# Research on Ensemble Learning-Based Feature Selection Method for Time-Series Prediction
## 3 Feature Selection Based on Ensemble Learning
### Determination of Feature Selection Methods  
To establish a suitable set of feature selection methods \(\{M_{1},M_{2},\ldots,M_{N}\}\), this paper introduces a \"good but different\" principle. According to this principle, individual learners should contribute to performance and exhibit differences among themselves. This ensures mutual complementarity and deficiency compensation during the ensemble process, ultimately enhancing the overall performance. When applying ensemble learning to feature selection, it is equally crucial to ensure that the feature selection methods meet the \"good but different\" criteria. This implies that they should exhibit diversity, independence, stability, reliability, efficiency, adaptability, and robustness. Adhering to these requirements is essential in guaranteeing that feature selection methods can offer robust support for ensemble learning, thereby improving the performance and reliability of the model. Grounded in the principles outlined above, this paper selects the Pearson correlation coefficient method, recursive feature elimination method, random forest method, gradient boosting decision tree, and XGBost algorithm as the foundational feature selection methods. The specific rationale was as follows:  
The Pearson correlation coefficient method (Pearson) is a statistical approach employed to measure the linear correlation between two continuous variables. It was deemed suitable as a feature selection method owing to its capability to identify the strength and direction of the linear relationship between features and the target variable. This method proves particularly valuable for features that manifest a linear relationship with the target variable. Pearson correlation coefficient is calculated using the following formula:  
\[\textbf{r}=\frac{\sum_{i=1}^{n}(X_{i}-X)(Y_{i}-\bar{Y})}{\sqrt{\sum_{i=1}^{n}( X_{i}-\bar{X})^{2}}\sqrt{\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{2}}} \tag{5}\]where r represents the Pearson correlation coefficient, \(X_{i}\) and \(Y_{i}\) denote the \(i\)-th observation of the two variables, and \(\bar{X}\) and \(\bar{Y}\) represent the means of the two variables, respectively. Recursive Feature Elimination (RFE) is a stepwise feature selection method that identifies the most informative feature subset by iteratively training models and eliminating the least important features. It is well suited as a feature selection method due to its ability to automatically identify and select important features, mitigate overfitting, enhance model interpretability, and the fact that it requires minimal manual intervention. Let \(X\) represent the feature set, and \(M\) be the metric used to evaluate the performance of the model. The process of Recursive Feature Elimination (RFE) can be expressed as follows:  
\[\text{RFE}(X)= \begin{cases}X&\text{if }|X|\leq\text{desired\_num\_features}\\ \text{RFE(remove\_least\_important\_feature(X))}&\text{otherwise}\end{cases}  \tag{6}\]  
where remove_least_important_feature(X) denotes the operation of removing the least influential feature in terms of model performance from the feature set \(X\). The Random Forest method (RF) is an ensemble learning approach that improves overall performance by aggregating predictions from multiple decision trees. It is well suited as a feature selection method due to its capability to estimate feature importance and utilize a voting method for classification problems. Random forests exhibit a notable level of robustness against outliers and noise, rendering them suitable for addressing complex data scenarios. The importance calculation for feature \(X_{i}\) is as follows:  
\[\text{Importance}(X_{i})=\frac{1}{N}\sum_{j=1}^{N}\text{Impurity\_Decrease}_{j}(X _{i}) \tag{7}\]  
where \(N\) is the number of decision trees in the random forest, and \(\text{Impurity\_Decrease}_{j}(X_{i})\) represents the decrease in impurity in the \(j\)-th decision tree due to the introduction of the feature \(X_{i}\).