## 5 Conclusions  
This paper demonstrates the development of an air pollutant proxy based on data-driven methods. The deployment of air pollutant proxy is beneficial to replace missing data, to substitute a real measurement that are typically expensive and required operators and to be embedded in low-cost sensors as a virtual sensor. This work presents two main contributions in air pollutant proxy development: (1) mutual information (MI) for selecting appropriate inputs and (2) the proxy based on a probabilistic ML that is a Bayesian neural network (BNN). The MI method demonstrates the effectiveness in finding other relevant variables to a pollutant to be modelled (as a proxy) where standard correlation method is unable to discover them. The proposed method guarantees to capture linear and nonlinear relationship among the variables. The most relevant variables as the proxy inputs are determined based on several highest MI scores obtained and based on proxy performance metrics as well as data loss monitoring. For practicality, the selected proxy inputs are then evaluated based on the number of instruments involved and their respective performance metrics. In this way, testing all combinations of measured variables can be avoided. A probabilistic ML method, named BNN, is then used to establish a pollutant proxy. The proposed method offers  
Figure 11: The average of the measurement and estimation of Ozone concentration from two different models on weekday and weekend. confidence interval for further analysis. The use of Bayesian method also prevents overfitting naturally due to the presence of regularisation and dropout method. In particular, establishing an O\({}_{3}\) concentration proxy is selected as a case study. The O\({}_{3}\) and other variables' databases are obtained from an extensive air pollution research campaign in Jeddah, Saudi Arabia. The MI application in the data results in the associated variables to O\({}_{3}\) which have also been reported in other studies separately. The variables are NO\({}_{2}\), NOx, WS and TNX. The overall implementation of our proposed methodology on the case study leads to adequate performance in general and the obtained results are also considerably better in comparison with the previous attempt using the same dataset. As this is a general methodology, the strategy is also promising to be adopted to estimate any other types of air pollutant concentration. Several future works can be continued from this approach. First, the proxy performance and accuracy can still be improved. The slight bias in the proposed proxy might take place due to the presence of autocorrelated effect in the data or the drifting in other measured variables. The former can be solved by applying dynamic ML models where they typically accommodate some previous steps of historical data in model inputs and outputs into the model. The latter can be checked by diagnosing the drift. For example, anomalies in the variables involved should be detected in the estimation given the absent of ground truth which can be performed using a predictive distribution analysis. If the drifting occurs, the remedy is then to develop an adaptive proxy which is capable to modify automatically its parameters based on the nature change in the data. Second, the capability of developed proxy has a potential to be enhanced for acting as a pollutant forecasting. Therefore, the proxy does not only estimate the current pollutant concentration, but also forecast it a couple of hours/days in advance, so better mitigation can be taken by policy makers. This can be done by applying MI between variables at different times for analysing and determining the dynamic relationships between air pollution variables. Then, the proxy can be upgraded into a dynamic model, such as Bayesian recurrent neural network (BRNN). In this case, a forecast proxy is fitted on newly available data prior to each prediction. Third, the developed proxy may not work well in other geographical locations.