## 3 Proposed Method
### The Proposed VAEGA Model
The sampled value of \(\varepsilon\) participates in the gradient descent, while the operation of sampling \(\varepsilon\) does not. Using the trained VAE network model, we input the test data. The feature information of the normal data can be represented by latent vectors in latent space and reconstructed with minimal loss. However, outliers are difficult to represent in the latent space, resulting in huge errors between the reconstructed data and the original input data. Therefore, we can combine the reconstruction probability error and latent space information as a measure of the outlier degree of the data, denoted as \(RP_{i}\), as shown in Equation (11). \[RP_{i}=\frac{1}{M}\sum_{m=1}^{M}N(x_{i}|\mu_{d}[i,m],\sigma_{d}[i,m]) \tag{11}\]  
For comparability, we normalize the anomaly score \(RP_{i}\) of each point to the range [0,1] and re-denote it as \(RP\_score_{i}\), as shown in Equation (12). As the anomaly score approaches 1, the abnormality of the data increases, making it more likely that it is an outlier. On the contrary, it indicates that the characteristics of the data are more normal. \[RP\_score_{i}=\frac{RP_{i}-minRP_{i}}{MaxRP_{i}-minRP_{i}} \tag{12}\]  
It is then compared to a threshold \(\varepsilon\), where \(\varepsilon\) is a metric used to control the model's sensitivity to outliers. If the abnormal score \(RP\_score_{i}\) of the candidate data is less than the threshold \(\varepsilon\), it means that the candidate data has a high degree of similarity with the normal sample and is marked as a normal value. If the abnormal score \(RP\_score_{i}\) of the candidate data is greater than or equal to the threshold \(\varepsilon\), it means that the difference between the current data to be tested and the normal sample is large. This is then recorded as an outlier. The second part of the model is the GA module, which takes as input the high-dimensional abnormal data obtained by the previous module and trains the genetic algorithm to search the abnormal subspace of high-dimensional outliers. The abnormal subspace of the outliers can provide the basis for the analysis of abnormal causes. Figure 4 shows the flow of its search algorithm. By analyzing the subspace of the outliers, we found that there are only two states of the subspace feature components, so the binary encoding rules were chosen to establish the mapping relationship between data subspaces and encoding strings. The problem of solving the abnormal subspace of data is transformed into the problem of searching for the optimal individual through genetic algorithm. Firstly, \(N_{Gen}\) individuals containing \(N_{dim}\) genes are randomly generated to form an initial population representing feasible solutions of the abnormal subspace, where \(N_{Gen}\) is the population size and \(N_{dim}\) is the dimension of the full data space. Each chromosome consists of {0,1}, where 1 means that the feature component was selected. The initial population generation is shown in \[p(z)=N(0,I) \tag{5}\]0  
To determine the fitness function for the target problem, the second part analyzes the target problem. We redefine the metric to be the fitness function of the genetic algorithm since our goal is to obtain the anomalies of the current candidate subspace to determine whether it has been eliminated. **Definition 1**. : _Subspace Outlying Degree, SOD. In this work, \(D^{k}\) (the distance between the point to be measured and the \(k\)-th nearest neighbor) is used as an outlier metric, which is called the subspace outlying degree. Since there is likely to be a high numerical outlier distance within a subspace, it is difficult to compare outliers between different subspaces.