## 3 Results and Discussion
### Feature Set  
Feature selection is essential since the dataset contains 145 features. Using a global search algorithm such as GA for feature selection will consume excessive time and generate space complexity. Due to the large number of features, the feature set filtered by GA may still contain a lot of redundant information. Therefore, a feature selection algorithm is proposed to quickly filter out a low-redundant feature set from 145 features and accurately classify alloy phases. As a summary for the feature selection method, firstly, the Embedded method, which runs faster, is constructed in combination with a specific machine learning method (the LightGBM algorithm is selected in Section 3.1) to perform a rough filtering of the feature set to obtain a simplified feature set. The simplified feature set is then further filtered using a global optimization algorithm such as GA. After that, the speed and accuracy of feature selection are balanced. In Section 3.1, the LightGBM was selected as the benchmark algorithm to build the classifier. At first, Embedded was used to evaluate the importance of features and correspondingly select features that are important to LightGBM to reduce feature redundancy. The alloy phase dataset was fitted using the LightGBM and ranked according to the importance of the features. Subsequently, the features are added to the feature set based on their importance level, and the feature group with the highest accuracy is selected. In Figure 2a, the LightGBM classifier achieves the best fit when the top 52 features in the importance ranking are included. Since the feature set used by the LightGBM classifier does not contain a large number of features, GA was used for feature selection. In Figure 2b, finally, after the global search of fifty-two features through GA, nine features, \(\delta\)T\({}_{\rm m}\), \(\delta\)X, \(\chi\), Col, AW, \(\Delta\)MN, max_R, L\({}_{7}\), and min_R, are finally retained. In Figure 2c,d, to further evaluate the features, the importance of nine features was ranked using Embedded and the correlation between the features was analyzed using the Pearson correlation coefficient (PCC) method. From Figure 2c, the importance of the parameters \(\delta\)T\({}_{\rm m}\) and \(\delta\)X associated with the mismatch ranks in the top two. It is worth mentioning that the parameters \(\delta\)T\({}_{\rm m}\)  
Figure 1: Confusion matrix of (**a**) LR, (**b**) KNN, (**c**) SVM, (**d**) BPNN, (**e**) AdaBoost, (**f**) RF, (**g**) LightGBM, (**h**) GBDT, and (**i**) XGBoost algorithms. Col, \(\Delta\)MN, AW, and L\({}_{7}\) were not or were rarely used in previous studies for the construction of alloy phase classifiers. These new parameters may have a new role in exploring the formation mechanism of the HEAs phase. To further observe whether the selected nine features contain a lot of redundant information, the PCC method was used to analyze the selected features. The PCC is formulated as:  
\[\text{PCC}=\frac{\sum_{i=1}^{n}(u_{i}-\overline{u})(v_{i}-\overline{v})}{\sqrt{ \sum_{i=1}^{n}(u_{i}-\overline{u})^{2}}\sqrt{\sum_{i=1}^{n}(v_{i}-\overline{v })^{2}}}, \tag{18}\]  
where the numerator of the equation is the covariance between feature \(u_{i}\) and feature \(v_{i}\), and the denominator is the product of the standard deviation of feature \(u_{i}\) and feature \(v_{i}\). In Figure 2d, the \(|\text{PCC}|\) between each feature is less than 0.8, which proves that our selected feature set has removed a large number of redundant features and can effectively improve the physical interpretability of the model. The nine selected features have no significant linear correlation by PCC analysis; no further feature selection is required. To further evaluate the classification effects of each parameter in linear space, a \(9\times 9\) scatter matrix plot (as shown in Figure 3) is drawn to visualize the alloy phase distribution data between any two features, with the diagonal lines showing the distribution of different alloy phases depending on each input features.