## 6 Quantification of Class Overlapping
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline \multicolumn{10}{c}{**Accuracy before Hyper-Tuning**} & \multicolumn{6}{c}{**Precision before Hyper-Tuning**} \\ \hline
**Datasets** & **GB** & **RF** & **DT** & **KNN** & **R-SVM** & **LR** & **GB** & **RF** & **DT** & **KNN** & **R-SVM** & **LR** \\ \hline IRIS & 95.12 & 97.75 & 95.12 & 92.37 & 94.37 & 87.1 & 93.75 & 96.38 & 93.75 & 91.01 & 93.01 & 85.74 \\ Glass & 94.97 & 95.7 & 91.73 & 86.58 & 89.18 & 84.63 & 93.6 & 94.33 & 90.36 & 85.22 & 87.82 & 83.27 \\ HAR & 88.83 & 87.68 & 86.58 & 81.86 & 83.26 & 80.6 & 87.46 & 86.31 & 85.21 & 80.5 & 81.9 & 79.24 \\ B\_Tissue & 96.09 & 96.95 & 93.24 & 87.12 & 88.83 & 80.83 & 94.72 & 95.58 & 91.87 & 85.76 & 87.47 & 79.47 \\ Car & 86.88 & 83.27 & 81.35 & 76.3 & 79.53 & 70.32 & 85.51 & 81.9 & 79.98 & 74.94 & 78.17 & 68.96 \\ CMC & 87.59 & 85.3 & 82.05 & 79.92 & 80.91 & 75.16 & 86.22 & 83.93 & 80.68 & 78.56 & 79.55 & 73.8 \\ Ecoli & 92.36 & 89.27 & 88.76 & 81.94 & 86.75 & 80.33 & 90.99 & 87.9 & 87.39 & 80.58 & 85.39 & 78.97 \\ Nursery & 83.32 & 84.05 & 82.37 & 79.08 & 81.2 & 70.97 & 81.95 & 82.68 & 81 & 77.72 & 79.84 & 69.61 \\ Opt\_Digits & 85.85 & 86.39 & 83.57 & 80.86 & 81.86 & 75.7 & 84.48 & 85.02 & 82.2 & 79.5 & 80.5 & 74.34 \\ Page\_Block & 86.23 & 83.75 & 84.79 & 73.11 & 82.41 & 73.5 & 84.86 & 82.38 & 83.42 & 71.75 & 81.05 & 72.14 \\ Satimage & 85.91 & 84.78 & 85.91 & 75.65 & 83.08 & 72.82 & 84.54 & 83.41 & 84.54 & 74.29 & 81.72 & 71.46 \\ Soyaben & 92.73 & 91.42 & 90.3 & 86.1 & 88.77 & 81.85 & 91.36 & 90.05 & 88.93 & 84.74 & 87.41 & 80.49 \\ Vehicle & 82.88 & 83.88 & 75.74 & 67.05 & 77.42 & 70.76 & 81.51 & 82.51 & 74.37 & 65.69 & 76.06 & 69.4 \\ Volcanoesa & 81.19 & 79.38 & 82.28 & 72.14 & 74.39 & 69.23 & 79.82 & 78.01 & 80.91 & 70.78 & 73.03 & 67.87 \\ Wine & 96.99 & 96.99 & 95.3 & 83.14 & 88.49 & 81.18 & 95.62 & 95.62 & 93.93 & 81.78 & 87. The overlapping level for the majority class negative instances in the overlapping region affects the classification performance and is calculated via Equation (12). \[Overlapping\ Degree(\%)=\frac{Overlapping\ Region}{Majority\ Class\ Area}*100 \tag{12}\]  
The overlapping region is the shared feature space of the majority and minority class samples with similar attributes, and the majority class area is calculated using the Euclidean distance and nearest neighbor rule. A class overlap region for two-class \(C_{i}\) and \(C_{j}\) can be described using Equations (13) and (14). \[Overlapping=if\left(P(x\mid C_{i})\right)\ \geq 0\ then \tag{13}\]  
\[\left(P(x\mid C_{j})\right)\ \geq 0,must\ be,where\ x\ ever\: overlapping\ region\ sample \tag{14}\]  
If the probability density of class \(C_{i}\) is greater than or equal to zero, the same must be true for class \(C_{j}\), where \(i\
eq j\), i.e., the sample of the class \(C_{i}\) has similar characteristics to the sample of class \(C_{j}\). To measure the overlap among the features of a different class in a multiclass dataset, it is necessary to evaluate the discriminative power of the features. If there are any features with discriminative characteristics, the problem is thus considered a simple problem. To measure the overlap among the different classes, we used [PERSON]'s maximum discriminative ratio [75], denoted by \(F1\) and given by Equation (15):  
\[F1=\frac{1}{1+max_{i=1}^{m}rf_{i}} \tag{15}\]  
where \(rf_{i}\) is the discriminative ratio for the feature \(f_{i}\) listed in the dataset. Originally, the largest discriminative ratio value was stored in \(F1\).