# Extreme Learning Machine Enhanced Gradient Boosting for Credit Scoring
## 4 Experimental Results
The critical difference (\(CD\)) can be defined as:  
\[CD_{\alpha}=q_{\alpha}\sqrt{\frac{K(K+1)}{12D}}, \tag{21}\]  
where \(CD_{\alpha}\) is the critical difference at significance level \(\alpha\), \(q_{\alpha}\) is the critical value at significance level \(\alpha\), which is computed from a studentized range. Figure 4: Comparison of testing AUC for the credit datasets. (**a**) Australian, (**b**) German, (**c**) Japanese, (**d**) Taiwan. To investigate the statistical difference among credit scoring models, we first compute \(\chi_{F}=6.81\) according to Equation (20), rejecting the null hypothesis at significance level 0.01. Next, we perform Nemenyi test for pair-wise comparison. \(q_{0.01}=3.82\), \(q_{0.05}=3.35\), \(q_{0.1}=3.12\) are first calculated to further computed CDs for different significance levels. According to Equation (21), we can get \(CD_{0.01}=7.99\), \(CD_{0.05}=7.01\), \(CD_{0.1}=6.52\). Figure 5 shows the average ranks of credit scoring models for the Nemenyi test. As can be seen from Figure 5, if we consider AugBoost-ELM as the comparison baseline, SVM, AdaBoost, DT, and KNN are statistically inferior to AugBoost at significance level 0.01; moreover, AugBoost-ELM outperforms LDA at the significance level 0.05 as well as it is superior to NN at the significance level 0.1. Besides, as illustrated in Figure 5, AugBoost-ELM, AugBoost-NN, and AugBoost-PCA rank in top 3, demonstrating the effectiveness of the augmentation mechanism for GBDT framework. The lower ranks of AugBoost-ELM and AugBoost-NN compared with AugBoost-PCA and AugBoost-RP further verify supervised augmentation is a better augmentation compared with unsupervised feature augmentation for GBDT. In this study, ELM accomplishes the supervised step-wise feature augmentation process for GBDT, which avoids the iterative error back-propagation process of NN. Compared with NN-based feature augmentation for GBDT, ELM-based supervised feature augmentation accelerates the training process of the GBDT-based framework and avoids falling into local minima, leading to the robust generation of augmented features. Based on the above analysis, we further investigate the training efficiency by comparing the training cost of AugBoost-based models on the four credit datasets. Table 9 shows the training cost comparison of AugBoost-based models on the four credit datasets. As can be seen from Table 9, unsupervised AugBoost-based models such as AugBoost-RP and AugBoost-PCA get faster training speed. However, combined with the performance comparison analyzed above, supervised AugBoost models are the better solution for accurate credit scoring. Besides, as can be seen from the efficiency comparison among AugBoost-NN and AugBoost-ELM, the training efficiency of AugBoost-ELM has been greatly improved. Compared with AugBoost-NN, AugBoost-ELM reduces the training time by 86.90%, 85.2  
Figure 5: Average ranks of credit scoring models for the Nemenyi test: the black solid line is \(CD_{0.01}+lowest\ rank\), the dotted line denotes the \(CD_{0.05}+lowest\ rank\), and the point line represents \(CD_{0.1}+lowest\ rank\). 98.58% for the Australian dataset, German dataset, Japan dataset, and Taiwan dataset, respectively. Even in the incorporation with 1080 Ti GPU, compared with AugBoost-NN, the efficiency improvement of AugBoost-ELM on large-scale datasets such as Taiwan is more significant than that on the small-scale datasets such as Australian, German, and Japan.