## 5 Discussion  
Constructing a reliable _HI_ is the first and most important step in order to estimate the accurate RUL for bearings; therefore, this has been the focus of many studies [24]. In general, these methods are classified into three categories: mechanical signal processing-based, model-based, and machine learning-based. In mechanical signal processing-based methods, after pre-processing of the vibration signal, statistical parameters are directly used to construct the _HI_. Due to the flexibility and simplicity of mechanical signal processing methods, these methods are widely used in industries. These methods also have an acceptable performance to detect early bearing defects and abnormal bearing health conditions. However, it has been experimentally shown that the indicator performance decreases in the presence of transient conditions caused by bearing's defects [1]. Compared to these methods, the proposed method is sensitive to initial degradation, and is consistent with the degradation process. Nevertheless, in this work, the data of the run to failure vibrations is divided into two parts: the first part is used to train the CAE model and the second part is used to construct the _HI_. However, nothing ensures that a sudden degradation or failure does not happen during the training phase. Therefore, the method proposed in this work is limited to those faults that cause particular vibration patterns. In the case of any sudden failure or extremely slow degradation, this method is not able to construct the _HI_.  
In contrast to the time or frequency techniques, which only represent the information in time or frequency domain, time-frequency techniques provide more information in both domains. In the present work, the CWT technique was used to pre-process the vibrational signals. The CWT method is a joint time-frequency analysis method that can decompose a time series into time and frequency spaces simultaneously. Therefore, the outputs of the CWT analysis are images that contain information on both time and frequency domains. When a defect appears in the bearing, it generates an impulsive force and excites resonances in the bearing and surrounding elements. With the progress of the defect over time, the frequency spectrum changes drastically. Since the faulty signals are non-stationary and transient in nature, using the CWT for pre-processing the vibration signals has better performance than time or frequency techniques in constructing the _HI_. Furthermore, in the proposed method, the _HI_ is constructed by comparing the images of normal and failure stages, which are acquired for an identical bearing. Therefore, the perpetual background noise will not affect the _HI_ accuracy. In addition, in this  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Metrics**} & \multirow{2}{*}{\
\begin{tabular}{c} **Proposed** \\ **Method** \\ \end{tabular} } & \multirow{2}{*}{**RMS**} & \multirow{2}{*}{**Variance**} & \multirow{2}{*}{**ApEn**} & \multirow{2}{*}{**Kurtosis**} & \multirow{2}{*}{**Skewness**} & \multicolumn{2}{c}{
\begin{tabular}{c} **EMD-SVD-K-Medoids** \\ \end{tabular} } \\ \hline _Mon_ & 0.7587 & 0.0113 & 0.0140 & 0.0051 & 0.0063 & 0.0016 & 0.0229 \\ \hline _Rob_ & 0.9484 & 0.8411 & 0.7565 & 0.8790 & 0.8875 & 0.4427 & 0.6971 \\ \hline _Corr_ & 0.9969 & 0.6363 & 0.4480 & \(-\)0.3814 & 0.3622 & \(-\)0.4282 & 0.6358 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of health indicators based on monotonicity _(Mon)_, correlation (_Corr_), and robustness (_Rob_) for subset 2 bearing 1.  
work, a deep learning model is used in extracting features, as well as for dimensionality reduction from the pre-processed vibration signals. This provides a more powerful capability of learning complex nonlinear relationships, which is able to extract the best-suited features automatically. Moreover, using the exponential function improves the smoothness and monotonicity of the preliminary designed _HI_, which leads to better RUL estimation.