## 5 Results and Discussions
### Simulation Study
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(p\)} & \multirow{2}{*}{**\%Out**} & \multicolumn{5}{c}{**\(T^{2}\)-FMCD**} \\ \cline{3-6}  & & **Accuracy** & **AUC** & **FP Rate** & **FN Rate** \\ \hline \multirow{5}{*}{5} & 5 & 0.994 & 0.996 & 0.007 & 0.001 \\ \cline{2-6}  & 10 & 0.994 & 0.996 & 0.007 & 0.002 \\ \cline{2-6}  & 20 & 0.994 & 0.996 & 0.007 & 0.001 \\ \cline{2-6}  & 30 & 0.995 & 0.996 & 0.007 & 0.001 \\ \cline{2-6}  & 40 & 0.995 & 0.995 & 0.007 & 0.001 \\ \cline{2-6}  & 50 & 0.500 & 0.500 & 0.007 & 0.992 \\ \hline \multirow{5}{*}{10} & 5 & 0.994 & 0.997 & 0.007 & 0.000 \\ \cline{2-6}  & 10 & 0.994 & 0.997 & 0.007 & 0.000 \\ \cline{2-6}  & 20 & 0.995 & 0.997 & 0.007 & 0.000 \\ \cline{2-6}  & 30 & 0.995 & 0.997 & 0.007 & 0.000 \\ \cline{2-6}  & 40 & 0.861 & 0.828 & 0.007 & 0.337 \\ \cline{2-6}  & 50 & 0.500 & 0.500 & 0.007 & 0.991 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance. Simulation of Robust Fast-MCD-Based \(T^{2}\) Charts. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(p\)} & \multirow{2}{*}{**\%Out**} & \multicolumn{5}{c}{**\(T^{2}\)-Conventional**} \\ \cline{3-6}  & & **Accuracy** & **AUC** & **FP Rate** & **FN Rate** \\ \hline \multirow{5}{*}{30} & 5 & 0.939 & 0.509 & 0.013 & 0.970 \\ \cline{2-6}  & 10 & 0.892 & 0.508 & 0.012 & 0.972 \\ \cline{2-6}  & 20 & 0.794 & 0.505 & 0.013 & 0.978 \\ \cline{2-6}  & 30 & 0.699 & 0.503 & 0.007 & 0.988 \\ \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{5}{*}{40} & 5 & 0.954 & 0.596 & 0.007 & 0.801 \\ \cline{2-6}  & 10 & 0.899 & 0.520 & 0.007 & 0.954 \\ \cline{2-6}  & 20 & 0.798 & 0.505 & 0.006 & 0.983 \\ \cline{2-6}  & 30 & 0.698 & 0.502 & 0.007 & 0.989 \\ \cline{2-6}  & 40 & 0.599 & 0.501 & 0.007 & 0.991 \\ \cline{2-6}  & 50 & 0.501 & 0.501 & 0.008 & 0.991 \\ \hline \multirow{5}{*}{50} & 5 & 0.952 & 0.576 & 0.006 & 0.841 \\ \cline{2-6}  & 10 & 0.897 & 0.514 & 0.007 & 0.966 \\ \cline{2-6}  & 20 & 0.797 & 0.504 & 0.007 & 0.986 \\ \cline{2-6}  & 30 & 0.699 & 0.503 & 0.007 & 0.988 \\ \cline{2-6}  & 40 & 0.599 & 0.501 & 0.007 & 0.991 \\ \cline{2-6}  & 50 & 0.500 & 0.500 & 0.008 & 0.993 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Based on the outlier detection, detailed results are obtained in Table 4. In terms of performance, robust Fast-MCD based Hotelling's \(T^{2}\) control charts work very well for conditions of low-dimensional (\(p=5\)) and medium dimensions (\(p=10\)) data, which is indicated by an AUC of more than 0.99 and an FN Rate below 0.01. The same good results can still be achieved in high dimensional conditions, but only at outlier percentages of \(5\%\) and \(10\%\). Meanwhile, in high-dimensional data, the outlier percentage is more than \(10\%\), and there is a drastic decrease in performance, where the AUC value is around 0.5, and the FN Rate is above 0.97. This constraint is based on previous research stating that the greater the observed quality characteristics, the lower the performance of the \(T^{2}\) charts. This limitation also applies to the robust Fast-MCD estimator and cannot be resolved entirely. Finally, the outlier detection in robust MRCD-based Hotelling's \(T^{2}\) control charts is carried out, and detailed results are obtained in Table 5. Table 5 presents the performance of Accuracy, AUC, FP rate, and FP rate of robust MRCD-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. Based on the outlier detection, detailed results are obtained in Table 5. In terms of performance, the robust MRCD-based Hotelling's \(T^{2}\) control charts work very well for almost all conditions, as shown by an AUC of more than 0.99 and an FN Rate of below 0.01. This diagram only experiences problems: all data with an outlier percentage of 50% and a slight decrease in data