## 3 Method
### A Novel Permutation-Based Group Feature Importance Measure  
We aimed to compute the importance of features by considering groups of features rather than individually. To achieve this, we proposed a new group feature importance measure that extended the permutation importance of individual features. Conventional methods typically compute the permutation importance of individual features by permuting them separately. However, in the present study, we encountered multidimensional groups consisting of multiple features. To handle this scenario appropriately, we propose permuting all columns within a common group at the same time, while keeping all other groups and target variables unchanged. This approach expands the importance of traditional feature permutation to a group feature level to capture the collaborative effect of feature groups. Suppose we obtain a new dataset including a feature matrix \(\mathbf{X}\) that has \(d\) individual features, which is much smaller than \(p\), after removing irrelevant individual features based on LASSO. As shown in Figure 3, the new feature matrix \(\mathbf{X}\) includes \(\mathbf{L}\) groups, denoted as \(\mathbf{X}=\{\mathbf{X}_{1},\mathbf{X}_{2},\dots,\mathbf{X}_{L}\}\), where \(\mathbf{X}_{l}\) is a \(n\times p_{l}\) matrix of features representing the \(l\)-th group feature of the new dataset. Each of these group features has \(p_{l}\) individual features, where \(\mathbf{x}_{11(I)}\) is an individual value of the individual feature. Each group feature contains relevant and informative individual features, because irrelevant features have been removed using the LASSO method. \(\mathbf{Y}=\{\mathbf{y}_{1}\), \(\mathbf{y}_{2},\dots,\)\(\mathbf{y}_{n}\}\) is the target variable. By permuting all the features within a common group, while preserving the other groups and target variables in their original forms (Figure 3), we accurately assessed the importance score of each group feature in the context of the entire dataset. This enabled us to measure the impact of each group on the prediction performance of the model and to determine the relative significance of the group features in contributing to the overall predictive power of the model. This score assesses the increase in the prediction error of the model when the values of the group features are permuted. The proposed method permutes the values of individual features within a specific group feature without altering the target variable. During this process, all other group features and target variable are kept in their original form, allowing us to assess the impact of that specific group on the prediction accuracy of the model. A significant decrease in prediction accuracy indicates that the group is strongly aligned with the target variable. The process of random-forest-based permutation importance of the group feature is as follows: The group feature \(\mathbf{X}_{l}\) is randomly permuted, and the association of this group feature with the target variable \(\mathbf{Y}\) is disrupted. When the permuted group feature \(\mathbf{X}_{l}\), along with the nonpermuted group features, is used to predict the target variable for out-of-bag observation, the prediction accuracy decreases considerably if the original \(\mathbf{X}_{l}\) is aligned with the target variable. Therefore, according to [PERSON] [46], the difference in prediction accuracy before and after permuting \(\mathbf{X}_{l}\) is averaged across all trees as a measure of the  
Figure 3: Permutation scheme for individual features (**left**) and for the group features (**right**). group feature importance. This concept is defined as follows: Consider a tree \(t\) within an RF composed of \(n_{tree}\) trees. Let \(\beta^{(t)}\) represent the out-of-bag (OOB) sample for the tree \(t\). Before permuting any features, the predicted class of the \(i\)-th observation by the tree \(t\) is denoted by \(\widehat{y}_{i}^{(t)}\).