## 4 Methodology
### Dataset Preparation with Anomaly Injection  
To simulate drift in the equipment's CPS behavior, we inject synthetic noise into the sensor data values collected for each component of the cleanroom's CPSs. Evaluating the performances of adaptive learning models requires distinct differences between training and testing data, which demonstrate changes in drift distributions. Consequently, we introduce one type of noise within the training data and a different noise distribution for the testing data. In the training dataset, Gaussian white noise is injected with a standard deviation (\(\sigma_{N}\)) of 0.1 to emulate anomalous operational variations. For the testing dataset,  
Figure 5: An overview of a scientific semiconductor laboratory (cleanroom) at UIUC.  
Figure 6: Sensor data distribution over one month for (**a**) Fumehood temperature, (**b**) Fumehood humidity and (**c**) Vacuum Pump temperature.  
a combination of Gaussian and Uniform distribution anomalies is introduced, each with a standard deviation (\(\sigma_{N}\)) of 0.5. This mixed approach in the testing set is designed to challenge the classifiers' ability to distinguish between normal operations and anomalies, where the latter follows a different distribution in the training and testing dataset.  
Using a mixed-noise model (Gaussian and Uniform) in the testing data, we ensure a variety of anomaly profiles: some will be obvious due to the higher deviation of the Uniform distribution, while others will closely resemble the training dataset's anomalies (Gaussian), albeit with a different standard deviation. This setup provides a comprehensive assessment of the classifier's performance, evaluating its ability to detect anomalies across various thresholds. It tests the classifier's robustness against clear anomalies and its precision in distinguishing between normal and anomalous conditions in a more ambiguous and realistic operational environment. Figure 7 illustrates the probability density functions of the added noise distributions. For each type of distribution, the mean (expected) value of the noise is set to zero, and the standard deviation varies from 0.1 to 0.5 to simulate different noise levels.  
Two hyperparameters control the extent of anomalies in both the training and testing sets: the noise ratio and the noise level. Specifically, these are defined as follows:  
* **Noise Ratio:** Determines the proportion of data points in the dataset that are affected by noise.
* **Noise Level:** Determines the intensity of the noise injected into the data. A higher noise level means that the anomalies will deviate more significantly from the original data points. For example, if noise_level = 0.1, the anomalies will be relatively close to the original data, while noise_level = 0.5 will introduce larger deviations. This parameter is used directly as the standard deviation for the Gaussian and Uniform distributions of injected noise.  
Figure 7: Probability density functions of the noise injected to simulate anomalies.  
For instance, in the Vacuum Pump dataset, which primarily involves temperature readings (a single-feature dataset), even a slight increase in noise level significantly degrades the model performance, due to the model's sensitivity to small deviations in a less complex feature space. Conversely, the Fumehood, equipped with multiple sensors for temperature, humidity and airflow, requires a more substantial injection of anomaly distribution to induce concept drift. This is because the presence of multiple features dilutes the impact of noise on a single sensor, necessitating stronger noise ratios and levels in the multiple-feature (multiple sensors) dataset in order to challenge the model's performance during the test phase. Then, Table 2 shows different values set for the noise level and noise ratio for each type of cyber-physical system's training and testing dataset.