## 2 Preliminaries
### High Degree F-Transform
For brevity, we show the complete results obtained for three classification datasets: the two datasets Red and White Wine Quality and the dataset Adult. ### Red and White Wine Quality Classification Dataset

The red and white wine quality datasets are two classification datasets used to classify the quality of red and white variants of the Portuguese vinho verde wine. The data points are vectors given by 11 physicochemical features. In Table 3 all the features are described. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Dataset** & **Number of Data Points** & **Number of Features** \\ \hline Adult & 48,842 & 14 \\ Balance Scale & 625 & 4 \\ Bank Marketing & 41,188 & 17 \\ Breast cancer & 286 & 9 \\ Echocardiogram & 132 & 12 \\ Ecoli & 336 & 7 \\ Heart disease & 303 & 14 \\ Hepatitis & 155 & 19 \\ Thyroid disease & 7200 & 21 \\ Wine quality—red wine & 1599 & 12 \\ Wine quality—white wine & 4898 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Datasets in the UCI machine learning datasets used in the comparison analysis. The last feature, called quality, is the output feature containing the class of wine quality in a scale between 0 (very bad) to 10 (excellent). The white wine quality dataset contains 4898 data points, and the red wine quality dataset contains 1599 data points. The two datasets are unbalanced, with over 70% of the data points classified with quality five and six. Paragraphs 4.1.1 and 4.1.2 show the results obtained, respectively, for the white and for the red wine quality datasets. #### 4.1.1 White Wine Quality Dataset--Classification Results

Now we show the results obtained on the white wine-quality dataset. In the preprocessing phase, the PCA algorithm was executed on the training set. In Figure 3 the scree plot is shown. The orange line highlights the elbow point. In the transformed training set, the size of the data points is reduced from eleven to five. Then, the F\({}^{1}\)-transform classification algorithm is executed on the transformed training set. Finally, the stored final F\({}^{1}\)-transform coefficients are used to classify the data points in the transformed test set. We compare the classification results with the ones obtained executing SVM, RF, ANN, and MFC on the original training set. Table 4 shows the accuracy, precision, recall, and F1-score measures computed on the training and test sets executing the five classification methods. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Feature Name** & **Description** & **Type of Field** \\ \hline fixed acidity & tartaric acid (g/dm\({}^{3}\)) & Continuous \\ volatile acidity & Acetic acid (g/dm\({}^{3}\)) & Continuous \\ citric acid & Citric acid (g/dm\({}^{3}\)) & Continuous \\ residual sugar & Residual sugar (g/dm\({}^{3}\)) & Continuous \\ chlorides & Sodium chloride(g/dm\({}^{3}\)) & Continuous \\ free sulfur dioxide & Free sulfur dioxide (mg/dm\({}^{3}\)) & Continuous \\ total sulfur dioxide & Total sulfur dioxide (mg/dm\({}^{3}\)) & Continuous \\ density & Density (g/cm\({}^{3}\)) & Continuous \\ pH & pH & Continuous \\ sulphates & Potassium sulphate (g/dm\({}^{3}\)) & Continuous \\ alcohol & Alcohol (vol.%) & Continuous \\ quality & Wine quality & List (an integer from 0 to 10) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Red and white wine datasets: Description of the features. Figure 3: Scree plot obtained for the white wine quality training set; the orange line shows the elbow point. The results in Table 3 show that the best performances are obtained by executing the F\({}^{1}\)-transform classification method. The F\({}^{1}\)-transform method improves the classification accuracy obtained by executing ANN by about 5% on the training and test sets; it improves the one obtained by executing MFC by about 7% on the training set and 6% on the test set. #### 4.1.2 Red Wine Quality Dataset--Classification Results

The PCA algorithm was executed on the training set. Figure 4 shows the scree plot. The orange line highlights the elbow point.