## 6 Experiments
### Performance Evaluation of the Approximate Attribute Reduction Algorithm  
To test the proposed approximate attribute reduction algorithm, we selected the Wine dataset. In Reference [11], the diversity of the reduction results obtained by an attribute reduction algorithm was evaluated. Compared with the algorithm in Reference [11], the proposed attribute reduction algorithm further relaxes the measurement criteria by introducing a correction coefficient. Therefore, the diversity obtained by the proposed algorithm is not evaluated in this paper.  
The experiments in this section consist of two main parts, which involve the impacts of the correction coefficient \(\varepsilon\) and approximation \(\delta\) on the proposed attribute reduction algorithm and a performance evaluation of the proposed attribute reduction algorithm. In the first experiment, different values of \(\varepsilon\) and \(\delta\) were used and the number of reduced attributes as well as the classification precision of a bagging ensemble classifier were determined. Since \(\varepsilon\) and \(\delta\) have a similar value range and the same function of relaxing the attribute importance criteria, the experiment set \(\varepsilon=\delta\) and tested them at the same time. In this experiment, the ensemble size was eight. Since \(\beta\in(0.5,1]\), the initial parameter values were set to \(\beta=0.9\). The test results are shown in Table 1.  
From Table 1, it can be seen that as \(\varepsilon\) and \(\delta\) increase, the number of attributes selected by the attribute reduction algorithm increases and the classification precision continues to improve, but the runtime also increases. This result corresponds to the definitions of \(\varepsilon\) and \(\delta\). When \(\varepsilon=\delta\geq 0.9\), the number of reduced attributes and classification precision remain unchanged and the runtime is stable with slight fluctuations, which indicate that, in the Wine dataset, the selected data attributes are stable under the definition of mutual information. Moreover, Table 1 shows that when classification precision is satisfied, the number of selected data attributes can be reduced by decreasing the values of \(\varepsilon\) and \(\delta\), which reveals a method to balance the runtime and classification precision of the algorithm.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\varepsilon=\delta\) & **Classification Precision (\%)** & **Runtime (ms)** & **Number of Reduced Attributes** \\ \hline
0.7 & 95.39 & 266.91 & 4 \\
0.8 & 97.20 & 267.79 & 5 \\
0.9 & 98.45 & 268.39 & 6 \\
1 & 98.45 & 268.40 & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Test results of the proposed attribute reduction algorithm with different \(\varepsilon\) and \(\delta\) (\(\beta=0.9\)) on the Wine dataset.  
Since the increase in values of the classification precision is more obvious than that of the runtime, we set \(\varepsilon=\delta=0.95\) in the subsequent experiments.  
When \(\beta=0.9\) and \(\varepsilon=\delta=0.95\), the test results of the approximate attribute reduction algorithm based on the positive region in Reference [11] (test algorithm 1), the attribute reduction algorithm based on a variable precision rough set in Reference [12] (test algorithm 2), and the algorithm proposed in this paper (test algorithm 3) are shown in Table 2.  
The data in Table 2 show that test algorithm 3 performs well with respect to both classification precision and runtime. The experiment results indicate that test algorithm 3 selects important attributes and reduces the effect of noise on the classification results and the computation complexity of the classification system. In particular, the classification precision is higher than of the precision in test algorithm 1, which indicates the performance of the attribute importance measurement method while the runtime is lower than that of test algorithm 2. This indicates the performance of the approximate attribute reduction method.