## 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{4}{c}{**T\({}^{2}\)-MRCD**} \\ \cline{3-6}  & & **Accuracy** & **AUC** & **FP Rate** & **FN Rate** \\ \hline \multirow{6}{*}{5} & 5 & 0.994 & 0.996 & 0.006 & 0.002 \\ \cline{2-6}  & 10 & 0.994 & 0.995 & 0.007 & 0.003 \\ \cline{2-6}  & 20 & 0.994 & 0.996 & 0.007 & 0.001 \\ \cline{2-6}  & 30 & 0.994 & 0.996 & 0.007 & 0.002 \\ \cline{2-6}  & 40 & 0.643 & 0.556 & 0.007 & 0.882 \\ \cline{2-6}  & 50 & 0.500 & 0.500 & 0.007 & 0.992 \\ \hline \multirow{6}{*}{10} & 5 & 0.994 & 0.997 & 0.006 & 0.000 \\ \cline{2-6}  & 10 & 0.994 & 0.997 & 0.006 & 0.000 \\ \cline{2-6}  & 20 & 0.995 & 0.997 & 0.007 & 0.000 \\ \cline{2-6}  & 30 & 0.995 & 0.996 & 0.007 & 0.000 \\ \cline{2-6}  & 40 & 0.853 & 0.818 & 0.007 & 0.357 \\ \cline{2-6}  & 50 & 0.500 & 0.500 & 0.007 & 0.991 \\ \hline \multirow{6}{*}{20} & 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 & 0.994 & 0.997 & 0.007 & 0.000 \\ \cline{2-6}  & 30 & 0.995 & 0.996 & 0.007 & 0.000 \\ \cline{2-6}  & 40 & 0.964 & 0.957 & 0.007 & 0.079 \\ \cline{2-6}  & 50 & 0.500 & 0.500 & 0.007 & 0.992 \\ \hline \multirow{6}{*}{30} & 5 & 0.994 & 0.997 & 0.006 & 0.000 \\ \cline{2-6}  & 10 & 0.994 & 0.997 & 0.007 & 0.000 \\ \cline{2-6}  & 20 & 0.995 & 0.996 & 0.007 & 0.000 \\ \cline{2-6}  & 30 & 0.995 & 0.997 & 0.007 & 0.000 \\ \cline{2-6}  & 40 & 0.692 & 0.617 & 0.007 & 0.760 \\ \cline{2-6}  & 50 & 0.501 & 0.501 & 0.008 & 0.991 \\ \hline \multirow{6}{*}{50} & 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 & 0.810 & 0.764 & 0.007 & 0.464 \\ \cline{2-6}  & 50 & 0.501 & 0.501 & 0.008 & 0.991 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Performance. Simulation of Robust MRCD-Based \(T^{2}\) Charts. Figure 2 shows the AUC values for all diagrams based on the number of variables where [PERSON]'s \(T^{2}\) charts perform poorly in almost all scenarios. Fast-MCD-based \(T^{2}\) charts can work very well in low dimensional data scenarios where \(p\leq 10\), but when the variables increase, the performance will decrease. Meanwhile, the MRCD-based \(T^{2}\) charts can overcome the weaknesses of previous diagrams where the proposed diagram can consistently deal with high-dimensional data conditions to reach \(p=50\). To illustrate the comparative performance of the three methods, visual representations for \(p=5\), \(20\), \(50\) are presented in Figures 3-5. A representative scenario, considered a potential outlier breakpoint, namely \(p=20\), with a \(30\%\) proportion of outliers. Figure 4 shows that the MRCD-based \(T^{2}\) charts can detect outliers well, while the conventional \(T^{2}\) and Fast-MCD-based \(T^{2}\) charts are less effective in identifying and detecting outliers. Figure 3: Statistics Plot of Simulation on \(p=5\) using (**a**) \(T^{2}\) (**b**) \(T^{2}\)-FMCD (**c**) \(T^{2}\)-MRCD. Figure 2: AUC Performance for All Charts Based on Number of Variables (**a**) \(p=5\), (**b**) \(p=10\), (**c**) \(p=20\), (**d**) \(p=30\), (**e**) \(p=40\) (**f**) \(p=50\).