## 3 Materials and Methods
### Data Collection
The data consists of the following estimates: rotation speed (rpm), surface temperature (celcius), ultrasound level (dB), 3D kurtosis, 3D peak acceleration, 3D RMS acceleration (g), 3D vibration speed (mm/s), and classification level. The rotation speed indicates the shaft rotation speed of the motor, while the surface temperature reflects the heating status of the motor. The ultrasound level indicates the ultrasonic vibrations in the system, and 3D kurtosis describes the sharpness of the signal's probability density function. Three-dimensional peak acceleration represents the highest acceleration in the vibration signal amplitude, while 3D RMS acceleration represents the overall acceleration level of the signal. Finally, 3D vibration speed represents the velocity derived from the acceleration signal, and the classification level determines whether the signal is produced by a healthy machine or indicates a mechanical fault. These parameters are used to detect anomalies and performance variations in the monitored machine [70]. To begin with, all the signals mentioned above will be represented with their discrete-time counterparts due to the analog-to-digital conversion inherent in the transmitter. A discrete-time zero-mean vector \(x\in\mathbb{R}^{N}\) is represented as x[n], where \(n\) denotes the total number of elements in the observed dataset, and the kurtosis of the sampled signal is estimated by employing the expectation operator \(E\{.\}\) as in Equation (9). \[K_{(x[n])}=\frac{\mathrm{E}\{x^{4}[n]\}}{\left(\mathrm{E}\{x^{2}[n]\}\right)^ {2}}. \tag{9}\]  
These features capture fault-prone behaviors and dynamic (and highly non-linear) interactions that are essential parts of the machinery's operational state. The temporal dynamics of these aspects are highlighted in Figure 3, which shows how variables including vibration acceleration, vibration velocity, kurtosis, rotational speed, and ultrasonic levels change over time. The trends and fluctuations highlight industrial systems' complex and dynamic behavior and the need for sophisticated pre-processing and analytical techniques to gain valuable insights from such complicated data. As mentioned in previous sections, the motor has been driven above 50 Hz during most of the operation time, rotating over its  
Figure 2: Industrial coupled suction (radial) fan that nominally rotates at 3000 rpm with the multi-sensor attached on the drive end. nominal speed and amplifying the imbalance in the rotor. The extra imbalance created in the coupled fan (even though standing on the vibration dampers it) produces improper clearance and concentrated load distribution in the bearing outer race [2], increasing metal-metal contact by destroying the lubrication film oil, which finally increases the ultrasound level. The original raw dataset comprised 3413 rows and 24 features. Data pre-preparation was applied to remove redundant readings and outliers, reducing the data sample points. The steps involved in data preparation were as follows: The raw data were pre-processed for missing values, and baseline scaling was applied to numerical features of the import for homogeneity, selecting the information from the features relevant to model training. Noise reduction methods were also implemented to improve signal clarity, given the complex nature of sensor data. Feature engineering has been crucial in extracting any pertinent signal from the raw data; for instance, higher-order statistics--such as but not limited to kurtosis--play an essential role in the fault detection process. Data cleaning ensured that outliers or irrelevant data points were removed, and thus the dataset was prepared for training machine learning models against unnecessary noise and inaccuracies. As illustrated in Figure 3, the vibration data solely consists of time-domain features without transforming the data into another domain to isolate and focus the study on determining the best-performing classification task on such multi-modal data collected from real working conditions [71].