## 3 Results
### The Experimental Preparation
#### 3.1.2 Evaluation Metrics  
The performance of feature selection algorithms in clustering is evaluated by using two generally used measurements: clustering accuracy (ACC) and normalized mutual information (NMI) [37]. Then, the metric ACC is denoted as follows:  
\[ACC=\frac{1}{n}\underset{i=1}{\overset{n}{\sum}}\delta(p_{i},map(q_{i})) \tag{28}\]  
The cluster label and the actual label of the sample \(x_{i}\) are represented by \(p_{i}\) and \(q_{i}\), respectively. If \(x=y\), \(\delta(x,y)=1\), otherwise \(\delta(x,y)=0\). \(map(\cdot)\) utilizes the Kuhn-Munkres algorithm to establish a one-to-one correspondence between cluster labels and their corresponding true labels is an optimal permutation mapping function [2]. ACC ranges from 0 to 1, and the higher the value, the greater the performance.  
The NMI is defined as follows:  
\[NMI=\frac{I(P,C)}{\sqrt{H(P)H(C)}} \tag{29}\]  
Two arbitrary variables are given, P and C, where the mutual information between \(P\) and \(C\) is represented by \(I(P,C)\). Then the entropy of \(P\) and \(C\) are represented as \(H(P)\) and \(H(C)\), respectively. During this experiment, \(P\) and \(C\) represent the clusters and true labels of the sample. It should be noted that the closer ACC and NMI are to one, the greater the clustering performance.  
To further verify whether the performance is superior compared to other algorithms, model performance evaluation methods are required: the Friedman test and the Nemenyi post hoc test. These two methods are particularly useful for comparing multiple algorithms.  
First, calculate the rank values corresponding to each model. The algorithms are ranked based on their test performance on each dataset (in this study, using ACC and NMI). If the test performance of the algorithms is identical, the ranks are averaged. Let there be N datasets and K models. Denote the rank of the \(i\)-th model on the \(j\)-th data set as \(rank_{i,j}\). Then, the rank of the \(i\)-th model is  
\[r_{i}=\frac{1}{N}\underset{j=1}{\overset{N}{\sum}}rank_{i,j} \tag{30}\]  
Then, calculate the corresponding statistical quantities. Assuming that the rank of each model follows a normal distribution, the corresponding chi-square statistic is given by  
\[\tau_{\chi^{2}}=\frac{k-1}{k}\frac{12N}{k^{2}-1}\underset{i=1}{\overset{k}{ \sum}}(r_{i}-\frac{k+1}{2})^{2}=\frac{12N}{k(k+1)}(\underset{i=1}{\overset{k}{ \sum}}(r_{i}^{2}-\frac{k(k+1)^{2}}{4}) \tag{31}\]  
When both \(k\) and \(N\) are relatively large, the chi-square statistic follows a chi-square distribution with \(k-1\) degrees of freedom. The improved statistic is  
\[\tau_{F}=\frac{(N-1)\tau_{\chi}^{2}}{N(k-1)-\tau_{\chi}^{2}} \tag{32}\]  
The test statistic \(\tau_{F}\) follows an F-distribution with degrees of freedom \(k-1\) and \((k-1)(N-1)\). The critical values can be obtained from standard F-distribution tables. If the null hypothesis is rejected, it indicates a significant difference in the performance of the algorithms, and a post hoc test is then employed to further distinguish betweenthe algorithms. A commonly used post hoc test is the Nemenyi test. The Nemenyi test calculates the critical value range for the difference in average ranks between the algorithms:  
\[CD=q_{\alpha}\sqrt{\frac{k(k+1)}{6N}} \tag{33}\]  
The \(q_{\alpha}\) for different confidence levels can be obtained from the table. If the difference in the mean ranks of two algorithms exceeds the critical value range (CD), the hypothesis that \"the two algorithms have the same performance\" is rejected with the corresponding confidence level.