## 4 Results and Discussion
### Al-Based Models Employed for Forecasting Tasks  
This subsection identifies and analyses the most accurate model for forecasting hourly PM\({}_{2.5}\) and PM\({}_{10}\) concentrations using a Deep-NARMAX model. The Deep-NARMAX framework implemented in this study explores different historical data configurations for PMs and other related variables to effectively predict future PM\({}_{2.5}\) and PM\({}_{10}\) levels. The model initialization includes the following tuning parameters:  
- nu = [1 5]: Range of time lags for the output variable;  
- ny = [1 5]: Range of time lags for the input variables;  
- ne = [1 5]: Range of time lags for the error term;  
- P\(\_\)temp = max([nu, ny]): Maximum time lag among variables;  
- degree = 2: Nonlinearity degree of the model;  
- term\(\_\)len = 5: Maximum length of the potential model terms;  
- threshold = 0.7: Gate weight threshold for feature selection.  
Figures 14 and 15 illustrate the performance of the top-performing Deep-NARMAX model for multi-step-ahead forecasting of hourly PM\({}_{2.5}\) and PM\({}_{10}\) concentrations. These  
Figure 13: A comparison of FS techniques in predicting PM\({}_{10}\) concentration in the training and testing stages.  
figures show the forecasted values, the corresponding forecast errors, and the associated performance metrics, including RMSE and R\({}^{2}\) values.  
The most accurate model for predicting PM\({}_{2.5}\) concentration at time t is defined by Equation (13) and incorporates the following key predictors:  
- PM\({}_{2.5}\) at time (t \(-\) 2);  
- CO at time (t \(-\) 1) and (t \(-\) 2);  
- Sun Brightness at time (t \(-\) 1);  
- p-Xylene at time (t \(-\) 1).  
The corresponding model is given by:  
Figure 14: The multistep-ahead time series forecasting of PM\({}_{2.5}\) concentrations using deep-NARMAX-model.  
Figure 15: The multistep-ahead time series forecasting of PM\({}_{10}\) concentrations using deep-NARMAX-model.  
\[\begin{split}\mathbf{y}(\mathbf{t})&=0.860362\mathbf{y}(\mathbf{t}-2)+40.88 858\mathbf{u}_{2}(\mathbf{t}-1)-0.249971\mathbf{u}_{2}(\mathbf{t}-2)\mathbf{y}(\mathbf{t}-2)\\ &-5.38573.10^{-03}\mathbf{u}_{14}(\mathbf{t}-1)\mathbf{y}(\mathbf{t}-2)-8.5159\mathbf{ u}_{10}(\mathbf{t}-1)\mathbf{u}_{2}(\mathbf{t}-2)\end{split} \tag{13}\]  
This model demonstrates strong predictive performance, with low error dispersion and an R\({}^{2}\) value approaching 0.85, indicating a high level of reliability and consistency in forecasting accuracy. These results highlight the robustness of the Deep-NARMAX framework in capturing the complex temporal dynamics of PM\({}_{2.5}\) concentrations.  
Similarly, the optimal model for PM\({}_{10}\) forecasting is defined in Equation (14) and uses the following predictors:  
* PM\({}_{10}\) at time (t \(-\) 1);
* CO at time (t \(-\) 1) and (t \(-\) 2);
* Relative humidity at time (t \(-\) 3);
* Sun Brightness at time (t \(-\) 3).  
The model is expressed as:  
\[\begin{split}\mathbf{y}(\mathbf{t})&=1.02542\mathbf{y}(\mathbf{t}- 1)-0.357789\mathbf{u}_{2}(\mathbf{t}-2)\mathbf{y}(\mathbf{t}-1)+51.89441\mathbf{u}_{2}(\mathbf{t}-2)\\ &-0.499499\mathbf{u}_{2}(\mathbf{t}-2)\mathbf{u}_{12}(\mathbf{t}-3)-0.072085\mathbf{ u}_{2}(\mathbf{t}-2)\mathbf{u}_{14}(\mathbf{t}-3)\end{split} \tag{14}\]  
This model also yields high predictive accuracy, with minimal error variation and an R\({}^{2}\) value nearing 0.89, further validating the Deep-NARMAX approach as a powerful tool for short-term air quality forecasting.  
In summary, both models exhibit very good generalization and accuracy, reinforcing the effectiveness of Deep-NARMAX in modelling non-linear dependencies and temporal lags in air pollutant data.