## 4 Results
### Discussion  
This subsection discusses the advantages of using the developed proxy in terms of uncertainty analysis, as well as its potential to be embedded in low-cost PM sensors, and bias-variance trade-off in modelling choice. Since we can directly estimate the uncertainty associated with our prediction using the (Bayesian) predictive distributions, we are able to quantify how many estimated BC data points fall within each degree of model uncertainty (standard deviation, symbolized as \(\sigma\)). Figure 6 shows the bar chart of the percentage of estimated BC data points that lie within \(\sigma\) to \(3\sigma\). It can be seen that about 81% and 96% of the estimated BC proxy data points lie within a very high confidence interval (\(\sigma\)) for both the WB and BB models. The remaining BB estimation (4%) lie within \(2\sigma\), whereas 16% and  
Figure 5: Regression plots between measured and estimated BC mass concentrations and error histogram between measured and estimated BC mass concentrations. (**a**) Regression plot. (**b**) Error histogram. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Measurement Locations**} & \multicolumn{2}{c}{**MAE (\(\mu\)g/m\({}^{3}\))**} & \multicolumn{2}{c}{**RMSE (\(\mu\)g/m\({}^{3}\))**} & \multicolumn{2}{c}{**R\({}^{2}\)**} \\ \cline{2-7}  & **WB** & **BB** & **WB** & **BB** & **WB** & **BB** \\ \hline Urban (Amman and Zarqa) & 1.834 & 1.777 & 2.111 & 2.061 & 0.76 & 0.77 \\ \hline Jordan (including urban) & 1.945 & 1.893 & 2.414 & 2.358 & 0.77 & 0.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean absolute error (MAE), root mean squared error (RMSE), and coefficient of determination (R\({}^{2}\)) of the proposed BC proxies evaluated on BC measured at urban areas across Jordan. 3% of the WB estimations lie on confidence intervals of 2\(\sigma\) and 3\(\sigma\), respectively. In this case, BB is slightly better than WB due to its model flexibility, therefore it is able to adequately mimic the real BC measurements. Nevertheless, both methods successfully estimate all BC measurements to within 3\(\sigma\) confidence intervals. It is also beneficial to embed the developed proxies into low-cost PM sensors. Low-cost sensor devices have become increasingly popular to fill the data gap between air quality networks [70]. However, BC measurement using low-cost sensors is often absent due to the complexity of the measurement technique [71]. This motivates the integration of the developed proxy for estimating BC mass concentrations using inputs from low-cost PM sensors. In addition to trace gases, PM\({}_{2.5}\) has been a standard measurement of many low-cost air quality sensors [47; 72; 73]. However, there has been no low-cost technologies available to monitor UFPs as of yet, such as PN\({}_{0.3-0.025}\)[74]. Therefore, the number of proxy inputs needs to be reduced to one, by taking only PM\({}_{2.5}\) measurements. In this case, the input of PN\({}_{0.3-0.025}\) is excluded, the equation then becomes:  
\[\log[\mathrm{BC}]=\beta_{1}+\beta_{2}\log[\mathrm{PM}_{2.5}]+\varepsilon \tag{9}\]  
The above formulation simplifies the BC proxy to be a linear regression model which can also be treated through a Bayesian formulation. The use of a linear model leads to an exact Bayesian inference computation as described in [PERSON] et al. [75]. Having an exact Bayesian solution also makes the proxy easy to be embedded in low-cost PM sensors. However, after applying it on this case study, the performance degrades, as shown in Table 4. Nevertheless, the function of low-cost PM sensors is typically not for scientific research, instead they are often used to provide an approximation of air pollution information to the public. Therefore, the results of the proxy embedded in low-cost PM sensors should be implementable for scaling-up the air pollution information.