## 6 Comparison of Regression Models and Recommendations
### Uncertainty Quantification and Robustness of the Selected Regression Model  
This section presents the quantification of the prediction uncertainty as well as the fit robustness for the proposed model, applied to both products. The analysis comprises two cases, (i) the regression with fitted asymptotic load level, and (ii) the regression with a fixed asymptotic value at 40%. First, the prediction uncertainty bounds are calculated, see blue dashed lines in Figure 14. The prediction bounds are calculated with the Curve Fitting Toolbox of Matlab at a confidence level of 95%. Moreover, a bootstrap resampling is performed for each load level. In particular, a single data point, failure time, is sampled for each load level, such that a new possible dataset is obtained that contains one failure time only at each tested sustained load level. In each realization the same failure time for a load level is allowed to be resampled. The procedure is repeated and finally 1000 new datasets for each product are generated. The number of realizations is selected as to exceed sufficiently the number of possible permutations (\(\sim\)100). In that way, a potential bias of the resampling method is eliminated. Then, the proposed regression models are fitted to each of these datasets independently and extrapolated to a time horizon of few hundred years.  
The individually fitted curves are shown in Figure 14 as grey solid lines. Finally, the 5 th and 95 th percentiles of the load level for a given failure time are calculated. The estimated percentiles are calculated from the distributions of the load levels that are predicted for a given failure time by the bootstrapped fits. The 5 th and 95 th are presented in Figure 14 as black dashed lines. Figure 14 shows that the prediction bounds are narrower for P1, see Figure 14a,b. Contrary to P1, the prediction bounds become wider in the case of P2, see Figure 14c,d. This can be explained by the different distribution of the failure times along the load levels domain for the two products, i.e., the failure times are more distinct for different load levels in the case of P1, while the failure times in the case of P2 are overlapping among different load levels. An additional conclusion is that the prediction bounds are narrower for the regression model with a fixed asymptotic load level, see Figure 14b,d. This is certainly expected, since the regression model has less parameters to be fitted, and, thus, the imposed prediction uncertainty should be lower. Furthermore, the regression model with the fixed asymptote does not exhibit the clustering of the fitted curves, as it is observed in Figure 14a,c. The clustering of the curves is a result of the random sampling of the points and of the high influence of the later failure times on the fitted asymptotic value.  
Figure 13: Summary of different fitting models for (**a**) Product 1, and (**b**) Product 2.  
The same conclusions can be also derived for the uncertainty bounds as obtained by the 5 th and 95 th percentiles of the bootstrapped fits. Furthermore, the comparison of the obtained uncertainties obtained by the two methods reveals that the 5 th percentile, i.e., lower load level for a given failure time, is similar to the lower bound of the prediction errors for a prediction time of 1 to 100 years, with slight differences for the different cases.