## 5 Results and Discussion
### Results
The variation of parameter \(a\) has the greatest impact on type II errors, so we compare the performance of different algorithms when parameter \(a\) is the smallest (\(a=0.7\)). This represents the most difficult situation to detect. The proposed method achieves a superior performance as compared to traditional methods. The type I error, type II error, and \(F_{2}\) score of LOF are 0.049, 0.001, and 0.951, respectively, and for MHD-C, which is the second best, are 0.081, 0.015, and 0.899, respectively. We subsequently compared the performance of various methods given known contamination rates. Under this condition, only the three outlier-detection algorithms were are considered. This is because when the contamination rate is known, the outlier-detection algorithms are advantageous compared to the MHD-C and PPOD algorithms, and the comparison results will be biased. Therefore, we chose to compare only outlier-detection algorithms. Tables 4-6 summarizes the type I errors, type II errors, and \(F_{2}\) scores of the three outlier-detection algorithms, respectively. Table 4 indicates that when the contamination rate is less than 0.3 (i.e., \(m_{o}=20\) or 40), the type I error of the LOF algorithm is marginally lower than that of the EE algorithm. When the contamination rate is 0.3, the type I error of the EE algorithm is marginally lower than that of the LOF algorithm. Moreover, the type I error of the IF algorithm is considerably higher than those of the other two algorithms. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & & & & & & & & & & \\ \cline{2-10}
**Method** & \(m_{0}\) & & **0.7** & & **0.9** & & **1.1** & & **1.3** & & **1.5** \\ \hline LOF & 20 & 0.001 & (0.002) & 0 & (0) & 0 & (0) & 0 & (0) & 0 & (0) \\ EE & & 0.001 & (0.002) & 0 & (0) & 0 & (0) & 0 & (0) & 0 & (0) \\ IF & & 0.029 & (0.008) & 0.011 & (0.006) & 0.006 & (0.004) & 0.003 & (0.004) & 0.002 & (0.003) \\ \hline LOF & 40 & 0.002 & (0.004) & 0 & (0) & 0 & (0) & 0 & (0) & 0 & (0) \\ EE & & 0.003 & (0.004) & 0 & (0) & 0 & (0) & 0 & (0) & 0 & (0) \\ IF & & 0.093 & (0.013) & 0.059 & (0.010) & 0.041 & (0.011) & 0.038 & (0.008) & 0.037 & (0.010) \\ \hline LOF & 60 & 0.006 & (0.007) & 0 & (0) & 0 & (0) & 0 & (0) & 0 & (0) \\ EE & & 0.005 & (0.007) & 0 & (0) & 0 & (0) & 0 & (0) & 0 & (0) \\ IF & & 0.189 & (0.012) & 0.148 & (0.016) & 0.129 & (0.012) & 0.130 & (0.013) & 0.124 & (0.012) \\ \hline \hline \end{tabular} Numbers in parentheses denote standard deviations.  
\end{table} 
Table 4: Type I error of the three outlier-detection algorithms under the assumption that the contamination rate is known. Table 5 presents the type II errors of the three outlier algorithms. We can see that the results have the same trend as those for the type I errors, that is, the type II error of the LOF algorithm is close to that of the EE algorithm, whereas the IF algorithm has a considerably higher type II error than the other two algorithms. When the coefficient \(a\) is higher than 0.9, the type II errors of the LOF and EE algorithms approach 0, while the IF algorithm still has a very high type II error. As presented in Table 6, the \(F_{2}\) scores of the LOF and EE algorithms are almost equal to 1 when the coefficient \(a\) is greater than or equal to 0.9. The IF algorithm has high type I and type II errors and poor \(F_{2}\) performance. It should be noted that the collected dataset may contain profiles with different sizes of coefficient \(a\). The evaluation framework employed in previous relevant studies [33; 55] is based on the assumption that only one size of coefficient \(a\) exists in each dataset. This approach might produce biased estimates of overall performance. The type I error obtained using the method of [PERSON] et al. [33] might be affected by the outlying profiles with the smallest value of coefficient \(a\). In contrast, the method proposed in this study can maintain a stable performance under variation of coefficient \(a\). To verify the aforementioned statement, we conducted an additional experiment.