# Visual Data and Pattern Analysis for Smart Education:
## 4 Proposed Methods
### Federated Learning-Based Authentication
Figure 4: Precision vs. recall. \(\mathcal{B}\) represents the proportion of true negatives (class 0) that the classifier incorrectly predicts as positives and can be calculated using the following equation:

\[\mathcal{B}=\frac{\tau}{\tau+\bar{t}} \tag{22}\]

At a false negative rate (FNR) of 0.1, PCT has a true negative rate (TNR) of 50 (1-0.50) and gradually enhances its TNR as the FNR increases, reaching 0.05 at a FNR of 1.0. MST-FaDe has a true negative rate (TNR) of 49 (1-0.51) at a false negative rate (FNR) of 0.1. Its TNR steadily improves as the FNR rises, reaching 0.05 with a false negative rate of 1.0 at an FNR of 0.1. Fuzzy SVM has a TNR of 0.48 (1-0.52) and gradually enhances its TNR as the FNR increases, reaching 0.05 at an FNR of 1.0. AISAR exhibits a similar pattern, beginning with a slightly higher TNR of 0.47 (1-0.53) at an FNR of 0.1 and improving to 0.04 at an FNR of 1.0. The proposed model starts with the highest initial TNR of 0.46 (1-0.54) at an FNR of 0.1 and consistently outperforms the other models, achieving the highest TNR of 0.01 at an FNR of 1.0. Table 5 represents the numerical results and Figure 5 illustrates the trade-off between TNR and FNR for each of the five models, demonstrating that the proposed model optimally balances minimising false negatives and maximising true negatives. #### 5.1.5 True Positive Rate vs. False Positive Rate

The true positive rate (\(\mathfrak{A}\)), also recognised as sensitivity or recall, and the false positive rate (\(\mathfrak{s}\)) are critical metrics in binary classification utilised to evaluate a classifier's performance. Both rates can be derived using the confusion matrix. \(\mathfrak{A}\) signifies the proportion of actual positives (class 1) that are accurately predicted as positives, as detailed in Equation (20). On the other hand, \(\mathfrak{s}\) represents the proportion of actual negatives (class 0) that the classifier erroneously predicts as positives. The relationship for \(\mathfrak{s}\) can be described as:

\[\mathfrak{s}=\frac{\mathfrak{f}}{\mathfrak{f}+\mathfrak{r}} \tag{23}\]

In practical terms, PCT starts with a true positive rate (TPR) of 0.45 and gradually elevates the TPR with a rising false positive rate (FPR), achieving 0.77 at an FPR of 1.0. The true positive rate (TPR) of MST-FaDe is initially set at 0.49, and it is progressively increased with an increasing false positive rate (FPR) to reach 0.84 at an FPR of 1.0. Fuzzy SVM starts with a TPR of 0.52 and gradually elevates the TPR with a rising FPR, achieving 0.90 at an FPR of 1.0. AISAR exhibits a similar trajectory, initiating with a TPR of 0.53 at an FPR of 0.1 and escalating to 0.91 at an FPR of 1.0. The proposed model begins with a TPR of 0.54 at an FPR of 0.1, consistently outperforming the prior models, and achieves the highest TPR of 0.98 at an FPR of 1.0. Table 6 represents the numerical outcomes, and Figure 6 illustrates the trade-off between TPR and FPR for all models, demonstrating that the proposed model consistently sustains a higher TPR while managing FPR, indicative of enhanced classification task performance. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**False Positive**} & \multicolumn{5}{c}{**True Positive Rate**} \\ \cline{2-6}  & **PCT** & **MST-FaDe** & **Fuzzy SVM** & **AISAR** & **Proposed** \\ \hline
0.1 & 0.45 & 0.49 & 0.52 & 0.53 & 0.54 \\ \hline
0.2 & 0.47 & 0.52 & 0.62 & 0.66 & 0.69 \\ \hline
0.3 & 0.50 & 0.65 & 0.74 & 0.77 & 0.79 \\ \hline
0.4 & 0.53 & 0.70 & 0.75 & 0.78 & 0.80 \\ \hline
0.5 & 0.58 & 0.72 & 0.76 & 0.79 & 0.81 \\ \hline
0.6 & 0.62 & 0.74 & 0.77 & 0.81 & 0.85 \\ \hline
0.7 & 0.65 & 0.76 & 0.81 & 0.83 & 0.86 \\ \hline
0.8 & 0.70 & 0.78 & 0.82 & 0.84 & 0.92 \\ \hline
0.9 & 0.74 & 0.80 & 0.86 & 0.87 & 0.96 \\ \hline
1 & 0.77 & 0.84 & 0.90 & 0.91 & 0.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Numeric results of true positive rates. Figure 6: True positive rates vs. false positive rates. #### 5.1.6 Specificity vs.