## 3 Methodology
### Ensemble Learning Classifiers
#### 3.4.5 Multi-Stacking Ensemble (MSE)  
This section introduces our proposed MSE classifier, which consists of three levels of classifiers. The first level includes two base classifiers: decision tree (DT) and histogram-based gradient boosting. From the initial dataset, this level trains the data independently and introduces the obtained predictions to the next level (i.e., MLP\({}_{1}\)). However, the second level incorporates a meta-classifier, MLP\({}_{1}\), along with an additional base classifier (i.e., RF). The MLP\({}_{1}\) will train the data obtained from level one, while the RF will train the data produced from ResNet. Then, the obtained predictions introduced from the second level will be provided to the third level (i.e., MLP\({}_{2}\)). Finally, the third level, which includes another meta-classifier (MLP\({}_{2}\)), trains the predictions resulting from level two and generates the final enhanced prediction.  
The training dataset (\(D\)) for the base classifiers is constructed using features extracted by ResNet. To mitigate overfitting during training, we employ a k-fold cross-validation technique. In this approach, the training data (\(D\)) are split into K distinct, equally sized subsets (\(D_{t1},\ D_{t2},D_{t3},\ldots,D_{tn}\)). One subset is reserved for testing, while the remaining \(k-1/k\) subsets are used to train the classifiers.  
In this study, the MSE employs two meta-learners. The input for the first meta-learner (MLP\({}_{1}\)) includes DT and hist-gradient boosting algorithms as base classifiers, while the input for the second meta-learner (MLP\({}_{2}\)) includes the output of MLP\({}_{1}\) and the base classifier of the RF algorithm. The parameters for the MSE are presented in Table 6, and Algorithm 2 outlines the general workflow of the MSE.  
```
Input: Train data \(D=\{x_{i},y_{i}\}_{(i=1)}\); \((x_{i}R^{x},\ \ y_{i}\in y)\), where \(x_{i}\) represents features and \(y_{i}\) represents the label.  Output: the final predictions output (\(F\)) // Prepare training set using cross-validation Step 1: Apply five-fold cross-validation in preparing the training set. // Splitting the dataset into five equally sized subsets. Step 2: Randomly Split \(D\) into \(K=5\) equally sized subsets. Step 3: for \(k\gets 1\to 1\) to \(K\)do \(\text{RF}=5\) Step 3.1: for \(k\gets 1\) to \(K\)do \(\text{where }K=5\)  Employ busy level classifiers (DT, hist-gradient boosting)  for \(\text{ }\text{ }\text{ }t\gets 1\) to \(T\)do \(\text{Learn a classifier }h_{\text{H}t}\) from \(DKF\backslash D_{\text{KK}}\) end for  Construct a training set for first meta-classifier (\(ml_{1}\)) MLP\({}_{1}\) for \(x_{i}\in D_{\text{KK}}\)do \(\text{Get record }[\hat{x}_{i},y_{i}]\), where \(\hat{x_{i}}=\{h_{\text{H}t}(x_{i}),h_{\text{H}t2}(x_{i}),\ \ldots,\ h_{\text{H}t}(x_{i})\}\) end for end for Step 3.2: Learn the first meta-classifier (\(ml_{1}\)) MLP\({}_{1}\)  Learn a new classifier \(\hat{h}\) from the collection of \(\{\ \hat{x}_{i},y_{i}\}\)  Step 3.3: Re-learn classifiers  for \(\text{ }\text{ }\text{ }t\gets 1\) to \(T\)do \(\text{Learn a classifier }(h_{i})\) based on \(DKF\) end for Step 3.4: \(H(x)=\hat{h}(h_{1}(x),h_{2}(x),\ \ldots,\ h_{T}(x))\)  Step 3.5: Employ busy level classifiers (\(H(x)\), RF)  for \(\text{ }\text{ }t\gets 1\) to \(T\)do \(\text{Learn a classifier }f_{\text{KF}t}\) from \(D\backslash D_{\text{KF}}\) end for Construct a training set for the second meta classifier (\(ml_{2}\)) MLP\({}_{2}\)  for \(z_{i}\in D_{\text{KF}}\)do \(\text{Get record }[z_{i},m_{i}]\), where \(z_{i}=\{f_{\text{KF}1}(z_{i}).f_{\text{KF}2}(z_{i}),\ \ldots,\ f_{\text{KF}t}(z_{i})\}\) end for end for Step 4: Learn the second meta-classifier (\(ml_{2}\)) MLP\({}_{2}\)  Learn a new classifier \(\hat{f}\) from the collection of \(\{\ \hat{z}_{i},m_{i}\}\)  Step 5: Re-learn classifiers  for \(\text{ }\text{ }t\gets 1\) to \(T\)do \(\text{Learn a classifier }(f_{\text{ }t})\) based on D end for Step 6: Return \(F(x)=\hat{f}(f_{1}(z),f_{2}(z),\ \ldots,\ f_{\text{ }t}(z))\)
```  
**Algorithm 2**Multi-Stacking Ensemble