# An Efficient AdaBoost Algorithm with the Multiple Thresholds Classification
## 4 Experiment
### Wilcoxon Rank-Sum Test of These Two Algorithms and Analysis of Experimental Results  
Comparisons of these two algorithms on 18 testing subsets are shown in Table 2. T_Ad is the abbreviation of AdaBoost with Threshold Classification, and MT_Ad is the abbreviation of AdaBoost with Multiple Thresholds Classification, in both Figures 1 and 2. From Table 2, AdaBoost based on Multiple Thresholds Classifications perform significantly better for most of the 18 datasets with different numbers of examples and attributes.  
Figure 1 shows frequency histogram of accuracies of these two algorithms. Obviously, accuracies of these two algorithms are not consistent with a normal distribution, so Wilcoxon signed-rank test is set to prove significant difference between the two algorithms.  
Null hypothesis and alternative hypothesis are set as:  
\(H_{0}\): there is no significant difference between the two algorithms;  
\(H_{1}\): AdaBoost with Multiple Thresholds Classification is better than that with Threshold Classification.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Dataset** & **T\_Ad** & **MT\_Ad** \\ \hline
1 & Abalone & 0.8138 & 0.8013 \\
2 & Breast Cancer Wisconsin (Diagnostic bcw) & 0.9854 & 0.9756 \\
3 & Breast Cancer Wisconsin (Diagnostic wdbc) & 0.9532 & 0.9649 \\
4 & Connectionist Bench (Sonar) & 0.6774 & 0.7097 \\
5 & Cryptenergy & 0.8148 & 0.8889 \\
6 & EEG Eye State & 0.5642 & 0.5735 \\
7 & Hill-Valley & 0.5495 & 0.5611 \\
8 & HTRU2 & 0.9868 & 0.9883 \\
9 & Immunotherapy & 0.7778 & 0.7778 \\
10 & Ionosphere & 0.9524 & 0.9524 \\
11 & Liver Disorders & 0.7212 & 0.7404 \\
12 & Molecular Biology (Splice) & 0.9587 & 0.9630 \\
13 & Raisin & 0.8852 & 0.8704 \\
14 & Seismic-bumps & 0.9665 & 0.9665 \\
15 & SPECTF Heart & 0.7059 & 0.7540 \\
16 & Statlog (Heart) & 0.8395 & 0.8519 \\
17 & Wholesale Customers & 0.8864 & 0.9167 \\
18 & Wine Quality & 0.9932 & 0.9875 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The accuracy of the two algorithms.  
Figure 1: Frequency histogram of accuracies of two algorithms: (**a**) AdaBoost with Threshold Classification; (**b**) AdaBoost with Multiple Thresholds Classification.  
The probability of \(H_{0}\), \(p(H_{0})\approx 0.0078\leq 0.05\) by Wilcoxon signed-rank test, so the null hypothesis \(H_{0}\) was rejected at the significance level 0.05, in other words, \(H_{1}\) is received. Specifically, AdaBoost based on Multiple Thresholds Classification performs better than that based on Threshold Classification by the result of Wilcoxon signed-rank test.  
Figure 2 shows the variation process of this two algorithms' convergence performances directly. In each panel, we depict the changes of training sets errors and testing sets errors with the growth of iterations of the 18 datasets, abscissa, and ordinate values indicate times of training iteration and errors of two classifiers, respectively. The blue straight line with triangles indicates the test data's error of the test data, and the dashed line with a cross indicates the error of the training data of the threshold classification method AdaBoost; the red line with a triangle indicates the error of the test data, and the dashed line with a cross indicates the error of the training data of the AdaBoost with Multiple Thresholds Classification.  
Figure 2: _Cont._  
Figure 2: Comparison of the two algorithms for datasets (18 datasets). (**a**) Cryotherapy dataset; (**b**) EEG eye state dataset; (**c**) HTRU2 dataset; (**d**) Immunotherapy dataset; (**e**) Ionosphere dataset; (**f**) Liver Disorders dataset; (**g**) Seismic-bumps dataset; (**h**) Sonar dataset; (**i**) Spectf dataset; (**j**) Splice dataset; (**k**) Statlog (Heart) dataset; (**l**) Wholesale Customers dataset; (**m**) Abalone dataset; (**n**) Breast Cancer Wisconsin (Diagnostic bcw) dataset; (**o**) Breast Cancer Wisconsin (Diagnostic wdbc) dataset; (**p**) Hill-Valley dataset; (**q**) Raisin dataset; (**r**) Wine Quality dataset.