# Visual Data and Pattern Analysis for Smart Education:
## 4 Proposed Methods
### Federated Learning-Based Authentication
#### 5.1.3 Precision vs. Recall Rate

A single equation can succinctly represent the relationship between precision and recall as:

\[\mathfrak{E}=\frac{\mathfrak{k}}{\mathfrak{k}+\mathfrak{r}} \tag{19}\]

Here, precision \((\mathfrak{E})\) is a metric representing the accuracy of a classification model, defined as the ratio of true positives (\(\mathfrak{k}\)) to the total predicted positives (\(\mathfrak{k}+\mathfrak{r}\)). True positives (\(\mathfrak{k}\)) refer to instances correctly identified as positive by the model, while false positives (\(\mathfrak{r}\)) denote instances that are incorrectly labelled as positive. In contrast, \(\mathfrak{K}\) can be expressed using a different equation:

\[\mathfrak{K}=\frac{\mathfrak{k}}{\mathfrak{k}+\mathfrak{I}} \tag{20}\]

Here, recall \(\mathfrak{K}\) of a classification model is represented as the ratio of true positives to total actual positives (\(\mathfrak{k}+\mathfrak{I}\)). False negatives (3) are instances that the model fails to identify as positive. Table 4 illustrates the numerical outcomes and Figure 4 illustrates the relationship between precision and recall. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Number of**} & \multicolumn{5}{c}{**F1-Score**} \\ \cline{2-6}  & **PCT** & **MST-FaDe** & **Fuzzy SVM** & **AISAR** & **Proposed** \\ \hline
5 & 42 & 43 & 45 & 46 & 50 \\ \hline
10 & 43 & 45 & 48 & 49 & 51 \\ \hline
15 & 45 & 48 & 52 & 53 & 56 \\ \hline
20 & 48 & 52 & 57 & 59 & 61 \\ \hline
25 & 52 & 57 & 62 & 63 & 65 \\ \hline
30 & 57 & 62 & 65 & 66 & 71 \\ \hline
35 & 62 & 65 & 72 & 74 & 76 \\ \hline
40 & 65 & 72 & 76 & 77 & 80 \\ \hline
45 & 72 & 76 & 79 & 80 & 88 \\ \hline
50 & 76 & 79 & 80 & 81 & 95 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numeric results of F1-score. Figure 3: Number of epochs vs F1-Score. PCT initiates with a low precision of 0.08, gradually improving precision as the recall rate rises, reaching an accuracy of 0.85 and a recall rate of 1.0; MST-FaDe initiates with a low precision of 0.10, increasing accuracy over time while maintaining a constant recall rate, reaching an accuracy of 0.90 and a recall rate of 1.0; Fuzzy SVM initiates with a low precision of 0.11, steadily enhancing precision as the recall rate ascends, achieving an accuracy of 0.95 and a recall rate of 1.0. AISAR begins with a reasonable precision of 0.12, which rapidly improves alongside the recall rate, attaining a precision of 0.96 at a recall rate of 1.0. The proposed model demonstrates the highest precision among the five models, commencing at 0.13 and progressively improving, culminating in a precision of 0.99 at a recall rate of 1.0. #### 5.1.4 True Negative Rate vs. False Negative Rate

The true negative rate (\(\mathcal{A}\)) and false negative rate (\(\mathcal{B}\)) are pivotal metrics used in binary classification to assess a classifier's performance. Typically, these rates are computed using the confusion matrix, which is a table that summarises the classifier's predictions. \(\mathcal{A}\), Specificity, represents the fraction of actual negatives (class 0) that the classifier correctly predicts as negatives. Here is the \(\mathcal{A}/\mathcal{B}\) equation, along with a brief explanation:

\[\mathcal{A}=\frac{\mathcal{Z}}{\mathcal{Z}+\tau} \tag{21}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Number of**} & \multicolumn{5}{c}{**Precision**} \\ \cline{2-6}  & **PCT** & **MST-FaDe** & **Fuzzy SVM** & **AISAR** & **Proposed** \\ \hline
0.1 & 0.08 & 0.10 & 0.11 & 0.12 & 0.13 \\ \hline
0.2 & 0.19 & 0.21 & 0.24 & 0.25 & 0.28 \\ \hline
0.3 & 0.25 & 0.29 & 0.31 & 0.35 & 0.39 \\ \hline
0.4 & 0.35 & 0.39 & 0.42 & 0.45 & 0.48 \\ \hline
0.5 & 0.45 & 0.48 & 0.51 & 0.53 & 0.59 \\ \hline
0.6 & 0.55 & 0.59 & 0.62 & 0.65 & 0.69 \\ \hline
0.7 & 0.68 & 0.70 & 0.73 & 0.76 & 0.79 \\ \hline
0.8 & 0.75 & 0.78 & 0.81 & 0.86 & 0.89 \\ \hline
0.9 & 0.77 & 0.80 & 0.82 & 0.87 & 0.90 \\ \hline
1 & 0.85 & 0.90 & 0.95 & 0.96 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numeric results of precision.