# Neural Network System for Predicting Anomalous Data in Applied Sensor Systems
## 3 Results
### Evaluation of the Effectiveness of the Results Obtained
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Metric** & **Analytical Expression** & **Resulting Value** & **Comment** \\ \hline Mean squared error & \(MSE=\frac{1}{|R|}\cdot\sum\limits_{i=1}^{|R|}(t_{i}-t_{i})^{2}\) & 0.218\% & It indicates that the neural-network-based anomaly predicting system achieves a high level of precision in predicting sensor data, with minimal average squared deviation between the predicted and actual values. \\ \hline Mean absolute error & \(MAE=\frac{1}{|R|}\cdot\sum\limits_{i=1}^{|R|}[t_{i}-t_{i}]\) & 0.232\% & It reflects the neural-network-based anomaly predicting system’s effectiveness in maintaining a low average absolute error between the predicted and actual sensor readings. \\ \hline Determination coefficient & \(R^{2}=1-\frac{\sum\limits_{i=1}^{|R|}\left(t_{i}-t_{i}\right)^{2}}{\sum\limits_{i=1 }^{|R|}\left(t_{i}-t_{i}\right)^{2}}\) & 0.983 & It signifies that the neural-network-based anomaly predicting system explains 98.3\% of the variance in sensor data, indicating a highly accurate model. \\ \hline Percentage of outliers correctly & \(POCR=\frac{(TP)_{weighted}}{(TP)_{weighted}+(TP)_{weighted}}\cdot 100\%\) & 98.73\% & It indicates that the neural-network-based anomaly predicting system accurately replaces nearly 99\% of detected outliers, demonstrating high efficacy in handling anomalies. \\ \hline True positive rate & \(TPR=\frac{TP}{TP+TN}\) & 0.818 & It indicates that the neural-network-based anomaly predicting system correctly identifies 81.8\% of actual anomalies, demonstrating a strong sensitivity in detecting anomalies. \\ \hline False positive rate & \(FPR=\frac{FP}{TP+TN}\) & 0.0112 & It indicates that the neural-network-based anomaly predicting system incorrectly classifies 5\% of normal data as anomalies, reflecting a low false positive rate. \\ \hline False negative rate & \(FNR=\frac{FN}{FN+TP}\) & 0.0101 & It indicates that the neural-network-based anomaly predicting system fails to detect only 1.4\% of actual anomalies, highlighting its high detection accuracy. \\ \hline \multirow{2}{*}{_AUC-ROC_} & \(AUC\)- & 0.818 & It demonstrates that the neural-network-based anomaly predicting system has a robust capability to distinguish between anomalous and normal data, with an 81.8\% area under the ROC curve. \\ \hline Mean response time & \(MRT=\frac{1}{N}\cdot\sum\limits_{i=1}^{N}(T_{i}^{\text{detect}}-T_{i}^{\text{ second}})\) & 1.611 s & It indicates that the neural-network-based anomaly predicting system achieves a high level of precision in correctly identifying anomalies, with 98.2\% accuracy. \\ \hline Mean response time & \(MRT=\frac{TP}{TP+TP}\) & 1.611 s & It indicates that the neural-network-based anomaly predicting system correctly identifies 97.9\% of detected anomalies, minimizing false positives. \\ \hline Anomaly detection accuracy & \(TPR=\frac{TP}{TP+TN}\) & 0.982 & It indicates that the neural-network-based anomaly predicting system achieves complete sensitivity, identifying all actual anomalies without any misses. \\ \hline \multirow{2}{*}{_Precision_} & \(Precision=\frac{TP}{TP+TP}\) & 0.979 & It signifies that the neural-network-based anomaly predicting system correctly identifies 97.9\% of detected anomalies, minimizing false positives. \\ \hline \multirow{2}{*}{_Recall_} & \multirow{2}{*}{_Recall_} & \multirow{2}{*}{_Recall_} & It indicates that the neural-network-based anomaly predicting system achieves complete sensitivity, identifying all actual anomalies without any misses. \\ \hline \multirow{2}{*}{_F1_} & \multirow{2}{*}{_Recall_} & \multirow{2}{*}{_Recall_} & It reflects that the neural-network-based anomaly predicting system balances high precision and recall, achieving an excellent overall performance in anomaly detection. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Effectiveness evaluation results (author’s research).