## 4 Results and Discussion
### Performance Analysis
As a result, the number of data points may be very less when we take many inputs into the ML training process. Using the remaining data that is less than 80% may not be optimal in data-driven approach because the use of more data is a key in proxy training. The good performance obtained may be because the number of validation data points becomes much fewer. It can also be seen that when the number of inputs is increased to be 10, the proxy performance deteriorates rapidly because there are not enough validation points at all. Furthermore, the use of many input variables is also impractical when deploying the proxy. The deployed proxy will require many instruments working simultaneously at the same location to supply the same type of input data used in the training. Using many inputs from instruments indicates that the developed proxy is becoming less independent from real measurements. The real value of having a pollutant proxy will then be less because the deploying cost would rise. Therefore,  
Figure 7: Mutual information level between ozone concentration and other measured variables on full day (**a**), day only (**b**) and night only (**c**). as a compromise, we determine to select the number of inputs to be four. In this case, the number of missing data excluded can be minimised and we can still gain relatively good proxy performance. After all, we decided to exclude the two separate proxies, i.e., day and night models. The main reason is that the proxy performances do not vary significantly between the single proxy and the coupled proxies as shown in Figure 8. Therefore, for simplification, we will present the results of single proxy only from now on. A brute-force method is used to search the most optimal architecture of the BNN structure. In this case, we found that the best network consists of one hidden layer with rectifier activation functions and linear output layer. The hidden layer comprises 100 neurons, but the \"drop-out\" method (with 50% initial drop-out rate) is also applied as explained earlier to remove unnecessary neurons. For optimisation, an adaptive learning rate optimisation algorithm (Adam) [62] is used for tuning the BNN parameters. Once the top four inputs are selected, we applied cross-validation methods to find the optimal training/testing data that generate the best BNN. However, we found that this implementation does not result in significant difference in term of performance. It is also preferred to have continuous training and testing data in this case. Hence, we determine the used training data is from 21 December 2011 until 21 July 2012 whereas the validation data is selected from 22 July 2012 until 10 October 2012. This selection constitutes 70% training and 30% testing data. In this work, we do not compare the proposed probabilistic ML method with other ML methods, as the focus emphasises the concept of proxy development. The concept should be generic to be applied on other types of air pollutant proxy with different ML methods. As discussed earlier, the objective of deploying proxy is to reduce the number of operated instruments in an air pollution station, and then to minimises the cost as well as the operators involved in the measurement processes. Table 1 displays the performance metrics of different inputs combination from the selected proxy inputs. Please note that the inputs of NO2 and NOx are not separated since they are measured from the same instrument. From Figure 5, it is found that TNX correlates highly with NOx which is another input of the proxy. Therefore, this variable can be discarded. It can be  
Figure 8: Proxy performance metrics and the percentage of data loss evaluated on testing data. In the subplot (**a**), the green, blue and red constitute the metrics of Mean Absolute Error (MAE), Root-Mean-Squared Error (RMSE) and coefficient of determination (R\({}^{2}\)), respectively. The subplot (**b**) presents the percentage of data loss due to the involvement of more input numbers.