## 3 Methodology
### Methods of Predictive Modeling
#### 3.4.2 Evaluation of Metrics of Prediction Models
However, as for the dependent factor carbon monoxide (CO), we observed a significant difference and effect with the two variables \"city\" and \"year\" based on their probability values being less than the alpha value. We found that the ozone pollutant (O\({}_{3}\)) is affected by the city's independent factor and has a probability value less than the alpha value (\(p\)-value = 5.385055 \(\times\) 10\({}^{-72}\)). This indicates a relationship between the city's characteristics and the concentration rate of O\({}_{3}\). It also has a lower probability value with the independent \"year\" factor (\(p\)-value = 4.28593 \(\times\) 10\({}^{-10}\)), which indicates that the year and city factors have a relatively considerable influence on the O\({}_{3}\) concentration rate in the air. As for the last pollutant, HCHO, we found that it is also affected by the independent factor of the city, as there is a relationship between the characteristics of the city and the rate of increase or decrease in the concentration of HCHO in the air. It is also affected by the independent \"year\" factor. To gain more insights into the differences in the pollutants across the three cities and the two years, we present the interaction plots of each pollutant alone in Figure 4. Regarding NO\({}_{2}\) (Figure 4a), the plot clearly shows that the highest concentration of NO\({}_{2}\) is observed in Al Riyadh, followed by Al Juhail, with Najran having the lowest levels. However, the analysis shows that the year does not affect the concentration of NO\({}_{2}\) in any of the three cities, suggesting that year-to-year variation is minimal for this pollutant. The SO\({}_{2}\) plot (Figure 4b) shows that the concentration in Al Juhail is higher compared to the other two cities, while the difference between Al Riyadh and Najran is not statistically significant. This finding is further supported by the Games-Howell post hoc test, which \begin{table}
\begin{tabular}{c c c} \hline \hline
**Air Pollutants** & \(p\)**-Value of City** & \(p\)**-Value of Year** \\ \hline NO\({}_{2}\) & 1.58370 \(\times\) 10\({}^{-167}\) & 9.72024 \(\times\) 10\({}^{-1}\) \\ SO\({}_{2}\) & 1.840551 \(\times\) 10\({}^{-45}\) & 1.75437 \(\times\) 10\({}^{-1}\) \\ CO & 1.332872 \(\times\) 10\({}^{-238}\) & 1.279 \(\times\) 10\({}^{-3}\) \\ O\({}_{3}\) & 5.385055 \(\times\) 10\({}^{-72}\) & 4.28593 \(\times\) 10\({}^{-10}\) \\ HCHO & 1.173732 \(\times\) 10\({}^{-32}\) & 7.6 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Probability values of [PERSON]’s ANOVA for all air pollutants. was used to compare the SO\({}_{2}\) concentration across the three cities and the two years. This test shows that the \(p\)-value for the comparison between Najran and Al Riyadh is higher than the significance level of 0.05, indicating no statistically significant difference between the two cities. The plot for CO (Figure 4c) shows that the highest concentration of CO is observed in Al Juhail, followed by Al Riyadh, with Najran having the lowest levels. Moreover, the analysis shows that the year has an effect only in Najran City, such that 2023 displays a higher concentration than 2022. This finding is further supported by the Games-Howell post hoc test, such that the \(p\)-value of the comparison of CO in Najran in 2022 and 2023 is \(6.917516\times 10^{-6}\), which is lower than the significance level of 0.05, indicating a significant difference between the concentration of the two years. Figure 4: Comparison of the air pollutant averages across the three cities and the two years: (**a**) NO\({}_{2}\); (**b**) SO\({}_{2}\); (**c**) CO; (**d**) O\({}_{3}\); and (**e**) HCHO. Regarding the O\({}_{3}\) plot (Figure 4d), it is clear that in the year 2022, each city displays a significant difference in the concentration of this pollutant. The same can be said for the year 2023. However, the differences in the concentrations across the years have a significant effect only in Najran and Al Riyadh.