## 4 Results and Discussion
### Application Results  
Once the ANN and ELM models' architectures have been defined, they are applied to the test set of data and their performances are evaluated by computing the four performance metrics on the \(n=36\) data points, using the 100 cross-validations procedure. The obtained results are compared to those obtained by the two benchmarked models: the MLR and the two hidden layer ANN developed in [54].  
Table 1 reports the average performance metrics values obtained by the application of the models on the training and validation sets of data (\(n=143\) data points) and the unseen test set of data (\(n=36\) data points).  
Figure 5: The influence of the number of hidden neurons on the prediction performance of the extreme learning machine (ELM) model.  
For clarification purposes, Figure 6 shows the measured and predicted values of the PV daily system conversion efficiency obtained by the optimized ANN and ELM models for the whole study period (\(n=179\) data points) with respect to those previously obtained by the MLR and the two hidden layer ANN in [54].  
The following can be observed;  
* The predicted values using all models are reasonably close to the measured values for the whole study period;
* In particular, the MLR model seems to provide less accurate predictions despite its easiness and flexibility compared to the other prediction models. In fact, all performance metrics values obtained by the MLR model are worse than for the other models. This can be justified by the fact that the behavior of the PV daily system conversion efficiency as a function of dust accumulation and ambient temperature is not strictly linear; thus, it cannot be accurately captured by the inherently linear MLR model, unlike the other nonlinear models (i.e., ANN and ELM);
* The effectiveness of having an optimum version of the ANN model is apparent in the four performance metrics compared to the two hidden layer ANN model proposed in [54];
* Furthermore, the effectiveness of the ELM model with respect to the other models is proved by all performance metrics. For instance, the optimum ELM model provides an enhancement in the conversion efficiency predictions compared to the MLR model with around 6.18%, 6.4%, 0.3%, and 42.92% for the \(R^{2}\), \(R^{2}_{adjusted}\), _Accuracy_, and _MSE_ performance metrics.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Training and Validation Data**} & \multicolumn{4}{c}{**Test Data**} \\  & \multicolumn{4}{c}{**(143 Data Points)**} & \multicolumn{4}{c}{**(36 Data Points)**} \\ \cline{2-9}  & \(R^{2}\) & \(R^{2}_{adjusted}\) & _Accuracy_ & _MSE_ & \(R^{2}\) & \(R^{2}_{adjusted}\) & _Accuracy_ & **MSE** \\  & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **MSE** \\ \hline MLR & 87.7 & 87.5 & 98.4 & 0.066 & 86.8 & 86.4 & 98.7 & 0.048 \\ \hline Two hidden layer ANN & 90 & 89.9 & 98.6 & 0.057 & 89.2 & 88.9 & 98.8 & 0.042 \\ \hline
**Optimized ANN** & **90.69** & **90.63** & **98.71** & **0.0502** & **90.55** & **90.27** & **98.87** & **0.0331** \\ \hline
**Optimized ELM** & **91.42** & **91.35** & **98.74** & **0.0462** & **92.16** & **91.93** & **98.99** & **0.0274** \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison results of the proposed prediction models to those proposed and developed in [43]. MLR: multivariate linear regression model.  
Figure 6: Measured and predicted values of the PV daily system conversion efficiency obtained by the prediction models for the whole study period.