## 4 Experimental Results  
In the experiments, we compare the proposed method, tree-based relative support distance (denoted by TRSD), with some previously suggested methods, specifically SVMs with candidate vectors selection, denoted by CVS [18], and SVM with Fisher linear discriminant analysis, denoted by FLD [25], as well as standard SVMs, denoted by SVM. For all comparing methods, we use LibSVM [15] since it is one of the fastest methods for training SVMs. The experiments are run on a computer with the following features: Core i5 3.4 GHz processor, 16.0 GB RAM, Windows 10 enterprise operating system. The algorithms are implemented in the R programming language. We use 18 datasets which are from UCI Machine Learning Repository [35] and LibSVM Data Repository [36] except the checkerboard dataset [37]: a9a, banana, breast cancer, four-class, German credit, IJCNN-1 [38], iris, mushroom, phishing, Cod-RNA, skin segmentation, waveform, and w8a. Iris and Waveform datasets are modified for binary classification problems by assigning one class to positive and the others to negative. Table 2 shows a summary of the datasets used in the experiments where Size is the number of instances in dataset and Dim is the number of features. Figure 4: Illustration of the distances \(r\) and \(h\) according to distinct adjacent regions for the iris data in Figure 1. (**a**) Distances for leaf \(L_{4}\) relative to \(L_{1}\) are shown. Notice the support vectors captured by leaf \(L_{4}\) to leaf \(L_{1}\) are in the (dotted) red ellipsoid. (**b**) Distances for leaf \(L_{4}\) relative to \(L_{2}\) are shown. Notice the support vectors captured by leaf \(L_{4}\) relative to leaf \(L_{2}\) are in the (dashed) green ellipsoid. (**c**) Relative support distances of the observations \(x_{i}\) in leaf \(L_{4}\) relative to \(L_{1}\) are shown. Notice the support vectors captured by leaf \(L_{4}\) to leaf \(L_{1}\) are in the (dotted) red ellipsoid. (**d**) Relative support distances of the observations \(x_{i}\) in leaf \(L_{4}\) relative to \(L_{2}\) are shown. Notice the support vectors captured by leaf \(L_{4}\) relative to leaf \(L_{2}\) are in the (dashed) green ellipsoid. For testing, we apply three-fold cross validation, repeated three-times, by shuffling each dataset and dividing it into three parts, and use two parts as the training dataset, the other part as a testing dataset with different seeds. We use the RBF kernel for training SVMs in all tested methods. For each experiment, cross validation and grid search are used for tuning two hyper-parameters: the penalty factor C and the RBF kernel parameter \(\gamma\) in Equation (3). Hyper-parameters are searched by a two-dimensional grid with \(C\in\{0.1,\,1,\,10,\,100\}\) and \(\gamma\in\{0.001,\,0.01,\,0.1,\,1,\,1/p\}\) where \(p\) is the number of features. Table 3 shows the values used for each dataset in the experiments. Moreover, we vary the fraction of data points in each region \(\beta\) from 0.1 to 0.3 with the interval of 0.1. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Dataset** & **Size** & **Dim** & \(\big{|}y_{i}=+1\big{|}\) & \(\big{|}y_{i}=-1\big{|}\) \\ \hline Iris-Setosa & 150 & 4 & 50 & 100 \\ Iris-Versicolor & 150 & 4 & 50 & 100 \\ Iris-Virginia & 150 & 4 & 50 & 100 \\ Breast Cancer & 683 & 10 & 444 & 239 \\ Four-class & 862 & 2 & 555 & 307 \\ Checkerboard & 1000 & 2 & 514 & 486 \\ German Credit & 1000 & 24 & 700 & 300 \\ Waveform-0 & 5000 & 21 & 1657 & 3343 \\ Waveform-1 & 5000 & 21 & 1657 & 3343 \\ Waveform-2 & 5000 & 21 & 1657 & 3343 \\ Banana & 5300 & 2 & 2924 & 2376 \\ Mushroom & 8124 & 112 & 3916 & 4208 \\ Phishing & 11,055 & 68 & 4898 & 6157 \\ w8a & 45,546 & 300 & 44,226 & 1320 \\ a9a & 48,842 & 123 & 37,155 & 11,687 \\ IJCNN-1 & 141,691 & 22 & 128,126 & 13,565 \\ Skin Segmentation & 245,057 & 3 & 50,859 & 194,198 \\ Cod-RNA & 488,565 & 8 & 325,710 & 162,855 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Datasets for experiments.