## 3 Method
### Removing Irrelevant Individual Features  
In high-dimensional data, many individual features may be irrelevant to the target variable, making them unsuitable for computing the permutation importance of the feature groups. To address this issue, we adopted the straightforward approach of removing individual features that were deemed irrelevant before calculating the permutation importance of the group features. Several existing approaches are available for identifying irrelevant individual features. In this study, we proposed employing the lasso method to remove irrelevant individual features from the group features. Using the lasso method, we effectively filtered out irrelevant individual features, ensuring that only the most informative features were considered when computing the permutation importance of the group features. This step significantly enhanced the relevance of the selected features to the target variable and rankings, which ultimately improved the classification performance of our proposed method in high-dimensional data analysis.  
[PERSON] [7] introduced the original lasso method, which has proven to be an effective technique for eliminating irrelevant individual features. This method has two primary objectives: regularization and loss functions. For regularization, the lasso method adds a penalty term that encourages some coefficients of the variables to be reduced to zero. This regularization step aids in controlling the complexity of the model and prevents overfitting. By shrinking some of the coefficients, we obtained a sparse feature space using the lasso method.  
Suppose there is a high-dimensional dataset with \(n\) instances and \(p\) individual features. In addition, \(\mathbf{Y}\) denotes a \(K\) dimensional response vector with \(K\) classes represented as \(\mathbf{Y}\)=\(\{\mathbf{y}_{1},\mathbf{y}_{2},\dots,\mathbf{y}_{n}\}\), where \(\mathbf{y}_{i}\) denotes the class label vector for the \(i\)-th instance. The LASSO method selects individual features by minimizing the following objective function [7]:  
\[Q(\beta|X_{O},Y)=-\sum_{i=1}^{n}\mathbf{y}_{i}\;log\;\widehat{\mathbf{y}}_{i}+\lambda \sum_{j=1}^{p}\left|\beta_{j}\right| \tag{1}\]  
where \(\lambda\) is a shrinkage parameter that controls the amount of penalty, and has great importance in this method. If \(\lambda\) is sufficiently large in Equation (1), more variables are forced to be exactly zero, which results in a greater dimension reduction. The coefficient \(\beta_{j}\) represents the feature weights for each class, and \(\widehat{\mathbf{y}}_{i}\) denotes the predicted probabilities of the \(i\)-th instance. The first term in the objective function represents the multiclass cross-entropy loss, which measures the dissimilarity between the true class labels and the predicted probabilities for each class. The second term is the \(L_{1}\) regularization term, which penalizes the absolute values of the coefficients for each class. This term promotes feature selection by driving some of the coefficients to zero, resulting in dimension reduction and the selection of relevant features. Based on this mechanism, the LASSO method can select features with non-zero coefficients. These features were deemed significant and were retained for further analysis, whereas the coefficients of the irrelevant features decreased to zero, effectively excluding them from the model.