## 3 Proposed Method
### The Proposed VAEGA Model
Therefore, to improve the comparability of the abnormal subspace SOD, the subspace abnormality SOD is defined as the ratio of \(D^{k}_{\mathrm{s}}(p)\) at a given point \(p\) to the average(avg) \(D^{k}_{\mathrm{s}}(Data)\) in the data set denoted as Data in the same subspace \(s\), as shown below:_  
\[p(z)=N(0,I) \tag{5}\]1  
The higher the ratio, the higher the \(D^{k}\) for the point sample \(p\) compared to other points, so the higher the outlier value of \(p\), and vice versa. Our definition of \(SOD\) derives from the definition of the outlier subspace. Given the input data set denoted as Data, the parameter denoted as \(n\) is the dimension of the data set, and \(k\) is the number of adjacent data points. If there is no subspace \(s^{\prime}\), such that \(SOD(s^{\prime},p)>SOD(s,p)\), then the abnormal subspace of a given data point \(p\) is \(s\). The fitness function SOD is calculated for all individuals in the current population and sorted by size. Determine whether the chromosome corresponding to \(SOD_{top}\) in the current generation population satisfies the stopping condition. If so, decode the chromosome to obtain the abnormal subspace of candidate outliers. Otherwise, the VAEGA model will utilize genetic operators to generate a new generation of subspace populations. The specific steps are as follows. We calculate the fitness function SOD of each chromosome and then use the current ratio of each individual's \(SOD_{i}\) value to the sum \(SOD_{sum}\) as the individual's selection probability \(P_{v_{i}}\), as shown in Equation (14). \[p(z)=N(0,I) \tag{5}\]2  
The larger the value, the higher the abnormal degree of the candidate outlier in the current subspace, and the higher the probability of the individual being selected to be inherited by the next generation. The selection of individuals is a random process, which means individuals with high SOD values may be lost in the selection process. Hence, we add a mechanism for optimal retention selection, which directs the top \(N\) outstanding individuals with SOD values in the previous generation into the new generation population. Then we use the non-replacement remainder rule to select population N chromosomes to inherit into the next generation population. In the new generation population, our work  
\[p(z)=N(0,I) \tag{5}\]3 Initial population of \(N_{Gen}\) individuals. randomly selects an even number of parent chromosomes without repetition and performs a uniform crossover operation according to the crossover probability \(P_{c}\). The resulting child chromosomes will be put into a new generation of populations. The binary uniform crossover operation is shown in \[p(z)=N(0,I) \tag{5}\]4  
Gene on parent chromosome 1: \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline
1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \\ \hline \end{tabular}   
Gene on parent chromosome 2: \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 \\ \hline \end{tabular}   
**Figure 7.** Chromosomes before uniform crossover. It is assumed that the random probability of genes at positions 2, 3, 5, 8, and 9 is greater than the crossover probability \(P_{c}\). Then, the genes at positions 2, 3, 5, 8, and 9 are exchanged to form two new child chromosomes in Figure 8.  
Gene on child chromosome 1: \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ \hline \end{tabular}   
Gene on child chromosome 2: \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 \\ \hline \end{tabular}   
**Figure 8.** Chromosomes after uniform crossover. To generate distinct evolutionary directions for populations, some children chromosomes are picked at random for single point mutations based on mutation probability \(P_{m}\). The new generated population continues to repeat the above process until the algorithm converges or the stopping condition is satisfied. By doing so, an optimal abnormal subspace solution that meets the search objective is obtained.