## 4 Experimental Results
### Comparison with Other Traditional Methods  
To verify the effectiveness and superiority of the proposed method, we considered a comparison among the proposed method and several traditional _HI_ methods. These methods were categorized in the two separate groups: five methods that are based on the time-domain features and a method that is based on the time-frequency domain feature. To compare, the vibration signal of the \"subset 2 bearing 1\" was used to construct the _HI_ in this section. (1) In the first group, several commonly used features from the time domain were extracted to construct the _HI_. The selected features include:  
* Root mean square (RMS), \[\sqrt{\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2}}\]
* Variance, \[\frac{1}{N-1}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}\]
* Kurtosis, \[\frac{1}{N}\sum_{i=1}^{N}\frac{(x_{i}-\overline{x})^{4}}{\sigma^{4}}\]
* Skewness, \[\frac{1}{N}\sum_{i=1}^{N}\frac{(x_{i}-\overline{x})^{3}}{\sigma^{3}}\]  
\(x_{i}\) is the vibrational signal series; \(\overline{x}\) and \(\sigma\) are the mean value and the variance of the series, respectively. Approximate entropy (ApEn). ApEn expresses the regularity of fluctuations over time-series data. The detailed steps to construct an _HI_ using the ApEn are introduced in [49]. During the failure stage, these equations were applied for each assessment interval to construct _HI_. Figure 11a-e illustrates the extracted _HI_ from these methods. The RMS and variance curves of the _HI_ are shown in Figure 11a,b, respectively. It can be noticed that these curves were insensitive during the early stage degradation, making it difficult for RUL prediction. The ApEn and kurtosis curves depicted in Figure 11c,d overcame the weakness of RMS and variance curves and recognized the infant mortality period. However, in these curves, oscillations and sudden changes near the end of the life of the bearing were obvious. This sudden change in the _HI_ curve may cause problems in predicting the RUL accurately. As can be seen from Figure 11e, the skewness curve contained severe noises, and no up-and-down trend was visible, especially during the end of failure period. (2) In the second group, the empirical mode decomposition (EMD) process was applied to decompose the vibrational signal into a series of intrinsic mode functions (IMFs). Afterward, the concept of singular value decomposition (SVD) was used to compute singular values (SVs) from the first two IMFs, known as defect feature vectors. Finally, the extracted feature vectors were taken as the input of the K-medoids algorithm to clustering normal and abnormal conditions and constructing the _HI_. The details steps can be found in [50]. The _HI_ constructed by the second method, EMS-SVD-K-medoids,is shown in Figure 11f. It is realized from Figure 11f that during the early stages, the \(HI\) curve had a smoothly increasing trend. However, after about \(50\times 10^{4}\) s, unpredictable stochastic fluctuations were obvious. To evaluate the performance of \(HI\) as defined in different methods, we employed three metrics, namely, correlation (_Corr_), monotonicity (_Mon_), and robustness (_Rob_) for different \(H\)Is. It was expected that a good \(HI\) would exhibit a monotonically increasing or decreasing trend, and that it would be robust to noise and stochastic fluctuations. _Mon_ was used to assess consistently increasing or decreasing trend of the \(HI\) curve. In _Mon_, the difference between the values of any two adjacent points of the \(HI\) curve is measured. For the rising monotonicity, the total number of positive values is more than the total number of negative values and _Mon_ is close to 1. On the other hand, for the turbulent and oscillation curves, the total number of positive values is close to the total number of negative values and the _Mon_ value is close to 0.