## 5 Results and Discussions  
### Simulation Study  
Performance evaluation of control charts can be done by conducting a simulation study of outlier detection. The outlier detection method is carried out by simulating 1000 generated data containing outlier proportions ranging from 5%, 10%, 20%, 30%, 40%, and 50%, with the criteria being normal data \(X_{normal}\sim N_{p}(\mathbf{0},\mathbf{I})\) and outlier data \(\mathbf{X}_{out}\sim N_{p}(\mathbf{3},\mathbf{I})\) as implemented to detect small outliers [22]. The number of variables generated is also regulated under several conditions, namely \(p=5\), \(p=10\), \(p=20\), \(p=30\), \(p=40\), and \(p=50\). The performance evaluation encompassed three types of multivariate charts, specifically the conventional Hotelling's \(T^{2}\), the robust Hotelling's \(T^{2}\) based on Fast-MCD, and the robust Hotelling's \(T^{2}\) based on MRCD as proposed. Outlier detection in conventional Hotelling's \(T^{2}\) control charts is carried out with detailed results obtained in Table 3. Table 3 shows the performance of Accuracy, AUC, FP rate, and FP rate of Hotelling's \(T^{2}\) control charts with bootstrap control limits in detecting outliers for each number of variables and various percentages of outliers added according to the procedures. Regarding performance, the conventional \(T^{2}\) charts only work pretty well with an AUC of more than 0.75 and an FN Rate of less than 0.5 on data with a 5% outlier percentage for low dimensions (\(p=5\)) and medium dimension (\(p=10\)). In contrast, performance decreases drastically in high dimensional data (\(p=30,\ 40,\ 50\)). In addition, data with outlier percentages of 10%, 20%, 30%, 40%, and 50% for any dimension all indicate poor performance. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(p\)} & \multirow{2}{*}{\%**Out**} & \multicolumn{4}{c}{\(T^{2}\)**-Conventional**} \\ \cline{3-6}  & & **Accuracy** & **AUC** & **FP Rate** & **FN Rate** \\ \hline \multirow{8}{*}{5} & 5 & 0.978 & 0.833 & 0.006 & 0.327 \\ \cline{2-6}  & 10 & 0.914 & 0.595 & 0.007 & 0.804 \\ \cline{2-6}  & 20 & 0.801 & 0.513 & 0.007 & 0.967 \\ \cline{2-6}  & 30 & 0.701 & 0.506 & 0.007 & 0.982 \\ \cline{2-6}  & 40 & 0.600 & 0.502 & 0.007 & 0.988 \\ \cline{2-6}  & 50 & 0.500 & 0.500 & 0.007 & 0.992 \\ \hline \multirow{8}{*}{10} & 5 & 0.972 & 0.782 & 0.006 & 0.430 \\ \cline{2-6}  & 10 & 0.905 & 0.553 & 0.006 & 0.888 \\ \cline{2-6}  & 20 & 0.800 & 0.510 & 0.007 & 0.972 \\ \cline{2-6}  & 30 & 0.699 & 0.504 & 0.007 & 0.985 \\ \cline{2-6}  & 40 & 0.599 & 0.501 & 0.007 & 0.991 \\ \cline{2-6}  & 50 & 0.500 & 0.500 & 0.007 & 0.991 \\ \hline \multirow{8}{*}{20} & 5 & 0.963 & 0.685 & 0.006 & 0.624 \\ \cline{2-6}  & 10 & 0.901 & 0.529 & 0.007 & 0.936 \\ \cline{1-1} \cline{2-6}  & 20 & 0.798 & 0.506 & 0.007 & 0.981 \\ \cline{1-1} \cline{2-6}  & 30 & 0.699 & 0.502 & 0.007 & 0.988 \\ \cline{1-1} \cline{2-6}  & 40 & 0.599 & 0.501 & 0.007 & 0.992 \\ \cline{1-1} \cline{2-6}  & 50 & 0.500 & 0.500 & 0.007 & 0.991 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance. Simulation of Conventional Hotelling’s \(T^{2}\) Charts. Then, the outlier detection in robust Fast-MCD-based Hotelling's \(T^{2}\) control charts is carried out, and detailed results are obtained in Table 4. Table 4 presents the performance of Accuracy, AUC, FP rate, and FP rate of robust Fast-MCD-based Hotelling's \(T^{2}\) control charts with bootstrap control limits in detecting outliers for each number of variables and various percentages of outliers added according to the procedures before.