## 5 Evaluation of the Models
### Model 4  
Figures 15 and 16 give the overall result and the precision of Model 4. The main advantage of this model, compared to the previous ones, is that the confidence levels give a very good fit to the data and the predictions are, in all cases, in line with the initial data.  
This model has more parameters than the previous ones. In particular, the introduction of the error parameter \(\varepsilon_{s}\) permits fine-tuning of the data of each ANSP. This means that it is flexible and capable of quantifying the dispersion of the data.  
In this model, the hyperparameters \(K_{js}\) (K, K1, K2, K3, and K4) and \(\hat{\theta}_{j}\) (\(\hat{\theta}\), \(\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\), and \(\hat{\theta}_{4}\) ) behave in a similar manner to the way they do in the other models. Figure 17 gives an analysis of the caterpillar plots of the \(\beta j_{s}\) parameters of the regression function. This enables us to have a more intuitive view of the influence of the variables \(xj_{s}\) on each ANSP.  
Figure 14: Model 3: Distributions of the \(K_{js}\) parameters for Model 3. From left to right and top to down: K (**a**), K1 (**b**), K2 (**c**), K3 (**d**), and K4 (**e**).  
Figure 16: Model 4: Residuals, predicted values vs. residuals, and Q-Q plot for Model 4.  
Figure 15: Model 4: Predicted values (red) vs. actual values (black), and 2.5% and 97.5% confidence intervals for Model 4.  
Additionally, in this model, there is another consideration with respect to the parameter \(\beta 1_{s}\) in that it corresponds to the coefficient of the quadratic term in the regression. This parameter can be used to reflect the dependence of the rate of occurrence of SMIs on the number of flight hours. In aviation this phenomenon is known as the stress effect and refers to the increase in accident rates as a function of the number of operations or flight hours. This has not been considered in the other three models. The value of the quadratic coefficient in the regression parameter is a measure of the importance of the stress effect. The higher the value of \(\beta 1_{s}\), the greater the positive correlation between the rate of occurrence of SMIs and the square of the number of flight hours and, consequently, the greater the stress effect. ANSPs that experience this effect are considered to be saturated, since variations in the number of flight hours have a significant effect on the occurrence of SMIs. Therefore, special attention must be paid to this factor.  
Analysis of the parameters \(\varepsilon_{s}\), the error term in the linear regression, and \(\sigma_{s}\), the variance proportional to the number of hours of operation, is especially relevant in this case. Figure 18 shows the caterpillar plots for these variables.  
Practically all the ANSPs have values of \(\sigma_{s}\) equal to zero with the exception of ANSPs 29, 18, 19, 21, 25, 7, and 6. All of these correspond to small service providers with low numbers of flight hours.  
Finally, the values of \(\theta_{j}\) (\(\vartheta\), \(\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\), and \(\theta_{4}\) ) for all the models are summarized in Table 2.  
Figure 17: Model 4: Distributions of the \(\beta j_{s}\) parameters. From left to right and top to down: K (**a**), K1 (**b**), K2 (**c**), K3 (**d**), and K4 (**e**).