## 4 Experimental Results and Discussion
Although these linear models obtained the best values of the performance indexes, they have to be validated before continuing to step III. The residuals of the model must be satisfied the requirements of a white noise process. Therefore, the residuals of both clusters were checked. On the one hand, Figure 4 left collects the residual diagnosis plot of Cluster 1. According to Figure 4 left, the residuals have a mean around zero whereas Figure 4 left shows no obvious violation of the normality assumption. Figure 4c,d left show no significant autocorrelation and residuals are thereby uncorrelated. On the other hand, Figure 4 right shows the residual diagnosis plot of Cluster 2, corresponding to SARIMA(2,1,2)(2,1,3)\({}_{5}\) model. As in the previous case, residuals satisfy the requirements of a white noise process. In conclusion, the ARIMA(2,2,3) and SARIMA(2,1,2)(2,1,3)\({}_{5}\) models were validated as linear models to Cluster 1 and 2, respectively, in the second step. Once the ARIMA model is applied, a set of residuals and predicted values are obtained for each cluster. These values, together with the original data, are used as inputs in the third step. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Best Configuration**} & \multicolumn{3}{c}{**Performance Indices**} \\ \hline
**Clusters (c)** & **Temporal Leap** & _nc_ & **CQ1** & **CQ2** \\ \hline
2 & 7 & 3 & 0.659 & 717.548 \\
3 & 7 & 3 & 0.627 & 682.549 \\
4 & 1 & 3 & 0.588 & 643.792 \\
5 & 1 & 3 & 0.601 & 658.248 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Clustering results of the SOM step, where \(c\) is the number of clusters tested and _nc_ is the size of the input vector. The temporal leap in the past is 1 or 7 days. Figure 4: Results of the residual diagnosis for Cluster 1 (**left**), and for Cluster 2 (**right**): (**a**) standard residual plot; (**b**) quantile–quantile plot of standard residual versus the normal distribution; (**c**) sample ACF of the residual and (**d**) partial ACF of the residual. Finally, in the third step a SVR model was applied to each cluster considering the two proposed hybrid approaches, which were configured depending on the input setting used. Considering each hybrid configuration separately, a most accurate SVR model was achieved in each cluster to fit the data. Two hybrid approaches with two prediction horizons were considered in this third step for each cluster. Therefore, four prediction schemes for each cluster were evaluated (two hybrid approaches for each cluster and two prediction horizons for each hybrid approach). The final prediction results of each hybrid approach were obtained by joining the prediction values achieved in the two clusters as a single predicted time series. That is:  
\[\hat{y}_{inspections}=\{\hat{y}_{Cluster1}\}\cup\{\hat{y}_{Cluster2}\}, \tag{20}\]  
The most accurate models for each hybrid approach are collected in Table 3. The prediction results in terms of performance indexes reflect those obtained once the predicted values of the two clusters were grouped. Table 3 is divided according to the prediction horizon used (\(ph=1\) or \(ph=7\) days). Additionally, for each prediction horizon, results are collected depending on the hybrid configuration applied. For the hybrid approach 2 (SOM-SARIMA-SVR-2), results are presented considering the inputs employed in the model (\(y\), \(e\) or \(p\)) in order to demonstrate which inputs are most relevant. Regardless of the parameter configuration, only the best value achieved in each performance index is depicted in both prediction horizons for a better understanding. The best overall value of each performance index and the best hybrid configuration are pointed out in bold. The SOM-SARIMA-SVR-2 (the one without \(p\) as input) provides the most accurate results for one-day ahead prediction, followed by the rest of possible models of the hybrid approach 2 (considering different inputs) and finally the hybrid approach 1, in that order. This hybrid approach achieved the best value in at least four performance indexes.