## 2 Preliminaries
### High Degree F-Transform
The components of the F1-transform (18) and (19) are given by:  
\[c_{\text{h}_{1}\text{h}_{2}\text{...h}_{n}}^{0}=F_{\text{h}_{1}\text{h}_{2} \text{...h}_{n}}=\frac{\sum_{\text{j}=1}^{\text{N}}\text{m}_{\text{j}}\cdot \text{A}_{\text{1h}_{1}}(\text{P}_{\text{j1}})\cdot\text{A}_{\text{2h}_{2}}( \text{P}_{\text{j2}})\cdot\ldots\cdot\text{A}_{\text{nh}_{n}}(\text{P}_{\text {jn}})}{\sum_{\text{j}=1}^{\text{N}}\text{A}_{\text{1h}_{1}}(\text{P}_{\text{ j1}})\cdot\text{A}_{\text{2h}_{2}}(\text{P}_{\text{j2}})\cdot\ldots\cdot\text{A}_{ \text{nh}_{n}}(\text{P}_{\text{jn}})} \tag{20}\]  
\[c_{\text{h}_{1}\text{h}_{2}\text{...h}_{n}}^{1\text{s}}=\frac{\sum_{\text{j}=1 }^{\text{N}}\text{m}_{\text{j}}\cdot(\text{P}_{\text{js}}-\text{x}_{\text{h}_ {\text{s}}})\cdot\text{A}_{\text{1h}_{1}}(\text{P}_{\text{j1}})\cdot\text{A}_ {\text{2h}_{2}}(\text{P}_{\text{j2}})\cdot\ldots\cdot\text{An}_{\text{nh}_{n} }(\text{P}_{\text{jn}})}{\sum_{\text{j}=1}^{\text{N}}(\text{P}_{\text{js}}- \text{x}_{\text{h}_{\text{s}}})^{2}\cdot\text{A}_{\text{1h}_{1}}(\text{P}_{ \text{j1}})\cdot\text{A}_{\text{2h}_{2}}(\text{P}_{\text{j2}})\cdot\ldots\cdot \text{A}_{\text{nh}_{n}}(\text{P}_{\text{jn}})} \tag{21}\]  
Let \(f_{\text{F},\text{n}}^{1}(\text{P}_{\text{j}})\) the inverse F1-transform computed by (16). The index of the class assigned to the data point \(\text{P}_{\text{j}}\) is given by an integer \(\text{m}_{\text{j}}\) in {1,2,...,C} given by:  
\[\text{m}_{\text{j}}=\left[\min_{\text{m}=1,\ldots,\text{C}}\left(\left|\text{ f}_{\text{F},\text{n}}^{1}(\text{P}_{\text{j}})-\text{m}\right|\right)\right] \tag{22}\]  
where [a] stands for the greatest integer containing the positive real number a. The classification accuracy CA is given by the ratio between the number of data points correctly classified and the total number of data points. If CA is less than the threshold \(\theta\), then finer fuzzy partitions with n + 1 fuzzy sets are created and the process is iterated. Otherwise, the F1-transform classifier ends, storing the coefficients of the final direct F1-transform: \(c_{\text{h}_{1}\text{h}_{2}\text{...h}_{n}}^{0}\) and \(c_{\text{h}_{1}\text{h}_{2}\text{...h}_{n}}^{1\text{1}}\cdots c_{\text{h}_{1} \text{h}_{2}\text{...h}_{n}}^{1\text{n}}\) computed, respectively by (18) and (19). A new input data x = (x1, x2,..., xn) will be classified computing the inverse F1-transform \(f_{\text{F},\text{n}}^{1}(\text{x})\) by (16) and assigning it the w_th_ class by (20). The F1-transform classification algorithm is schematized below (Algorithm 1):  
```
1. Execute the PCA feature reduction algorithm
2. Use the Scree plot method to reduce the number of features
3. Create the reduced dataset
4. Set the accuracy threshold \(\theta\)
5. n=3
6. CA:=0 // initialize the CA index to 0
7. While CA < 0
8. Create the n-size fuzzy partitions of the feature domains

## 4 Experimental Results

We tested the F1-transform classification method on various classification datasets extracted from the UC Irvine Machine Learning Repository. Table 2 shows the size and the number of features of the classification datasets used in these comparison tests. To measure the classification performances, we compared the F1-transform method with the Support Vector Machine (SVM) [24], Random Forest (RF) [25], Artificial Neural Network (ANN) [26], and the MFC classification algorithm (MFC) [14]. Our comparison tests were executed using an Intel Core I7 processor having a 5.4 GHz clock frequency. After removing ambiguous data points, each dataset was randomly segmented into a training and a test set, containing 80% and 20% of the data points, respectively. While running RF the number of decision trees is set to 100. The Radial Basis kernel function is used to execute SVM, setting the parameters c and gamma, respectively, to 1.0 and 0.1. ANN is executed by constructing a network having two hidden layers, and setting to 100 the maximum number of epochs.