**4. Experimental Validations**

The vibration data collected from accelerated life tests (ALT) of rolling bearings were employed to validate the effectiveness of the proposed approach, the experimental setup is shown in Figure 6a. The experimental data were acquired and published by the IEEE-PHM Association [51,52]. The experimental platform included National Instruments (NI) data acquisition card, pressure regulator, cylinder pressure, force sensors, motor, speed sensor, torque-meter, accelerometers and tested bearing, etc. During the process of the experiment, the rotating speeds of the bearing were set to 1650 r/min and 1800 r/min, and 17 bearings were chosen during all those experiments. The sampling frequency was 25.6 kHz, and 2560 sample points (i.e., 0.1 s) were recorded each 10 s. The experimental tests were stopped if the amplitude of time-domain data exceeded 20 g. As shown in Figure 6b, the severe wear in bearing elements and the severe spalling failure in inner race were observed after dismantling.

**Figure 6.** Experimental setup. (**a**) Overview of experimental setup; (**b**) Normal and degraded bearings [51,52].

Figure 7a shows the whole lifetime data of a tested bearing in the time-domain, and the time-domain waveform of the tested bearing in the normal stage and the time-domain waveform of the tested bearing in the catastrophic failure stage are shown in Figure 7b,c, respectively. In the normal stage, the failure impulses cannot be observed in the time-domain, the range of the amplitude is [−2 g, 2 g]. However, in the failure stage, the range of the amplitude is [−50 g, 50 g], and there exist obvious impulses with time interval of Δ*T* = 0.0059 *s*, which is corresponding to the fault frequency of 169 Hz.

**Figure 7.** Raw signal of a tested bearing. (**a**) The historical signals of the whole lifetime; (**b**) The time-domain waveform of the bearing in the normal stage; (**c**) The time-domain waveform of the bearing in the failure stage.

In this experiment, four bearings (bearing 1, bearing 2, bearing 3 and bearing 4) are randomly selected and employed as testing targets to evaluate the prediction performance. Each bearing is degraded during the accelerated life tests without implanting any artificial fault in advance.


Depending on the diversity of operating conditions and the manufacturing accuracy of tested bearings, the effective running time may be different for different bearings. Therefore, the degradation trajectories of the health indicators are different for each tested bearing. Figure 8 shows the peak-to-peak values of the whole lifetime of bearing 1. Accordingly, the health indicators, i.e., equivalent vibration intensity (EVI) [9,11], Kurtosis and EVI of bearing 2, bearing 3 and bearing 4 are illustrated in Figure 9a–c, respectively. It is seen that the amplitudes of bearings 1, 3 and 4 have gradual increasing trends, in addition, the whole test life of bearing 3 is the shortest due to harsh operating conditions, which indicates that the extremely failures are occurred before the experiment stops, thus, representing abrupt degradation processes, whereas the EVI amplitudes of bearings 2 show gradual increases; it might be concluded that the design/manufacturing quality and fatigue resistance strength are much higher than others under the same operating conditions.

**Figure 8.** The health indicator curve of peak-to-peak value of bearing 1. (**a**) The peak-to-peak value of the whole lifetime of bearing 1; (**b**) The peak-to-peak value of the point 2001 from to point 2803.

Taking bearing 1 as an example, as shown in Figure 8b, because of the abrupt degradation time series is the most interesting and difficult part; thus, datasets 2001 to 2803 are selected as survey regions, where the dataset from 2001 to 2703 are selected as a historical curve, and the remaining 100 datasets, i.e., dataset 2704 to dataset 2803, are a predicted region. Specifically, the proposed APSD method is adopted to process the peak-to-peak curve of bearing 1, since the standard deviation (SD) of the peak-to-peak is unknown, it can be calculated by the following equation, i.e., ∧ *σ* = *MAD*(*y*)/0.6745 = 2.1750. Therefore, the related parameter specification of the APSD approach are summarized in Table 2. The decomposition results are presented in Figure 10, Figure 10a is the low frequency component and Figure 10b is the high frequency component, respectively. It should be noted that the low frequency component almost coincides with the actual peak-to-peak trends, especially in the abrupt degradation regions, which means the degradation processes and trend could be reflected by the low frequency component.

**Figure 9.** The health indicator curves of different bearings. (**a**) The equivalent vibration intensity curve of bearing 2; (**b**) The Kurtosis curve of bearing 3; (**c**) The equivalent vibration intensity curve of bearing 4.

Furthermore, the WNN algorithm and ARMA-RLS model are introduced for prediction. For the prediction of a low frequency component, the number of input neurons, hidden neurons, output neurons, training iterations, learning rate for WNN are set at 7, 10, 1, 300 and 0.01, respectively. For the prediction of high frequency components, the modelling orders, i.e., *p* and *q*, are calculated by Akaike information criterion (AIC) criterion, and then all the parameters of ARMA are fed into the RLS learning algorithm; to recursively estimate and update the related parameters, the forgetting factor is set to 0.99. The prediction curves of the LFC and HFC for rolling bearing are illustrated in Figure 11a,b, respectively. As can be seen in Figure 11, the prediction data generated by the ANN and ARMA-RLS models are able to reasonably trace the variation trend of original peak-to-peak time series. The final predicted result can be correspondingly obtained by integrating the predicted LFC and HFC components, as shown in Figure 12.

**Table 2.** The parameters setting of APSD method for bearing 1.


**Figure 10.** Decomposition results of low frequency component (LFC) and high frequency component (HFC) based on asymmetric penalty sparse decomposition. (**a**) The low frequency component (LFC); (**b**) The high frequency component (HFC).

