## 2 Preliminaries
### High Degree F-Transform
In fact, the classification accuracy of F\({}^{1}\)-transform improves that of MFC by a value between 2% and 7% and the CPU times are similar to those obtained by running MFC. In a nutshell, in all tests performed F\({}^{1}\)-transform produces classification accuracy better than that obtained with other classifiers and compatible execution times. In particular, the results in Table 8 show that:

* The F1-transform classifier improves the accuracy obtained with MFC by a percentage in the range between 2% and 7%;
* Even if the computation times of the multidimensional F\({}^{1}\)-transform are higher than those of the multidimensional F-transform, the proposed classification algorithm has execution times similar to those of MFC. A possible critical point of F\({}^{1}\)-transform consists of the choice of the elbow point in the scree plot obtained by executing the PCA method in the preprocessing phase. In all the tests performed in the scree plot obtained, there is a sudden change in slope which allows the elbow point to be easily recognized. It could happen that this trend change in the scree plot is not so evident, to the point of not allowing the precise definition of the elbow point. In these cases, the solution could be to run F\({}^{1}\)-transform multiple times, each time choosing a different number of main components and selecting the one that produces the greatest accuracy. However, this solution can significantly increase the CPU times of the algorithm. \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}{*}{Diabetes} & Accuracy & 0.73 & 0.73 & 0.76 & 0.75 & 0.77 \\  & CPU time (s) & 44.77 & 43.46 & 47.09 & 45.02 & 44.91 \\ \hline \multirow{2}{*}{Echocardiogram} & Accuracy & 0.75 & 0.73 & 0.75 & 0.74 & 0.78 \\  & CPU time (s) & 31.67 & 30.93 & 33.10 & 31.28 & 31.32 \\ \hline \multirow{2}{*}{Ecoli} & Accuracy & 0.79 & 0.78 & 0.81 & 0.79 & 0.83 \\  & CPU time (s) & 32.65 & 32.38 & 34.32 & 32.77 & 32.82 \\ \hline \multirow{2}{*}{Heart disease} & Accuracy & 0.83 & 0.82 & 0.85 & 0.85 & 0.87 \\  & CPU time (s) & 37.79 & 37.56 & 40.18 & 38.51 & 38.47 \\ \hline \multirow{2}{*}{Hepatitis} & Accuracy & 0.88 & 0.89 & 0.91 & 0.89 & 0.93 \\  & CPU time (s) & 42.61 & 42.49 & 44.26 & 42.68 & 42.73 \\ \hline \multirow{2}{*}{Thyroid disease} & Accuracy & 0.95 & 0.94 & 0.97 & 0.96 & 0.99 \\  & CPU time (s) & 103.34 & 102.85 & 113.93 & 102.98 & 102.87 \\ \hline \multirow{2}{*}{Wine quality—red wine} & Accuracy & 0.68 & 0.70 & 0.72 & 0.69 & 0.76 \\  & CPU time (s) & 67.52 & 66.81 & 69.54 & 68.15 & 68.23 \\ \hline \multirow{2}{*}{Wine quality—white wine} & Accuracy & 0.75 & 0.73 & 0.78 & 0.76 & 0.83 \\  & CPU time (s) & 58.60 & 58.46 & 59.87 & 58.72 & 58.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: _Cont._

## 5 Conclusions

We propose a classification method based on the multidimensional F\({}^{1}\)-transform in which the PCA technique is applied in the preprocessing phase to reduce the size of the features. The main aim of this research is to improve the accuracy of the MFC classification algorithm based on the multidimensional F-transform, without further increasing the processing times. We compared the performances of the proposed classification method both with those of MFC and with those of the well-known SVM, Random Forest, and ANN classification algorithms; the comparative tests were performed on classification datasets of the UCI machine learning repository. The results show that the multidimensional F\({}^{1}\)-transform classification algorithm increases the classification performances obtained executing SVM, Random Forest, ANN, and MFC; in addition, the execution times are comparable with the ones obtained running MFC. We intend to carry out future research to test the use of the classification method based on the multidimensional F\({}^{1}\)-transform on a more extensive and varied set of datasets, adapting the method to the management of ambiguous data points. Conceptualization, B.C. and F.D.M.