## 4 Results
### Performance Analysis  
This subsection discusses the performance analysis of the developed BC proxies. Figure 4 shows a fraction of the time-series results of the BC proxies tested on data measured in Amman city center on 29 May 2014 from 21:44 to 23:42. The blue, red, and green lines represent the real BC measurement and the estimated BC quantity via the BB and WB proxies, respectively. The red and green light areas are \(2\sigma\) and \(3\sigma\) predicted uncertainty of their respected estimations. Even though the BC proxies do not fit perfectly the data points of BC measurement, it can be seen that both BC estimations track well the pattern of the BC measurement. Furthermore, the uncertainty estimations of \(2\sigma\) nearly cover the entire region of the BC measurement. Nevertheless, the time-series result only displays a snapshot in a particular area at particular time, it does not represent the overall performance metric of the proxy development results. Figure 3: The procedure of the modelling processes for the white-box (WB) and black-box (BB) proxies. Three additional metrics are used to describe the overall quality of the BC proxy performance. The first metric is called the 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 (\(\boldsymbol{y}\)). The second metric is the root mean squared error (RMSE), which is also known as the standard deviation of the residuals (prediction errors). The third metric is called the coefficient of determination, denoted by \(\mathrm{R}^{2}\). It provides a measure of how well the observed outcomes are replicated by the proxy, based on the proportion of total variation of outcomes explained by the proxy. The summary of these three performance metrics is shown in the Table 2.  
The proposed BC proxies are implemented on the data obtained from the measurements performed across urban areas (i.e., Amman and Zarqa) and throughout Jordan (including urban areas). The repeated k-fold cross-validation is used to establish training and test data sets. Using the performance metrics shown in Table 2, the performance of the proxies are then evaluated and the results are presented in Table 3. The low values of MAE and RMSE indicate that the proxy performance is better than the high values of these metrics. On the other hand, the high \(\mathrm{R}^{2}\) values indicate the proxy performance is better than the lower values. It can be seen here that in both cases and across different areas in Jordan, all performance metrics indicate that the BB model outperforms the WB \begin{table}
\begin{tabular}{c c} \hline \hline
**Performance Metrics** & **Formulation** \\ \hline Mean Absolute Error & \(\mathrm{MAE}=\frac{\sum_{i=1}^{n}|\hat{y}_{i}-y_{i}|}{n}\) \\ \hline Root Mean Squared Error & \(\mathrm{RMSE}=\sqrt{\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}{n}}\) \\ \hline Coefficient of Determination & \(\mathrm{R}^{2}=\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}{\sum_{i=1}^{n}(y_{i }-\hat{y})^{2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The performance metrics used for evaluation of the developed BC proxies. The real measurement value, the mean of the measurement data points, and the predicted proxy value are symbolized by \(y\), \(\hat{y}\), and \(\hat{y}\), respectively. The notations of \(i\) and \(n\) are the point number and the total predicted values from the proxies, respectively. Figure 4: Time-series data of BC mass concentrations in the Amman city center. The blue dot is the measured BC, whereas the red dashed line and the light green region are the measured BC and its \(2\sigma\) uncertainty. model. However, it can be observed that the difference in the performance metric values are relatively small indicating that the performance of the WB and BB models are similar.