## 2 Materials and Methods  
### Problem Definition  
In supervised binary classification tasks, the dataset typically consists of a collection of labeled data pairs, denoted as \(D=\{(X_{1},y_{1}),(X_{2},y_{2}),...,(X_{n},y_{n})\}\), where \(n\) is the number of samples, \(X_{i}=(x_{1},x_{2},...,x_{m})\) represents the feature vector of the \(i\)-th sample with \(m\) features, and \(y_{i}\in\{0,1\}\) indicates the label of the current sample. In ensemble learning, the final result of the ensemble model is usually obtained by combining the outputs of multiple base learners. Let \(H=\{h_{1},h_{2},...,h_{J}\}\) denote the set of well-trained base learners, where \(J\) is the number of base learners. Each base learner can be considered as a function, expressed as \(h_{j}(X)=(\hat{y}_{j},prob_{j})\). In the resulting fusion of the bagging ensemble methods, two common strategies are voting and averaging. The voting method is expressed as  
\[H(X)=\arg\max_{c\in C}\sum_{j=1}^{J}w_{j}\cdot\mathbb{I}(h_{j}(x)=c) \tag{1}\] \begin{table}
\begin{tabular}{c p{142.3 pt}} \hline \hline
**Author(s)** & **Key Influence** \\ \hline [PERSON] (1997) [12] & Introduced instance-based learning classifiers as Level-0 base learners, using prototypes for classification instead of traditional models. \\ \hline [PERSON] (1999) [14] & Proposed Stacking with Correspondence Analysis and Nearest Neighbor (SCANN), using correspondence analysis to remove redundancy between base learners and a nearest neighbor method as the meta-learner, improving model diversity. \\ \hline [PERSON] et al. (1999) [13] & Evaluated stacking ensemble accuracy with conflict-based estimates, using tree-based and rule-based classifiers as base learners. \\ \hline [PERSON] et al. (2010) [17] & Explored classifier ensemble methods in incremental learning, proposing Bagging++, which integrates new incremental base learners into the original ensemble model using a simple voting strategy. \\ \hline [PERSON] et al. (2011) [18] & Demonstrated that higher divergence (heterogeneity) between base classifiers leads to stronger model performance, although they used the original bagging voting strategy without exploring a new fusion strategy. \\ \hline [PERSON] et al. (2015) [7] & Enhanced understanding of stacking method variants and their applications. \\ \hline [PERSON] et al. (2018) [10] & Enhanced ensemble performance but lacked interpretability in base learner weight distribution. \\ \hline [PERSON] et al. (2020) [16] & Addressed the class imbalance problem in binary classification tasks using heterogeneous ensembles and various sampling methods (undersampling and oversampling). \\ \hline [PERSON] et al. (2020) [19] & Proposed an ensemble selection method based on classifier prediction confidence and reliability, optimizing the empirical 0-1 loss to effectively combine static and dynamic ensemble selection, outperforming traditional strategies in experiments on 62 datasets. \\ \hline [PERSON] and [PERSON] (2021) [11] & Improved credit card fraud detection but lacked interpretability in base learner contributions. \\ \hline [PERSON] et al. (2023) [9] & Improved computational efficiency and diagnostic accuracy through feature selection, dimensionality reduction, and ensemble learning. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Related Work and Key Influences. where \(w_{j}\) represents the weight of the \(j\)-th learner, and \(\mathbb{I}(\cdot)\) is the indicator function, which takes the value \(1\) when \(h_{j}(x)=c\) and \(0\) otherwise. The weighted averaging method is expressed as  
\[H(X)=\sum_{j=1}^{J}w_{j}\cdot h_{j}(x) \tag{2}\]  
where \(w_{j}\) is the weight of the \(j\)-th learner, and \(w_{j}\in[0,1]\).