## 2 Related Deep Learning Algorithms
### Results of Experiments
Additionally, this detailed statistical analysis provides evidence that the deep learning methods are robust and reliable, with the Modified Loss Variational Autoencoder showing exceptional stability in its performance metrics. Our in-depth statistical analysis demonstrates the robustness and reliability of the deep learning methods. In particular, the Modified Loss Variational Autoencoder exhibits exceptional stability across all performance metrics (accuracy, specificity, and sensitivity), indicating its superiority in anomaly detection. This consistency is crucial for practical applications in network security, where reliable performance is essential. The correlation analysis between model complexity (as represented by 'Hidden_dim' and 'Latent_dim') and performance metrics reveals the following insights:  
Hidden Dimension Correlations:  
* Autoencoder Accuracy: Positive correlation (0.39), suggesting that larger hidden dimensions might be associated with higher accuracy. * Variational Autoencoder Accuracy: A moderate positive correlation (0.51) indicates that as the hidden dimension increases, there is a tendency for accuracy to improve. * Modified Variational Autoencoder Accuracy: A positive correlation (0.33) implies that larger hidden dimensions are somewhat associated with higher accuracy in the modified model. Latent Dimension Correlations:  
* Autoencoder Accuracy: A slight negative correlation (\(-0.12\)) suggests that larger latent dimensions do not necessarily lead to higher accuracy. * Variational Autoencoder Accuracy: A positive but weak correlation (0.11) indicates that increases in the latent dimension have a limited association with accuracy improvement. * Modified Variational Autoencoder Accuracy: A negative correlation (\(-0.19\)) suggests that larger latent dimensions may not be favorable for accuracy in the modified model. These correlations suggest that increasing the hidden dimensions generally correlates with better performance across all models. However, the effect of increasing the latent dimension is less clear, with either a very slight positive or negative correlation with accuracy. To better understand the observed performance differences, we analyzed how the architectural design of each model influenced its anomaly detection capabilities. Autoencoders (AEs) were particularly sensitive to the size of their latent space. Smaller latent dimensions led to underfitting, as the model could not retain enough information about the input. In comparison, larger latent dimensions increased the risk of overfitting, \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Autoencoder** & \
\begin{tabular}{c} **Variational** \\ **Autoencoder** \\ \end{tabular} &
\begin{tabular}{c} **Modified Loss** \\ **Variational Autoencoder** \\ \end{tabular} \\ \hline Mean & 0.9888 & 0.9944 & 0.9987 \\ \hline Standard Deviation & 0.0030 & 0.0033 & 0.0006 \\ \hline Min & 0.9803 & 0.9836 & 0.9967 \\ \hline Max & 0.9924 & 0.9976 & 0.9994 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Accuracy summary. where the model memorized training data rather than learning generalizable patterns. This limitation arises from the deterministic nature of AEs, which, while computationally efficient, struggle to capture complex or non-linear data distributions. Consequently, AEs demonstrated lower sensitivity in detecting subtle anomalies, making them less suitable for tasks requiring nuanced discrimination between normal and anomalous patterns. Additionally, fluctuations in the performance of AEs were observed as the model struggled to balance the trade-off between reconstruction accuracy and generalization, particularly in high-dimensional latent spaces. Variational Autoencoders (VAEs) overcame some challenges by introducing the Kullback-Leibler divergence in their cost function, which encouraged a more structured and organized latent space. This regularization allowed VAEs to generalize better across data distributions and capture inherent variability in the data.