## 4 Experimental Settings and Results
### Feasibility Validation of MAF-NBC  
A first experiment was performed to validate the feasibility of the MAF-NBC method by checking the convergence of ANN weights by the iterative update process. This experiment was conducted on the representative _Vowel_ data set, which has three discrete attributes and seven continuous attributes. The experimental results are the average values corresponding to 10 independent ANN training. Ten ANNs with 50 hidden-layer nodes were constructed with random weights in the \([-1,\,1]\) range, a balance factor \(\lambda=0.50\), and learning rates \(\xi=\zeta=0.01\). Figure 2 depicts the variation trends of the 1-norms for the input layer and output layer weights as the iteration number increased. Figure 3  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline Data sets & Abalone\_3\_4 & Abalone\_9\_10 & Adult * & Band * \\ Number of samples & 423 & 1073 & 5000 & 540 \\ Number of continuous attributes & 6 & 6 & 9 & 18 \\ Number of discrete attributes & 2 (2, 4) & 2 (2, 4) & 5 (5, 7, 5, 4, 2) & 5 (3, 5, 2, 3, 3) \\ Number of classes & 2 & 2 & 2 & 2 \\ \hline data sets & Bd & Bp & Heart & Ionosphere \\ Number of samples & 569 & 198 & 270 & 351 \\ Number of continuous attributes & 23 & 25 & 9 & 25 \\ Number of discrete attributes & 7 (4, 6, 5, 3, 2, 4, 3) & 8 (6, 3, 5, 4, 5, 3, 4, 2) & 4 (4, 3, 3, 4) & 7 (4, 6, 5, 3, 4, 3, 5) \\ Number of classes & 2 & 2 & 2 & 2 \\ \hline Data sets & Libras & Page blocks & Parkinsons & Ring \\ Number of samples & 360 & 547 & 195 & 500 \\ Number of continuous attributes & 85 & 7 & 17 & 15 \\ Number of discrete attributes & 15 (4, 5, 7, 5, 8, 4, 5, 6, 4, 5, 6, 7, 3, 4, 5) & 3 (3, 5, 4) & 5 (4, 5, 6, 5, 3) & 5 (3, 4, 5, 3, 4) \\ Number of classes & 15 & 5 & 2 & 2 \\ \hline Data sets & Segment & Sonar & SPECTF & Vehicle \\ Number of samples & 2310 & 208 & 267 & 846 \\ Number of continuous attributes & 14 & 50 & 36 & 6 \\ Number of discrete attributes & 5 (3, 6, 5, 4, 3) & 10 (6, 4, 3, 5, 6, 7, 5, 3, 4, 6) & 8 (4, 6, 5, 5, 3, 6, 7, 4) & 2 (4, 3) \\ Number of classes & 7 & 2 & 2 & 4 \\ \hline Data sets & Vowel & Wine & WineQR & WineQW \\ Number of samples & 528 & 178 & 1599 & 489 \\ Number of continuous attributes & 7 & 10 & 8 & 8 \\ Number of discrete attributes & 3 (3, 3, 4) & 3 (4, 5, 4) & 3 (4, 4, 3) & 3 (3, 3, 3) \\ Number of classes & 11 & 3 & 6 & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Details of 20 mixed-attribute data sets.  
presents the variation trends of encoding error and attribute dependence as the iteration number increased.  
Figure 2 shows that the sums of absolute values corresponding to the input layer and output-layer weights become more and more stable with the number of iterations. In addition, Figure 3 shows that the encoding error and attribute dependence decrease gradually and then remain unchanged. These experimental results reveal that the updating rules of Equations (15) and (16) are effective for the determination of optimal ANN weights.  
In fact, Equation (17) is the approximation of the expected loss \(\mathbb{E}\overline{\chi}\left[\frac{\partial\mathbb{L}(\overline{\chi}_{A, \theta})}{\partial\mathbb{E}_{A,t}}\right],\) where \(\overline{\chi}\) is the domain of the objective function \(\mathbb{L}(\bullet,\alpha,\beta)\). Although the gradient of input layer weights cannot be analytically calculated due to the existence of the attribute dependence term, Equation (17) uses the mean of the objective function values corresponding to all samples in the one-hot encoded attribute data set to approximate the gradient. The convergence of output layer weights guarantees the convergence of input layer weights, because the update of \(\alpha_{rl}\) depends on the update of \(\beta_{lr}\). This experiment demonstrates the feasibility of transforming the one-hot encoded attributes into independent encoded attributes.