## 3 HDELC Algorithm
### Classifier Selection and Ensemble Models
#### 3.2.1 Classifier-Selection Process  
After reconstructing the original feature space in Section 3, we obtain the best feature subspace that can be used for classifier selection. The primary ensemble parameter \(\alpha\) is set in this section. \(\alpha\), as the critical parameter for the subspace inheritance strategy proposed in this paper, is an essential coefficient for partitioning the high-dimensional feature subset and selecting the classifier size in this work, and will be presented in this section. The input to the classifier ensemble algorithm based on high-dimensional data is the original set of classifiers, and the size of the original set of classifiers is determined using \(\alpha\). If the ensemble learning scale were to increase indefinitely, it would not improve the classification prediction, so the number of classifiers is first bounded, defined as  
\[\widehat{\left|\varphi\right|}=2\alpha,\ \alpha\in[1,2,3] \tag{18}\]  
The number of classifiers in the original set of classifiers is the number of feature subspaces to be partitioned under the feature space reconstruction:  
\[\widehat{\left|\omega\right|}=\widehat{\left|\varphi\right|}=2\alpha \tag{19}\]  
where \(\omega\) is the set of characteristic subspaces, and \(\widehat{\left|\omega\right|}\) is the number of subspaces in the set of characteristic subspaces. At the start of the model, the classifier with the highest accuracy \(\varphi^{1}\) is selected as the first classifier:  
\[\varphi^{1}=argmax_{\varphi\in\widehat{\varphi}}\{AC_{1},AC_{2},\ldots,AC_{H}\} \tag{20}\]  
The high-dimensional ensemble algorithm will continue to gradually assign feature subspaces to classifiers until the second to \(H\)th classifiers are all assigned (\(H\) is the predefined number of selected classifiers). By ranking the feature information entropy in the previous section, the set of feature subspaces with the highest information entropy ranking in the reconstructed sample is initially assigned to the classifier with the highest accuracy, where \(\widehat{\varphi}=\left\{\varphi^{1},\varphi^{2},\ldots,\varphi^{H}\right\}\) denotes all classifiers in the ensemble model, and the mean and mean square error of each classifier are calculated based on its performance on the training set as follows:  
\[\varepsilon=\frac{1}{H}.\frac{\sum_{n=1}^{n}\left|f(x_{i})-y_{i}\right|}{n} \tag{21}\]  
where \(f(x_{i})\) is the model prediction, \(y_{i}\) is the actual value, and \(n\) is the number of high-dimensional data samples. The trade-off between accuracy and diversity is controlled due to the need to ensure high levels of accuracy in classifier integration, while increasing the diversity of classifiers in the integrated system. Therefore, for each classifier in \(\widehat{\varphi}\) corresponding to the feature subspace \(\omega_{j}\), we consider the following loss function \(\Delta_{l}(\varphi)\):  
\[\Delta_{l}(\varphi)=\theta\frac{1}{Acc_{j}}+(1-\theta)\frac{1}{\widehat{ \varphi}\left(\omega_{j},\omega_{h}\right)} \tag{22}\]where \(\theta\) is a weighted parameter between 0 and 1 that controls for the relative importance of accuracy and diversity in total losses. \(Acc_{j}\) is the accuracy of the \(j\)th classifier for high-dimensional data samples, \(\theta(\omega_{j},\omega_{h})\) denotes the similarity of the subspace \(\omega_{j}\) corresponding to classifier \(\varphi_{j}\), to the subspace \(\omega_{h}\) corresponding to classifier \(\varphi_{h+1}\), and classifier \(\varphi_{h+1}\) is the classifier chosen for the next iteration. Therefore, the total loss \(\Delta_{i}(\varphi)\) is a combination of the loss of accuracy and the loss of diversity balanced by \(\theta\) so that the model can reasonably balance these two aspects. By adjusting the weighting parameters, it is possible to control the relative importance of accuracy and diversity in the optimization process so that the model strikes a balance between the two, training the model and evaluating its performance on a validation set.