## 4 Case Study  
Our experiments were conducted on Google Colab, utilizing the default CPU environment with 12.7 GB of system RAM. The implementation was developed in Python 3.10, with key libraries including TensorFlow 2.15, Keras 2.15, Scikit-learn 1.2.2, and pandas 2.0.3, along with other relevant libraries. To evaluate the performance of our proposed model, we applied three open-source datasets. We express our sincere gratitude to the authors who provided the datasets free of charge. The first one is the Skoltech Anomaly Benchmark (SKAB) dataset, which can be found on Kaggle [30]. The datasets contain a multivariate time series collected from the sensors installed on the testbed. Columns in each data file are as follows:  
* datetime--Represents dates and times of the moment when the value is written to the database (YYYY-MM-DD hh:mm:ss);
* Accelerometer1 RMS--Shows a vibration acceleration (amount of g units);
* Accelerometer2 RMS--Shows a vibration acceleration (amount of g units);
* Current--Shows the amperage on the electric motor (ampere);
* Pressure--Represents the pressure in the loop after the water pump (bar);
* Temperature--Shows the temperature of the engine body (the degree Celsius);
* Thermocouple--Represents the temperature of the fluid in the circulation loop (the degree Celsius);
* Voltage--Shows the voltage on the electric motor (volt);
* RateRMS--Represents the circulation flow rate of the fluid inside the loop (liter per minute);
* anomaly--Shows if the point is anomalous (0 or 1);
* changepoint--Shows if the point is a changepoint for collective anomalies (0 or 1). In the SKAB dataset, \"normal\" corresponds to system operation under expected conditions, where sensor readings remain within predefined ranges, whereas \"anomalies\" are manually induced faults, such as sensor malfunctions or component failures, intentionally introduced to cause deviations from expected values. The training data of the dataset contain normal data only. Also, since our study is only for anomaly detection and not for changepoint detection, the last column (changepoint) will be removed in the data preprocessing stage. The size and details of the dataset are summarized in Table 1. Figure 6 illustrates a subset of the training data and one of the test datasets from the SKAB dataset, with the red regions in the figure labeled as anomalies. \begin{table}
\begin{tabular}{l c c c c} \hline
**Dataset** & **Data Shape** & **Data Amount** & **\# of Anomalies** & **Anomaly Rate** \\ \hline Training dataset & (9405, 8) & 9405 & 0 & 0 \\ \hline Test dataset 1 & (1155, 8) & 1155 & 410 & 0.3550 \\ \hline Test dataset 2 & (1147, 8) & 1147 & 402 & 0.3505 \\ \hline Test dataset 3 & (1090, 8) & 1090 & 347 & 0.3183 \\ \hline \end{tabular}
\end{table} 
Table 1: Details of SKAB datasets. The second and third datasets are the Soil Moisture Active Passive (SMAP) and Mars Science Laboratory (MSL) datasets from NASA [31]. These datasets are also publicly available and contain multivariate time series collected from satellite and space exploration missions. They are widely used for benchmarking anomaly detection models due to their diverse set of anomaly types and realistic operational scenarios. For the SMAP dataset, \"normal\" represents the normal operation of the satellite's soil moisture monitoring system, where telemetry readings align with expected operational conditions. \"Anomalies,\" on the other hand, are events derived from expert-labeled Incident Surprise Anomaly (ISA) reports, including system failures such as unexpected sensor readings or communication errors. Similarly, in the MSL dataset, \"normal\" denotes the proper functioning of the Mars Science Laboratory during its mission, as indicated by telemetry data that reflect standard operational behavior. \"Anomalies\" are identified through expert analysis of ISA reports and include irregularities such as sensor faults, communication disruptions, or mechanical malfunctions.