## 2 Methods
### KNN-Is and SMOTE in Big Data
\begin{table}
\begin{tabular}{l c c c c c c c} \hline
**Name** & **Dataset** & \(p\) & **Features** & **TPR** & **TNR** & **AUC** & \(G-mean\) \\ \hline Imbalanced & & – & 24,832 & 0.51402 & 0.97645 & 0.74524 & 0.70846 \\ SMOTE & & 2.00 & 24,832 & 0.85047 & 0.86215 & **0.85631** & **0.85629** \\ \cline{3-8} \(D_{d_{g}}\) & \multirow{2}{*}{NewIndian2} & 0.50 & 1000 & 0.38318 & 0.97800 & 0.68059 & 0.61217 \\  & & 0.33 & 2000 & 0.35748 & 0.98247 & 0.66997 & 0.59263 \\ \cline{3-8} \(D_{d_{g}+\text{SMOTE}}\) & & 1.00 & 500 & 0.91121 & 0.72806 & 0.84664 & 0.84417 \\  & & 0.50 & 2000 & 0.89252 & 0.79477 & 0.84365 & 0.84223 \\ \hline Imbalanced & & – & 24,832 & 0.45977 & 0.98814 & 0.72395 & 0.67403 \\ SMOTE & & 2.00 & 24,832 & 0.75096 & 0.93049 & 0.84073 & 0.83592 \\ \cline{3-8} \(D_{d_{g}}\) & \multirow{2}{*}{NewIndian3} & 0.25 & 50 & 0.36398 & 0.99114 & 0.67756 & 0.60063 \\  & & 0.33 & 2000 & 0.32184 & 0.99566 & 0.68575 & 0.56608 \\ \cline{3-8} \(D_{d_{g}+\text{SMOTE}}\) & & 0.25 & 100 & 0.93870 & 0.84378 & **0.89124** & **0.88997** \\  & & 0.66 & 500 & 0.96169 & 0.81988 & 0.89078 & 0.88796 \\ \hline Imbalanced & & – & 24,832 & 0.61268 & 0.99672 & 0.80470 & 0.78145 \\ SMOTE & & 2.00 & 24,832 & 0.80282 & 0.99397 & 0.88109 & 0.87761 \\ \cline{3-8} \(D_{d_{g}}\) & \multirow{2}{*}{NewIndian5} & 0.33 & 50 & 0.63380 & 0.99394 & 0.81387 & 0.79370 \\  & & 0.75 & 500 & 0.62676 & 0.99492 & 0.81084 & 0.78967 \\ \cline{3-8} \(D_{d_{g}+\text{SMOTE}}\) & & 0.25 & 2000 & 0.90141 & 0.93381 & **0.91761** & **0.91747** \\  & & 0.25 & 500 & 0.88732 & 0.94725 & 0.91729 & 0.91680 \\ \hline Imbalanced & & – & 24,832 & 0.57078 & 0.99121 & 0.78099 & 0.75217 \\ SMOTE & & 2.00 & 24,832 & 0.86758 & 0.96665 & 0.91711 & 0.91578 \\ \cline{3-8} \(D_{d_{g}}\) & \multirow{2}{*}{NewIndian6} & 0.75 & 1000 & 0.58447 & 0.98722 & 0.78585 & 0.75961 \\  & & 0.25 & 2000 & 0.57078 & 0.99121 & 0.78099 & 0.75217 \\ \cline{3-8} \(D_{d_{g}+\text{SMOTE}}\) & & 0.66 & 1000 & 0.96804 & 0.89132 & **0.92968** & **0.92889** \\  & & 0.75 & 2000 & 0.95434 & 0.89182 & 0.92308 & 0.92255 \\ \hline Imbalanced & & – & 24,832 & 0.92373 & 0.99461 & 0.95917 & 0.95852 \\ SMOTE & & 2.00 & 24,832 & 0.94915 & 0.99249 & 0.97082 & 0.97058 \\ \cline{3-8} \(D_{d_{g}}\) & \multirow{2}{*}{NewIndian8} & 0.33 & 500 & 0.95763 & 0.99510 & 0.97637 & 0.97619 \\  & & 0.25 & 500 & 0.94915 & 0.99543 & 0.97229 & 0.97202 \\ \cline{3-8} \(D_{d_{g}}\) + SMOTE & & 1.00 & 50 & 1.00000 & 0.98890 & **0.99454** & **0.99464** \\  & & 0.50 & 500 & 0.99153 & 0.99184 & 0.99168 & 0.99168 \\ \hline Imbalanced & & – & 24,832 & 0.59375 & 0.98456 & 0.78915 & 0.76458 \\ SMOTE & & 2.00 & 24,832 & 0.86111 & 0.91658 & 0.88885 & 0.88841 \\ \cline{3-8} \(D_{d_{g}}\) & \multirow{2}{*}{NewIndian10} & 2.00 & 100 & 0.58333 & 0.97583 & 0.77958 & 0.75448 \\  & & 2.00 & 1000 & 0.58333 & 0.97012 & 0.77673 & 0.75227 \\ \cline{3-8} \(D_{d_{g}+\text{SMOTE}}\) & & 2.00 & 2000 & 0.93056 & 0.89225 & **0.91140** & **0.91120** \\  & & 0.66 & 500 & 0.91319 & 0.90500 & 0.9910 & 0.90909 \\ \hline \end{tabular}
\end{table} 
Table 4: Experimental results. The best _AUC_ and \(G-mean\) results are highlighted in purple and grey for each dataset, respectively. The proposed approach of combining dissimilarity with fractional norms followed by SMOTE (\(D_{d_{p}}\)+SMOTE) demonstrated the most promising performance in terms of both AUC and G-mean. Notably, this improvement was achieved while maintaining a lower dimensionality compared to the original datasets. It demonstrates that integrating these techniques with \(p<1\) can synergistically enhance the model's performance on the minority class. In order to determine whether there are significant differences in the results, we used a nonparametric statistical test called the Friedman test. Additionally, a post-doc test was conducted using [PERSON]'s method to determine which algorithms perform better, equally, or worse.