## 2 Preliminaries
### High Degree F-Transform
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Phase** & **Method** & **Accuracy** & **Precision** & **Recall** & **F1-Score** \\ \hline \multirow{4}{*}{Training} & SVM & 0.68 & 0.68 & 0.67 & 0.67 \\  & RF & 0.70 & 0.68 & 0.66 & 0.67 \\  & ANN & 0.72 & 0.71 & 0.69 & 0.72 \\  & MFC & 0.69 & 0.69 & 0.68 & 0.69 \\  & F1-transform & 0.76 & 0.74 & 0.75 & 0.75 \\ \hline \multirow{4}{*}{Test} & SVM & 0.69 & 0.69 & 0.67 & 0.68 \\  & RF & 0.69 & 0.68 & 0.67 & 0.68 \\ \cline{1-1}  & ANN & 0.72 & 0.73 & 0.72 & 0.72 \\ \cline{1-1}  & MFC & 0.70 & 0.70 & 0.71 & 0.70 \\ \cline{1-1}  & F1-transform & 0.75 & 0.76 & 0.75 & 0.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Red wine dataset: Classification performance comparison. Then, the F\({}^{1}\)-transform classification method is applied to the reduced training set. The comparison results with SVM, RF, ANN, and MFC are shown in Table 7.

Even in this case, the best performances are obtained executing F\({}^{1}\)-transform. The F\({}^{1}\)-transform method improves the classification accuracy obtained by executing ANN by about 4% on the training set and 3% on the test set; it improves the one obtained by executing MFC by about 5% on the training set and 4% on the test set. Table 8 shows, for each dataset, the final classification accuracy and the CPU time measured executing the five algorithms. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Phase** & **Method** & **Accuracy** & **Precision** & **Recall** & **F1-Score** \\ \hline \multirow{4}{*}{Training} & SVM & 0.87 & 0.85 & 0.84 & 0.84 \\  & RF & 0.85 & 0.84 & 0.83 & 0.83 \\  & ANN & 0.88 & 0.86 & 0.86 & 0.86 \\  & MFC & 0.86 & 0.85 & 0.85 & 0.85 \\  & F\({}^{1}\)-transform & 0.93 & 0.89 & 0.90 & 0.89 \\ \hline \multirow{4}{*}{Test} & SVM & 0.87 & 0.86 & 0.85 & 0.85 \\  & RF & 0.85 & 0.83 & 0.82 & 0.67 \\ \cline{1-1}  & ANN & 0.88 & 0.87 & 0.86 & 0.86 \\ \cline{1-1}  & MFC & 0.87 & 0.86 & 0.86 & 0.86 \\ \cline{1-1}  & F\({}^{1}\)-transform & 0.92 & 0.91 & 0.90 & 0.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Adult dataset: Classification performance comparison. Figure 5: Adult training set—Scree plot; the orange line show the elbow point. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Dataset** & **Method** & **SVM** & **RF** & **ANN** & **MFC** & **F\({}^{1}\)-tr** \\ \hline \multirow{2}{*}{Adult} & Accuracy & 0.87 & 0.85 & 0.88 & 0.86 & 0.93 \\  & CPU time (s) & 858.22 & 849.74 & 915.19 & 881.86 & 877.37 \\ \hline \multirow{2}{*}{Balance scale} & Accuracy & 0.95 & 0.92 & 0.96 & 0.95 & 0.98 \\  & CPU time (s) & 26.48 & 26.19 & 28.41 & 27.63 & 26.92 \\ \hline \multirow{2}{*}{Bank Marketing} & Accuracy & 0.77 & 0.75 & 0.78 & 0.77 & 0.82 \\  & CPU time (s) & 832.31 & 823.09 & 944.56 & 867.83 & 864.95 \\ \hline \multirow{2}{*}{Breast cancer} & Accuracy & 0.90 & 0.88 & 0.91 & 0.90 & 0.94 \\  & CPU time (s) & 31.67 & 31.78 & 33.59 & 32.06 & 31.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Classification Accuracy and CPU time comparison results. In all cases the F\({}^{1}\)-transform method improves the final accuracy obtained executing SVM, RS, ANN, and MFC; the final accuracy obtained using F\({}^{1}\)-transform improves that obtained using MFC by a value ranging between 2% (dataset Diabetes) and 7% (dataset Adult). The CPU times employed by executing F\({}^{1}\)-transform are compatible with the ones employed by executing the other four classification algorithms. In fact, even if the construction of the multidimensional F\({}^{1}\)-transform in the proposed algorithm requires more computational expenditure than that necessary to construct the multidimensional F-transform in the MFC algorithm, the application of the PCA algorithm in the preprocessing phase allows for the reduction of the size of the data points, reducing, thus, the computational complexity. The results show that the F\({}^{1}\)-transform classification method improves the classification accuracy of all other classification methods, with CPU times comparable with those of MFC.