## 4 Methodology
### Optimization Objective for the Feature Extraction of Imbalanced Sequences  
Due to the fact that the number of LTE-R cells with degraded communication performance is much smaller than the number of normal cells, the detection of LTE-R cells with degraded communication performance is a typical imbalanced sequence classification problem. Traditional neural networks for classification often aim to minimize empirical risk during training, such as the cross-entropy loss function. However, when training imbalanced data, the classification information that minority class samples can provide is limited due to the insufficient number of minority class samples. If empirical risk minimization is chosen as the optimization objective of model training, it causes model bias. Therefore, when conducting model training in this paper, we considered optimization objectives based on the Fisher criterion to solve the imbalanced data problem. The [PERSON] criterion was proposed by [PERSON] in 1936 [36]. The basic idea of the Fisher criterion is to find a certain projection subspace in a feature space so that all feature points can have the best classification performance in that subspace. The basic principle of the Fisher criterion is shown in Figure 5.  
Figure 5 shows the projection of a two-dimensional dataset onto a one-dimensional vector. By comparing different projection vectors (\(W_{1}\) and \(W_{2}\)), it can be seen that the classification performance of the data samples after projection onto \(W_{1}\) is better than that after projection onto \(W_{2}\). It can also be seen that the basic idea of the [PERSON] criterion is that the results of a projection function should satisfy the following conditions: the between-class scatter after projection should be as large as possible and the within-class scatter should be as small as possible. In fact, the feature extraction network proposed in this paper can be regarded as a projection function, so the optimization objectives for training the network include the following two aspects:  
Figure 4: The working principle of min pooling. Figure 5: The basic principle of [PERSON] criterion. * Within-class scatter. Considering that the problem studied in this paper is a binary classification problem and assuming that the two classes are \(\Omega_{1}\) and \(\Omega_{2}\), then \(X^{i}=\left\{x_{1}^{i},x_{2}^{i},\ldots,x_{N_{i}}^{i}\right\}\) is the number of samples in class \(\Omega_{i}\). So, the within-class scatter matrix of class \(\Omega_{i}\) is as shown in Formula (4). \[S_{w}^{i}=\frac{1}{N_{i}}\sum_{k=1}^{N_{i}}\left(x_{k}^{i}-m^{i}\right)\left(x_ {k}^{i}-m^{i}\right)^{T}\] (4) In (4), \(m_{i}\) is the mean vector of \(\Omega_{i}\). The overall within-class scatter matrix is as shown in Formula (5). \[S_{w}=S_{w}^{1}+S_{w}^{2}\] (5) The trace of \(S_{w}\) is the overall within-class scatter of \(\Omega_{1}\) and \(\Omega_{2}\). Assuming that \(S_{w}\) is a \(M\times M\) matrix, \(s_{ij}\) is the element of \(S_{w}\) at row \(i\) and column \(j\) and \(tr(S_{w})\) represents the trace of \(S_{w}\). The trace of \(S_{w}\) is as shown in Formula (6). \[tr(S_{w})=\sum_{i=1}^{M}s_{ii}\] (6)
* Between-class scatter. The between-class scatter matrix between \(\Omega_{1}\) and \(\Omega_{2}\) is as shown in Formula (7). \[S_{b}=\left(m^{1}-m^{2}\right)\left(m^{1}-m^{2}\right)^{T}\] (7) The trace of \(S_{b}\) is the between-class scatter between \(\Omega_{1}\) and \(\Omega_{2}\). After obtaining the within-class scatter and between-class scatter, these two values can be combined by division to form the loss function for training feature extraction networks in order to reach the objectives of minimizing within-class scatter and maximizing between-class scatter. The loss function for training the proposed feature extraction network is as shown in Formula (8). \[l_{loss}(\theta)=\frac{tr(S_{w})}{tr(S_{b})} \tag{8}\]  
During the model training process, the smaller the value of the loss function, the better the corresponding model performance.