## 3 Methodology
### Data Acquisition and Analyzing  
In the Industry 4.0 era, by emerging technologies such as data mining, internet of things, and cloud computing, the online data acquisition and processing becomes more pervasive than ever. The goals of data acquisition and data processing in this work are to create a dataset from the healthy conditions and to construct the \(HI\) throughout the failure stage. In the absence of sudden and unexpected failures, the degradation process of the bearing generally includes two stages: the normal operation stage and the failure stage, shown in Figure 2. In practical experiences, most of the bearing lifetime is passed in a stable and healthy stage; therefore, it is possible to acquire sufficient healthy data to train the deep learning model. During a normal operation stage of the bearing, a sliding window is used to capture the vibrational signal, and for each window, the power spectrum of the CWT is used to convert a 1D vibrational signal into a 2D image. The transformed image contains both time and frequency domain information and can represent the non-stationary and transient evolution of the signal.  
Figure 1: Proposed methodology for the construction of a bearing health indicator (\(HI\)).  
In practical application, a component is evolved from the normal stage to the failure stage gradually (excluding sudden and unexpected failures) through a series of degradation states. In addition, there are high uncertainties about the ambient conditions and component properties. Therefore, defining a fixed failure threshold that clearly separates the normal stage from the failure stage is not feasible. To address this issue, in this paper, we introduced an adaptive failure threshold approach, as depicted in Figure 3. According to the theory of statistical process control (SPC), measured vibration signals under the normal operation follow a normal distribution (with the mean \(\mu\) and the variance \(\sigma\)) [41]. With the transfer from the normal operation to the failure operation, the distribution pattern of the vibration signals might vary from normal distribution to unknown distribution and, consequently, the mean and variance change.  
Figure 3: The flowchart of defining a failure threshold and establishing a healthy dataset. The \(P\) variable is used to count the abnormal consecutive assessment intervals.  
Figure 2: The lifespan of a bearing is divided into the normal stage and failure stage, and the healthy dataset is established from the normal stage data.  
In this paper, the Pauta criterion [42] was employed to determine the failure threshold. If the mean of the acquired vibrational signal amplitude for each assessment interval is within the \([\mu+3\sigma,\ \mu-3\sigma]\) range, the stage is recognized as a normal stage. Otherwise, the measured data are recognized as abnormal. In order to obtain an adaptive failure threshold, we established a healthy dataset \(\overline{X}=\ (\overline{x}_{1},\overline{x}_{2},\ldots,\overline{x}_{n})\) by collecting the mean amplitude of the vibrational data points in the normal stage. Then, a reference range \([\mu+3\sigma,\ \mu-3\sigma]\) is computed by using the \(n\) data. When a new datum \(\overline{x}_{m}\) is observed, the Pauta criterion is used to determine whether the new data belongs to the healthy dataset. If the \(\overline{x}_{m}\) falls within the \([\mu+3\sigma,\ \mu-3\sigma]\) range, it is added to the healthy dataset, i.e., \(\overline{X}=\ (\overline{x}_{1},\overline{x}_{2},\ldots,\overline{x}_{n},\ \overline{x}_{m})\), and the reference range is updated. Otherwise, the datum is recognized as an abnormal one and the original healthy dataset remains unchanged. If for more than, e.g., 500 consecutive assessment intervals the means of the vibrational amplitudes are not in the reference range, the failure threshold is recognized, and the collected healthy dataset is not changed anymore.