## 4 Results and Discussion
### Performance Analysis  
The results of the previous subsection justify our choice to utilise the strength of MI method for proxy inputs selection as proposed in Section 2. This subsection presents several results obtained from the proposed proxy development. Figure 7 presents the rank of MI score of the measured variables correlated to O\({}_{3}\) concentration. The variables with high MI score indicate the higher association with the O\({}_{3}\) concentration. Based on the bar chart, we can select the appropriate inputs for training the pollutant proxy. We first explore the possibility to divide the proxies into two: day and night proxies as previously proposed in our previous work [20], and then also generate one single full day proxy. Therefore, MI between O\({}_{3}\) concentration and other measured variables are applied on data sets for entire day, day only and night only. Figure 7 shows the MI level between O\({}_{3}\) and other measured variables for two main types of proxy. The subplot (a) is the MI level for a full day analysis, whereas the subplots (b) and (c) constitute the MI level for day and night measurements, respectively. The analysis for the subplot (a) is used to select the inputs for the single O\({}_{3}\) proxy, whereas information on the subplots (b) and (c) are utilised to select the divided O\({}_{3}\) proxy. It is noted that the bar chart results also support the scientific findings discovered earlier in the previous section. All proxies use at least two inputs at the first modelling attempt and then the inputs fed into BNN training following the order as shown in Figure 7. In order to evaluate the developed proxies, three types of performance metric are used. The first metric is called mean absolute error (MAE). It has a simple interpretation as the average absolute difference between the predicted proxy values (\(\boldsymbol{\hat{y}}\)) and the real measurement data points (\(\boldsymbol{y}\)). It can be defined mathematically as  
\[\text{MAE}=\frac{\sum_{i=1}^{n}|\hat{y}_{i}-y_{i}|}{n} \tag{8}\]  
Figure 6: The scatter plot between O\({}_{3}\) concentration and several selected variables, including NOx, wind speed, _m,p_-Xylene (TNX) and NO\({}_{2}\). R\({}^{2}\) is Pearson correlation coefficient (PCC), whereas MI is mutual information score. The concentration unit of trace gases and TNX is in part per billion (ppb) whereas the unit of wind speed is in m/s. where \(\hat{y}\), \(y\) and \(n\) are the predicted value from proxy, the real measurement value and the number of data points, respectively. The second metric is root-mean-squared error (RMSE) that is also known as the standard deviation of the residuals (prediction errors). It is given by  
\[\text{RMSE}=\sqrt{\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}{n}} \tag{9}\]  
The third metric is called coefficient of determination, denoted by R\({}^{2}\). It provides a measure of how well observed outcomes are replicated by the proxy, based on the proportion of total variation of outcomes explained by the proxy. It can be defined as  
\[\text{R}^{2}=1-\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}{\sum_{i=1}^{n}(y_ {i}-\bar{y})^{2}} \tag{10}\]  
where \(\bar{y}\) is the mean of the number of measurement data points, \(\mathbf{y}\). Figure 8 shows proxy performance metrics and the percentage of data loss. The Figure 8a demonstrates the proxy performance after incorporating different number of inputs into BNN, including MAE, RMSE and R\({}^{2}\). It is known that incorporating more inputs into ML models improves their performance generally, as shown in Figure 8a. Figure 8b demonstrates that after incorporating four input variables, the model performance still improves, but the number of data loss increases significantly, i.e., to be almost 80% because the need to find some common training data from all variables used in the training process. In a real-world problem, this is not always possible, some data points may be missing from several measurements at different time frames.