# Pattern Classification Based on RBF Networks with Self-Constructing Clustering and Hybrid Learning
## 6 Experimental Results
### Experiment 2--Multi-Label Data Sets  
In this experiment, we show the performance of different methods on multi-label classification datasets. Exact match ratio (EMR), labeling F-measure (LFM), and Hamming loss (HL) are adopted for performance evaluation [57]. Let \(N_{t}\) be the number of patterns. EMR, LFM, and HL are defined as  
\[\text{EMR}=\frac{\sum_{i=1}^{N_{t}}\left(\hat{\mathbf{v}}_{i}^{(i)} \hat{\mathbf{x}}_{i}^{(i)}\right)}{N_{t}}\] \[\text{LFM}=\frac{\sum_{i=1}^{N_{t}}\frac{2\sum_{n=-1}^{N_{t}}\left[ \frac{1+y_{i}^{(i)}}{2}\right)\left(\frac{1+y_{i}^{(i)}}{2}\right)}{2}}{\sum_{ n=-1}^{n}\left[\frac{1+y_{i}^{(i)}}{2}+\frac{1+y_{i}^{(i)}}{2}\right]}\] \[\text{HL}=\frac{\sum_{i=1}^{N_{t}}H_{d}\left(\hat{\mathbf{v}}_{i}^{(i) }\hat{\mathbf{x}}_{i}^{(i)}\right)}{mN_{t}}\]  
where \(I(\mathbf{a},\mathbf{b})=1\) if \(\mathbf{a}\) is identical to \(\mathbf{b}\) and \(I(\mathbf{a},\mathbf{b})=0\) otherwise, and \(H_{d}(\mathbf{a},\mathbf{b})\) is the Hamming distance between \(\mathbf{a}\) and \(\mathbf{b}\). Note that higher values in EMR and LFM, and lower value in HL, indicate a better classification. Several benchmark data sets, taken from the MULAN (Multi-Label Learning) library [30], are used in this experiment. The characteristics of these data sets are shown in Table 7. A pattern may belong to more than one category in average, so the cardinality of a data set can be greater than 1. For instance, the cardinality of \"Yeast-ML\" is 4.237, indicating each pattern belonging to 4.237 categories in average. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Data Set** & \(\mathbf{a}\) & \(\mathbf{\beta}\) & \(\mathbf{\Lambda}\) & & \(\mathbf{\alpha}\) & \(\mathbf{\beta}\) & \(\mathbf{\Lambda}\) \\ \hline Liver Disorders & 4 & 2 & 0 & Heart & 4 & 8 & 0.1 \\ Ionosphere & 4 & 2 & 0 & Sonar & 4 & 8 & 0 \\ Iris & 4 & 4 & 0 & Lung Cancer & 6 & 2 & 0 \\ Soybean & 4 & 2 & 0 & Glass & 2 & 2 & 0.1 \\ Ecoli & 4 & 16 & 0 & Yeast-SL & 4 & 16 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Pre-defined constants for RBF-ISCC for single-label data sets. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Data Set** & **\# Features** & **\# Categories** & **\# Patterns** & **Cardinality** \\ \hline Birds & 260 & 19 & 645 & 1.014 \\ Scene & 294 & 6 & 2407 & 1.074 \\ Emotions & 72 & 6 & 593 & 1.869 \\ Flags & 19 & 7 & 194 & 3.392 \\ Yeast-ML & 103 & 14 & 2417 & 4.237 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Multi-label data sets for experiments. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Data Set** & **LinearSVC** & **SVC-rbf** & **SVM** & **NN** & **RBF** & **RBF-ISCC** \\ \hline Liver Disorders & 0.7179 & 0.6739 & 0.7036 & 0.6134 & 0.5594 & **0.747** \\ Heart & 0.7766 & 0.7766 & **0.7917** & 0.7500 & 0.7029 & 0.7761 \\ Ionosphere & 0.9064 & 0.9162 & 0.9162 & 0.8827 & 0.8468 & **0.9169** \\ Sonar & 0.8221 & 0.8726 & **0.8822** & **0.8822** & 0.8091 & 0.8333 \\ Iris & 0.8889 & **0.9697** & **0.9697** & **0.9697** & **0.9697** & **0.9697** \\ Lung Cancer & 0.4111 & 0.1429 & 0.1111 & **0.6667** & 0.1111 & 0.5000 \\ Soybean & **1.0000** & **1.0000** & **1.0000** & **1.0000** & **1.0000** & **1.0000** \\ Glass & 0.6482 & 0.6799 & 0.5062 & 0.6333 & **0.7361** & 0.7250 \\ Ecoli & **0.9144** & 0.8998 & 0.5905 & 0.7027 & 0.7586 & 0.9139 \\ Yeast-SL & 0.5205 & **0.6989** & 0.2537 & 0.5183 & 0.4755 & 0.7499 \\ \hline Ave-MAPRE & 0.7606 & 0.7621 & 0.6725 & 0.7619 & 0.6971 & **0.8132** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Testing macro-averaged precision (MAPRE) obtained for single-label data sets.