## 5 Experimental Results
### Evaluation Methods  
In the framework of imbalanced data-sets, standard metrics such as overall accuracy are not the most appropriate, since they do not distinguish between the classification rates of different classes, which might lead to erroneous conclusions [45]. Therefore we adopt _minimum accuracy per class_, _F-measure_, _average accuracy_ and _diversity_ as performance measures in our experiments. * **Recall**, also called per class accuracy, is the percentage of instances correctly classified in each class. [10] strongly recommends using the dedicated performance measure _Recall_ to evaluate classification algorithms, especially when dealing with multi class imbalance problems. Let \(n_{ii}\) and \(n_{ij}\) represent the true prediction of the \(i\)th class and the false prediction of the \(i\)th class into \(j\)th class respectively. The per class accuracy for class \(i\) can be defined as (6). \[Recall_{i}=\frac{n_{ii}}{\sum_{j=1}^{L}n_{ij}}\] (6) where \(L\) stands for the number of classes
* **Average accuracy** is a performance metric that gives the same weight to each of the classes of the problem, independently of the number of examples it has. It can be calculated as the following equation: \[AverageAccuracy =\frac{\sum_{i=1}^{L}Recall_{i}}{L}\] (7)
* **F-Measure** is one of the most frequently used measurements to evaluate the performance of an algorithm for imbalance data classification. It is a family of metrics that attempts to measure the trade-offs between precision, which measures how often an instance that was predicted as positive is actually positive, and recalls by outputting a single value that reflects the goodness of a classifier in the presence of rare classes [72]. \[F-measure=\frac{2}{\frac{\sum_{i=1}^{L}Recall_{i}\sum_{i=1}^{L} Precision_{i}}{\sum_{i=1}^{L}Recall_{i}+\sum_{i=1}^{L} Precision_{i}}}\] (8) where \(Precision_{i}\) can be computed by \(\frac{n_{ii}}{\sum_{i=1}^{L}n_{ii}}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**Data** & **EX** & **AT** & **CL** & \(\mathbf{C_{1}}\) & \(\mathbf{C_{2}}\) & \(\mathbf{C_{3}}\) & \(\mathbf{C_{4}}\) & \(\mathbf{C_{5}}\) & \(\mathbf{C_{6}}\) & \(\mathbf{C_{7}}\) & \(\mathbf{C_{8}}\) & \(\mathbf{C_{9}}\) & \(\mathbf{C_{10}}\) \\ \hline Car & 1600 & 6 & 4 & 62 & 66 & 359 & 1113 & & & & & & \\ Cleveland & 297 & 13 & 5 & 13 & 35 & 35 & 54 & 160 & & & & & \\ Covtype\_data & 8000 & 54 & 7 & 33 & 139 & 241 & 278 & 481 & 2985 & 3843 & & & \\ Glass & 214 & 10 & 6 & 9 & 13 & 17 & 29 & 70 & 76 & & & & \\ Hayes-roth & 160 & 4 & 3 & 31 & 64 & 65 & & & & & & & \\ Newthyroid & 215 & 5 & 3 & 30 & 35 & 150 & & & & & & & \\ Optdigit & 1642 & 64 & 10 & 20 & 40 & 180 & 187 & 191 & 196 & 197 & 197 & 210 & 224 \\ Page-blocks & 5472 & 10 & 5 & 28 & 87 & 115 & 329 & 4913 & & & & & \\ Penbased & 1100 & 16 & 10 & 105 & 105 & 106 & 106 & 106 & 114 & 114 & 115 & 115 \\ Pendigit & 3239 & 16 & 10 & 20 & 20 & 362 & 379 & 394 & 396 & 397 & 408 & 426 & 437 \\ Segment & 2000 & 19 & 7 & 279 & 280 & 281 & 286 & 289 & 291 & 294 & & & \\ Statlog & 5000 & 36 & 6 & 485 & 539 & 540 & 1061 & 1169 & 1206 & & & & \\ Urbanlandcover & 300 & 147 & 9 & 11 & 13 & 19 & 28 & 30 & 45 & 46 & 47 & 61 & \\ Vehicle & 684 & 17 & 4 & 50 & 199 & 217 & 218 & & & & & & \\ Wilt & 4839 & 5 & 2 & 261 & 4578 & & & & & & & & \\ Wine & 178 & 13 & 3 & 48 & 59 & 71 & & & & & & & \\ Wine quality-red & 1599 & 11 & 6 & 10 & 18 & 53 & 199 & 638 & 681 & & & & \\ Wine quality-white & 4898 & 11 & 7 & 5 & 20 & 163 & 175 & 880 & 1457 & 2198 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Imbalanced data sets. * **KW Diversity**[73] is a performance metric that gives the same weight to each of the classes of the problem, independently of the number of examples it has.