265/1 project NSF-EPSRC/ShiRAS. Towards Safe and Reliable Autonomy in Sensor Driven Systems, a joint project with the USA National Science Foundation under Grant NSF ECCS 1903466. Other funders are NSFC (61703387) and the Global Challenges Research Funds (QR GCRF--Pump priming awards (Round 2), project entitled: \"Collaborating with North Pakistan for monitoring and reducing the air pollution (X/160978)\".  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not Applicable.  
**Acknowledgments:** We are grateful to UK EPSRC for funding this work through EP/T013265/1 project NSF-EPSRC-ShiRAS. Towards Safe and Reliable Autonomy in Sensor Driven Systems. This work was also supported by the USA National Science Foundation under Grant NSF ECCS 1903466. We also appreciate the support of NSFC (61703387). We are also grateful to the Global Challenges Research Funds (QR GCRF - Pump priming awards (Round 2), entitled: \"Collaborating with North Pakistan for monitoring and reducing the air pollution (X/160978))\". We also thank Urban FLOWS Observatory, the University of Sheffield for providing the air quality sensors for collecting air pollution data in Pakistan.  
**Conflicts of Interest:** The authors declare no conflict of interest.