# Incorporating Grey Total Influence into Tolerance Rough Sets for Classification Problems
## 5 Computer Simulations
### Statistical Analysis  
The nonparametric Friedman test [49] was used to statistically analyze the aforementioned classification methods. Using the null hypothesis, whereby the ranks of the classification methods were identical on average, the \(F_{F}\) statistic, distributed as an \(F\) distribution with \(k_{1}-1\) and \((k_{1}-1)\) (\(k_{2}-1\)) degrees of freedom, can be defined as follows [20]:  
\[F_{F}=\frac{(k_{2}-1)\chi_{F}^{2}}{k_{2}(k_{1}-1)-\chi_{F}^{2}} \tag{43}\]  
where \(r_{j}\), \(k_{1}\), and \(k_{2}\) are the average rank of method \(j\), the number of methods, and the number of datasets considered, respectively. \(\chi_{F}^{2}\) is defined as  
\[\chi_{F}^{2}=\frac{12k_{2}}{k_{1}(k_{1}+1)}[\sum_{j=1}^{k_{1}}r_{j}^{2}-\frac{k _{1}(k_{1}+1)^{2}}{4}] \tag{44}\]  
\(F_{F}\) is 14.09 because \(k_{1}=12\), \(k_{2}=10\), and \(\chi_{F}^{2}=67.13\). The null hypothesis was rejected at the 5% level as F\({}_{\rm F}\) was above the critical value of 1.98 (i.e., F(9, 99)).  
Subsequently, the Nemenyi test [50] was used to detect significant differences among the classification methods. The classification accuracies of the two methods were significantly different and the differences in their average ranks were greater than \(CD\):  
\[CD=q_{\alpha}\sqrt{\frac{k_{1}(k_{1}+1)}{6k_{2}}} \tag{45}\]  
where \(CD\) is a critical difference, and \(CD\) = 4.89 because \(q_{0.10}\) = 3.03 at the 10% level. We summarize the results as follows:  
1. GTI-TRSC-SU significantly outperformed TRSC-CO (9.15 \(-\) 2 = 7.15), TRSC-SU (9.55 \(-\) 2 = 7.55), RSES-H (7.90 \(-\) 2 = 5.90), RSES-O (8.80 \(-\) 2 = 6.00), RIONA (8.80 \(-\) 2 = 6.80), and HLM (9.20 \(-\) 2 = 7.2).
2. GTI-TRSC-CO significantly outperformed TRSC-CO (9.15 \(-\) 1.80 = 7.35), TRSC-SU (9.55 \(-\) 1.80 = 7.75), RSES-H (7.90 \(-\) 1.80 = 6.10), RSES-O (8.80 \(-\) 1.80 = 6.20), RIONA (8.80 \(-\) 1.80 = 7.00), and HLM (9.20 \(-\) 1.80 = 7.40).
3. There was no significant difference between GTI-TRSC and GTRSC for both set and concept approximations. Even so, GTI-TRSC outperformed GTRSC on seven out of ten datasets.
4. Although GTI-TRSC did not significantly outperform the FTRSC, the difference between GTI-TRSC-CO and FTRSC-SU was slightly less than \(CD\) (6.35 \(-\) 1.80 = 4.50). Therefore, it is reasonable to conclude that GTI-TRSC-CO was superior to FTRSC-SU. Even so, it is interesting to investigate the applications that can render GTI-TRSC and FTRSC significantly different.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multicolumn{6}{c}{**Classification Methods**} \\ \cline{2-7}  & **FRSC-SU** & **FRSC-CO** & **GTRSC-SU** & **GTRSC-CO** & **GTI-TRSC-SU** & **GTI-TRSC-CO** \\ \hline Australian approval & 88.0 & 87.7 & 89.3 & 89.1 & 91.0 & 90.9 \\ Glass & 69.1 & 69.4 & 70.1 & 69.9 & 79.8 & 79.7 \\ Hepatitis & 85.6 & 84.3 & 86.0 & 87.0 & 88.7 & 89.8 \\ Iris & 95.7 & 96.2 & 96.1 & 96.3 & 96.3 & 96.4 \\ Diabetes & 75.7 & 75.9 & 76.5 & 76.0 & 77.9 & 81.6 \\ Sonar & 78.8 & 79.5 & 83.0 & 82.8 & 86.7 & 87.8 \\ Stdug Heart & 83.9 & 84.1 & 84.4 & 84.0 & 86.9 & 86.1 \\ TTT & 97.3 & 97.8 & 98.5 & 98.5 & 98.9 & 98.5 \\ Voting & 96.6 & 96.3 & 96.0 & 96.2 & 97.4 & 97.7 \\ Wine & 93.2 & 95.1 & 97.4 & 97.9 & 97.9 & 98.1 \\ Average rank & 6.35 & 5.60 & 4.40 & 4.55 & 2.00 & 1.80 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Classification accuracy of TRSC variants. FTRSC: Flow-based TRSC; GTRSC: Grey-tolerance-rough- set-based classifier.