## 7 Results and Discussion
### PCA and MRE Performance Comparison with HMM and Proposed EHMM  
In this section, the performance of PCA and MRE is verified for the proposed EHMM and conventional HMM (that is, when the K-MC and GMM components are excluded from the HMM process). The prediction models' performance is shown in Table 4. The table shows that PCA-EHMM and MRE-EHMM outperform PCA-HMM and MRE-HMM, respectively. For example, at \(\chi=80\%\), PCA-EHMM outperforms PCA-HMM by 28.01%, 28.16%, 28.22%, and 28.12% in terms of \(\mathcal{S}_{e}\), \(\mathcal{S}_{p}\), \(\mathcal{P}\), and \(\mathcal{F}1\), respectively. Likewise, at \(\chi=80\%\), MRE-EHMM outperforms MRE-HMM by 28.02%, 28.14%, 28.22%, and 28.13% in terms of \(\mathcal{S}_{e}\), \(\mathcal{S}_{p}\), \(\mathcal{P}\), and \(\mathcal{F}1\), respectively. This indicates that sequentially initialising the HMM process with the K-MC and GMM techniques is vital for improving its performance and the prediction system as a whole. Figure 5: PCA-EHMM F1-score versus dimension performance. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\chi\) (\%)} & **MRE-** & **PCA-** & **MRE-** & **PCA-** & **PCA-** & **MRE-** & **PCA-** & **MRE-** & **PCA-** \\  & **EHMM** & **EHMM** & **EHMM** & **EHMM** & **EHMM** & **EHMM** & **EHMM** & **EHMM** \\ \cline{2-10}  & \multicolumn{2}{c}{\(\mathcal{S}_{e}\) (\%)} & \multicolumn{2}{c}{\(\mathcal{S}_{p}\) (\%)} & \multicolumn{2}{c}{\(\mathcal{P}\) (\%)} & \multicolumn{2}{c}{\(\mathcal{F}1\) (\%)} \\ \hline
70 & 99.68 & 99.69 & 99.70 & 99.70 & 99.62 & 99.63 & 99.65 & 99.66 \\
75 & 99.80 & 99.79 & 99.80 & 99.82 & 99.72 & 99.71 & 99.76 & 99.75 \\
80 & 99.81 & 99.81 & 99.81 & 99.82 & 99.73 & 99.72 & 99.77 & 99.77 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MRE-EHMM and PCA-EHMM performance comparison; \(h=5\) and \(k=9\). This section compares the performance of PCA-EHMM and MRE-EHMM with other relevant ML models in the literature. As highlighted in Section 2, RF, LR, DT, and KNN have been extensively used for detecting credit card frauds. RF combines different decision trees to limit overfitting [33; 34], LR predicts the class probabilities based on some certain dependant variable [22], DT utilises a tree-like model for feature classification [35], and KNN classifies transactions based on the K-nearest data point [36]. Table 5 shows the performance of PCA-EHMM and MRE-EHMM in comparison to the RF, LR, DT, and KNN models. From Table 5, the proposed MRE-EHMM and PCA-EHMM techniques achieved better \(\mathcal{S}_{e}\), \(\mathcal{S}_{p}\), \(\mathcal{P}\), and \(\mathcal{F}1\) performance in comparison to the RF, LR, DT, and KNN models. Furthermore, this improvement in performance is noticeable regardless of changes in \(\chi\). Thus, the proposed MRE-EHMM and PCA-EHMM techniques are more suited to detecting online banking fraudulent transactions, followed by RF, LR, and DT, with KNN having the worst performance. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\chi\) (\%) & **Predictor** & \(\mathcal{S}_{e}\) (\%) & \(\mathcal{S}_{p}\) (\%) & \(\mathcal{P}\) (\%) & \(\mathcal{F}1\) (\%) \\ \hline
70 & MRE-EHMM & 99.68 & 99.70 & 99.62 & 99.65 \\  & MRE-HMM & 71.22 & 70.91 & 71.05 & 71.13 \\  & PCA-EHMM & 99.69 & 99.70 & 99.63 & 99.66 \\  & PCA-HMM & 71.20 & 70.96 & 71.07 & 71.13 \\ \hline
75 & MRE-EHMM & 99.80 & 99.80 & 99.72 & 99.76 \\  & MRE-HMM & 71.65 & 71 61 & 71.44 & 71.54 \\  & PCA-EHMM & 99.79 & 99.82 & 99.71 & 99.75 \\  & PCA-HMM & 71.66 & 71.60 & 71.45 & 71.56 \\ \hline
80 & MRE-EHMM & 99.81 & 99.81 & 99.73 & 99.77 \\  & MRE-HMM & 71.79 & 71.67 & 71.51 & 71.64 \\  & PCA-EHMM & 99.81 & 99.82 & 99.72 & 99.77 \\  & PCA-HMM & 71.80 & 71.66 & 71.50 & 71.65 \\ \hline \end{tabular}
\end{table} 
Table 4: MRE and PCA performance comparison with EHMM and HMM; \(h=5\) and \(k=9\).