## 5 Results and Discussion
### Results  
To assess the performance of the proposed method, its performance was compared with those of other methods through simulation. Before moving to the presentation of the results, we describe the choices of parameters of different methods used in this study. The splrep() function from the Scipy library [51] was used to find the B-spline representation of the observed profile data. The key parameters affecting the interpolation include the degree \(d\) of the spline, the number of knots \(t\), and the smoothing condition \(s\). In this study, we fixed \(d\) to 3 (the most commonly used value) and \(t\) to 5 to get a smoother profile. Moreover, we set \(s\) to 0.0 (interpolating). Figure 7 shows the original profile and the smoothed profile for different values of coefficient \(a\). In determining the distance threshold, the current study set the type I error \(\alpha=0.05\) in order to be consistent with the study of [PERSON] et al. [55]. Figure 8 depicts histograms of the calculated distances for different contamination rates. In the normality test (Anderson-Darling test), a \(P\)-value greater than the significance level (usually 0.05) indicated the distances had a normal distribution. The results of the normality test conducted in this study indicated that \(\overline{x}_{d}+Z_{\alpha}\times s_{d}\) was an appropriate distance threshold. Most of the parameters of the three outlier-detection algorithms were set to default values, which could provide a satisfactory performance. For the  
Figure 8: Results of normality test (coefficient \(a=0.7\)). Figure 7: Original profiles and smoothed profiles. (**a**–**d**) are examples of profiles with coefficients \(a\) of 0.5, 0.7, 1.1 and 1.5, respectively. In subplots (**a**–**d**), the blue lines denote the original profiles and red lines represent the smoothed profiles. When the true contamination rates are 0.1, 0.2, and 0.3, the contamination rates estimated using the proposed approach are 0.15, 0.24, and 0.33, respectively. As shown in Table 1, regardless of the contamination rate, when the coefficient \(a\) is equal to 0.7, the type I error of LOF is smaller than that of MHD-C. The type I error of MHD-C is rapidly enhanced when the coefficient \(a\) is large. When the coefficient \(a\) is greater than 0.7, the type I errors of LOF are larger than those of MHD-C. The type I error of the PPOD algorithm approaches 0.05 only when the coefficient \(a\) is equal to 0.7. For other values of coefficient \(a\), the PPOD algorithm yields a stable type I error but is much greater than 0.05. In all cases, the type I errors of the LOF and EE algorithms are close to the prespecified 0.05. The type I error of the IF algorithm is significantly larger than the prespecified 0.05. As mentioned earlier, using PCA-processed data in the IF algorithm did not improve its performance. Taking the case with parameter \(a=0.7\), \(m_{0}=20\) as an example, the type I error of IF is 0.089, while that without PCA is 0.072. Except for the IF algorithm, we notice that the type I errors of other methods start to decrease as \(m_{0}\) becomes larger. From Table 2, it is readily apparent that regardless of the contamination rate and the size of coefficient \(a\), the type II errors of the LOF and EE algorithms are less than or equal to those of MHD-C. LOF and EE perform almost equally well. The PPOD algorithm produces a stable type II error when the coefficient \(a>0.7\), but performs worse than MHD-C when \(a=0.7\). In all outlier-detection methods, IF has the highest type II error. Moreover, the type II errors of the different considered algorithms increase with the contamination rate. Table 3 shows the \(F_{2}\) score of each considered method. For methods MHD-C and PPOD, this study can only use the mean values of the type I and type II errors to estimate the \(F_{2}\) score due to a lack of availability of original data. Therefore, the standard deviation of the \(F_{2}\) value is not provided in Table 3. Several trends can be observed from this table.