**Figure 11.** Prediction results of LFC and HFC based on WNN and ARMA-RLS method, respectively. (**a**) Predicted result of LFC based on WNN method; (**b**) Predicted results of HFC based on ARMA-RLS method.

As benchmarking approaches for the prognosis of bearing degradation trajectories, the ARMA, FARIMA and WNN, largest Lyapunov (LLyap) exponent models are employed for comparison. All the predicted results with the actual values are illustrated in Figure 12. As shown in Figure 12, all the benchmarking approaches could generally track the peak-to-peak fluctuation trend well except FARIMA because the memory function might be eliminated due to difference and inverse difference operating. Importantly, compared with the predicted results generated by benchmark approaches, the predictions created by the proposed method are closer to the actual values, and the specific quantitative comparison of the prediction performance are presented in following section.

**Figure 12.** Comparison of the forecasting for peak-to-peak series using benchmarking methods and proposed approach.

Five assessment criteria of prediction results are introduced and calculated including mean absolute error (*MAE*), average relative error (*ARE*), root-mean-square error (*RMSE*), normalized mean square error (*NMSE*) and maximum of absolute error (*Max-AE*), which are denoted by,

$$MAE = \frac{1}{N} \sum\_{k=1}^{N} \left| \mathbf{x}(k) - \mathbf{x}(k) \right| \tag{70}$$

$$ARE = \frac{1}{N} \sum\_{k=1}^{N} \frac{\left| \mathbf{x}(k) - \mathbf{x}(k) \right|}{\mathbf{x}(k)} \tag{71}$$

$$RMSE = \sqrt{\sum\_{k=1}^{N} \frac{1}{N-1} \left[ x(k) - x(k) \right]^2} \tag{72}$$

$$NMSE = \frac{\sum\_{k=1}^{N} \left[ \mathbf{x}(k) - \mathbf{x}(\overset{\wedge}{k}) \right]^2}{\sum\_{k=1}^{N} \left[ \mathbf{x}(k) - \overline{\mathbf{x}(k)} \right]^2} \tag{73}$$

$$Max - AE = \max\left( \left| x(k) - x(\overset{\wedge}{k}) \right| \right) \tag{74}$$

where *N* is the number of time points, *x*(*k*) and *x*(*k*) are actual data and predicted data, respectively. Generally speaking, obviously, the smaller *MAE*, *ARE*, *RMSE*, *NMSE* and *Max-AE* values, which means lower prediction errors and higher prediction accuracies. The performance of the proposed methodology is evaluated by computed prediction errors. The computed prediction errors are summarized in Table 3. From the results in Table 3, it is noticed that the proposed method has relatively higher accuracy than FARIMA, WNN and L-Lyap methods. In addition, it is observed that the ARMA model has smaller errors among these models, but the tracking trend of LFC and fluctuation trend of HFC cannot be reflected at all; see black line in Figure 12. This indicates that the presented prediction approach has certain application potentials.

∧

Meanwhile, to better evaluate the performances of all the methods from a statistical perspective, box plots were introduced. The box plots of all the errors based on benchmark approaches and

proposed method are shown in Figure 13. The results show that the median values of the prediction errors converge to 0 based on proposed method, which demonstrate that the predicted data generated by the proposed approach are close to their true values.


**Table 3.** The computed prediction errors of proposed method and benchmark methods.

**Figure 13.** Quantitative performance evaluation based on absolute error boxplot.

Furthermore, the prediction performances of the remaining bearings are illustrated in this section. The model parameter settings of the APSD, WNN, ARMA-RLS are similar to those in the aforementioned steps, the specific parameters are omitted here for the sake of simplification. The decomposition results, i.e., the LFC and HFC, of the bearing 2, bearing 3 and bearing 4 are respectively presented in Figure 14a–f; overall, it can be found that the low frequency component almost coincides with the actual health indicators trends, especially in the abrupt degradation regions.

**Figure 14.** *Cont.*

**Figure 14.** Decomposition results of LFC and HFC based on asymmetric penalty sparse decomposition. (**a**) The low frequency component of bearing 2; (**b**) The High frequency component of bearing 2; (**c**) The low frequency component of bearing 3; (**d**) The High frequency component of bearing 3; (**e**) The low frequency component of bearing 4; (**f**) The High frequency component of bearing 4.

The final predicted results of bearing 2, bearing 3 and bearing 4 are shown in Figure 15a–c, respectively. It is clear that all the predicted data converge to the actual data as time goes on. Interestingly, one can find that, the abrupt degradation points cannot be followed starting point, for example, the point 89 and point 90 in Figure 15b and point 72 in Figure 15c, nevertheless, subsequent points after those abrupt points could be restored, the reason is that the modeling parameters are updated by the artificial neural network and recursive least squares learning algorithm step by step, and the error is eliminated to the minimal range. The proposed two-step prediction method is more robust to the data collection errors. In conclusion, the proposed method performs best in the degradation trend prediction of the rolling bearings.

Additionally, Figure 16 summarizes the box plots of the absolute error (AE) of the remaining three bearings. Analogously, the box plots show that the median values of the absolute error of those three bearings converge to 0, which further reflects the effectiveness of the proposed method. Therefore, the proposed decomposition algorithm and prediction models have accurate prediction results.

**Figure 15.** The predicted results of remaining three bearings. (**a**) The predicted results of bearing 2; (**b**) the predicted results of bearing 3; (**c**) the predicted results of bearing 4.

**Figure 16.** Quantitative performance evaluation based on absolute error boxplot of remaining three bearings.
