## 6 Conclusions  
Profile monitoring has attracted considerable research interest recently. Outliers in historical datasets can have detrimental impacts on phase I parameter estimation and \begin{table}
\begin{tabular}{l c c c c c} \hline
**Contamination Rate** & **IF** & \multicolumn{2}{c}{**EE**} & \multicolumn{2}{c}{**LOF**} \\ \hline
0.1 (actual) & 0.013 & (0.005) & 0.001 & (0.002) & 0.001 & (0.002) \\
0.15 (estimated) & 0.059 & (0.003) & 0.056 & (0) & 0.053 & (0.003) \\ \hline
0.2 (actual) & 0.040 & (0.014) & 0.000 & (0.001) & 0.000 & (0.001) \\
0.24 (estimated) & 0.073 & (0.011) & 0.050 & (0) & 0.049 & (0.003) \\ \hline
0.3 (actual) & 0.095 & (0.015) & 0.001 & (0.003) & 0.000 & (0.002) \\
0.33 (estimated) & 0.118 & (0.019) & 0.043 & (0) & 0.041 & (0.003) \\ \hline \end{tabular} Numbers in parentheses denote standard deviations.  
\end{table} 
Table 7: Type I error of the three outlier-detection algorithms when the dataset is contaminated by profiles with different magnitudes of coefficient \(a\). \begin{table}
\begin{tabular}{l c c c c c} \hline
**Contamination Rate** & **IF** & \multicolumn{2}{c}{**EE**} & \multicolumn{2}{c}{**LOF**} \\ \hline
0.1 (actual) & 0.120 & (0.041) & 0.005 & (0.015) & 0.005 & (0.015) \\
0.15 (estimated) & 0.035 & (0.029) & 0 & (0) & 0 & (0) \\ \hline
0.2 (actual) & 0.160 & (0.054) & 0.001 & (0.006) & 0.001 & (0.006) \\
0.24 (estimated) & 0.094 & (0.042) & 0 & (0) & 0 & (0) \\ \hline
0.3 (actual) & 0.223 & (0.034) & 0.001 & (0.004) & 0.001 & (0.004) \\
0.33 (estimated) & 0.174 & (0.044) & 0 & (0) & 0 & (0) \\ \hline \end{tabular} Numbers in parentheses denote standard deviations.  
\end{table} 
Table 8: Type II error of the three outlier-detection algorithms when the dataset is contaminated by profiles with different magnitudes of coefficient \(a\). \begin{table}
\begin{tabular}{l c c c c c} \hline
**Contamination Rate** & **IF** & \multicolumn{2}{c}{**EE**} & \multicolumn{2}{c}{**LOF**} \\ \hline
0.1 (actual) & 0.880 & (0.041) & 0.995 & (0.015) & 0.995 & (0.015) \\
0.15 (estimated) & 0.877 & (0.026) & 0.909 & (0) & 0.913 & (0.005) \\ \hline
0.2 (actual) & 0.840 & (0.054) & 0.999 & (0.006) & 0.999 & (0.006) \\
0.24 (estimated) & 0.871 & (0.040) & 0.962 & (0) & 0.963 & (0.002) \\ \hline
0.3 (actual) & 0.778 & (0.034) & 0.999 & (0.004) & 0.999 & (0.004) \\
0.33 (estimated) & 0.810 & (0.047) & 0.980 & (0) & 0.981 & (0.002) \\ \hline \end{tabular}
\end{table} 
Table 9: \(F_{2}\) score of the three outlier-detection algorithms when the dataset is contaminated by profiles with different magnitudes of coefficient \(a\). phase II monitoring performance. In this study, we presented a phase I analysis procedure for nonlinear profiles by using unsupervised outlier-detection algorithms. The proposed method involves three major processes: profile smoothing, contamination rate estimation, and outlier detection. The performance of the proposed method was evaluated on a nonlinear profile taken from previous studies. The evaluation results indicate that the proposed method is generally more effective than the existing methods for profile monitoring. These improvements are primarily attributed to the accurate estimation of the contamination rate of the dataset. The LOF algorithm outperformed other competing methods in terms of type I error, type II error, and \(F_{2}\) score. When considering the smallest and hardest-to-detect variation, the type I and type II errors 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 also compared the performance of outlier-detection methods and found that the LOF algorithm outperforms the other two algorithms. The performance of the LOF algorithm is promising when evaluated with known or unknown contamination rates. The results indicate that the proposed method can significantly reduce type II errors, which can help practitioners rapidly remove relevant assignable causes.