## 5 Experiments and Results
### Performance Comparisons between CWC with Different Parameters and PIMOC  
To validate the influences of parameters on CWC, the control parameter \(\eta\) in Equation (5) is chosen as \(\eta=10,\ 50,\ 100\) and \(\mu\) is set as 80%, 85%, 90%, 95%, and 99%. Each experiment is repeated 10 times with different \(\eta\) and \(\mu\). The average results are shown in Table 1.  
From Table 1, it is obvious that PICP increases with the increase of \(\mu\) when \(\eta\) is fixed, and PICAW also becomes larger, which makes the accuracy of the PIs worse. On the other hand, \(\eta\) has a great effect on the quality of PIs with the fixed \(\mu\). A small \(\eta\) is helpful to improve the accuracy of PIs with a small width of PIs, but it is difficult to ensure that PICP reaches its expected \(\mu\) and meets the reliability requirements of the PIs. A large \(\eta\) can enhance the reliability of the PIs, and the PICP is always higher than its expected value. However, this will lead to a large width of the PIs, which then makes the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\eta\) & **Name** & **\(\mu\) = 0.80** & **\(\mu\) = 0.85** & **\(\mu\) = 0.90** & **\(\mu\) = 0.95** & **\(\mu\) = 0.99** \\ \hline \multirow{2}{*}{\(\eta=10\)} & PICP & 0.785 & 0.836 & 0.892 & 0.944 & 0.985 \\  & PICAW & 0.137 & 0.152 & 0.203 & 0.235 & 0.256 \\ \hline \multirow{2}{*}{\(\eta=50\)} & PICP & 0.822 & 0.864 & 0.923 & 0.963 & 0.995 \\  & PICAW & 0.152 & 0.198 & 0.232 & 0.251 & 0.284 \\ \hline \multirow{2}{*}{\(\eta=100\)} & PICP & 0.844 & 0.872 & 0.947 & 0.981 & 1.000 \\  & PICAW & 0.175 & 0.218 & 0.259 & 0.288 & 0.323 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparative results with different \(\eta\) and \(\mu\) of CWC. CWC: coverage width-based criterion; PICP: prediction interval coverage probability; PICAW: prediction intervals covered-normalized average width. solving process into a local optimum. Thus, the controlling parameter \(\eta\) is an uncertain factor in PI construction with CWC, which is unfavorable for solving this optimization problem in high quality. In order to compare the prediction results between CWC (with the ABC-WNN as the prediction model) and PIMOC (with the MOABC-WNN as the model), the experiments with a group of \(\mu\) (from 80-99%) in different \(\eta\) are performed and the results are shown in Figure 2.  
As is shown in Figure 2, the prediction results of PIMOC are better than CWC both in the criteria of PICP and PINAW. With different \(\eta\), the PICP and PICAW of CWC change greatly. During the optimization of CWC, the solutions that cannot satisfy the given confidence level \(\mu\) will be penalized by \(\eta\) and determined whether to be saved or not according to \(\eta\). Perhaps a large \(\eta\) can ensure that the requirement of the confidence level \(\mu\) should be met, but some good solutions which have a PICP slightly less than its expected value, but a good PI width, will be weeded out. However, if a small \(\eta\) is chosen and then the solutions with a slightly worse confidence level, but a small width of the PIs is saved, those solutions dissatisfy the requirement of the confidence level and will not be penalized sufficiently. As a result, the confidence level of all the solutions will be lowered. Thus, a suitable \(\eta\) is of great significance to CWC, but it is always chosen empirically. In PIMOC, this multi-objective optimization problem is solved by adopting a multi-objective evolution algorithm directly instead of transforming it into a single-objective optimization problem, which avoids the choosing of \(\eta\) and it is a benefit for improving the quality of PIs both in accuracy and reliability. For a better explanation, the PIs of CWC are shown in Figures 3-5 with three different \(\eta\) (the value of \(\eta\) is same as it in Table 1) and the PIs of PIMOC are shown in Figure 6, where Pareto optimal sets are sorted in ascending order according to PICP and the first, fifth, tenth, fifteenth, and twentieth PIs are plotted to make a comparison with the PIs of CWC.