## 4 Experimental Evaluations
### Case 1: The EVI Time Series with Weak LRD Property  
Figure 5a shows the EVI curve of bearing 3, from the 801 st-point to 1200 th-point. The Hurst exponent is calculated via the R/S approach, as shown in Figure 5b. For the time series \(X(t)\), the R/S statistic can be expressed by,  
\[\frac{R(n)}{S(n)}=\frac{1}{S(n)}\Bigg{\{}\text{max}\!\sum_{i=1}^{n}\left[X(t_{ i})-\frac{1}{n}\!\sum_{i=1}^{n}X(t_{i})\right]-\text{min}\!\sum_{j=1}^{n} \left[X(t_{j})-\frac{1}{n}\!\sum_{j=1}^{n}X(t_{j})\right]\Bigg{\}} \tag{12}\]  
Figure 4: The equivalent vibration intensity (EVI) curve and kurtosis curve of the bearing degeneration. (a) EVI curve of rolling bearing 3; (b) kurtosis curve of rolling bearing 1.  
where \(S(n)\) is the standard deviation (SD); i.e., \(S(n)=\sqrt{\frac{1}{\operatorname*{\sum}\limits_{i=1}^{n}{\left({X(t_{i})-\frac{1 }{n}\sum\limits_{i=1}^{n}{X(t_{i})}}\right)}^{2}}}\), and \(R(n)\) is the range in each segment. The Hurst \(H\) can be generated from the fitted straight line of \(\log(R(n)/S(n))\) vs. \(\log(S(n))\). The resulting \(H=0.5455\) is close to 0.5, thus the fractional differencing parameter \(d=H-0.5=0.0455>0\) indicates that the EVI time series is to conform to the long-range dependence property. However, the LRD feature is very weak, that is to say, the performance in the future degradation trend has equal probability to be below and above the performance in the current degradation trend. Usually, the prediction accuracy generated by the FOC models may be decreased greatly in this case [13; 14; 16]. Further, the EVI prognostic result with the proposed method will be described and the results are also compared with the classical ARMA, \(f\)-ARIMA, and FBM models.  
Initially, the LRD model (e.g.,\(f\)-ARIMA) is defined as \(\Phi(B)(1-B)^{d}x_{t}=\Theta(B)\epsilon_{t}\), the particular modeling process can refer to [15; 30; 31], the AR coefficients \(\phi(1)\), \(\phi(2)\), \(\phi(3)\),..., \(\phi(p)\), MA coefficients \(\theta(1)\), \(\theta(2)\), \(\theta(3)\),..., \(\theta(q)\), wherein the parameters \(p\) and \(q\) can be calculated by the Akaike information criterion (AIC) [32; 33]. The AR coefficients, MA coefficients, and the fractional differencing parameter \(d\) are modeled as probability distributions following a certain distribution (called particles). The initial distribution of model parameters can determine the performance of sequential importance sampling (SIS) of PFF. In the update stage, the unknown parameters combined with the state transition probability \(p(x_{k}\,|\,x_{k-1})\) can be estimated recursively. Once the update measurements stop, the posterior distribution function \(p(x_{k+l}\,|\,x_{k})\) can be calculated to predict a one-step ahead EVI time series based on the latest updated parameters distribution. Furthermore, multi-step (here the step length is \(L=40\)) ahead prognostics can be obtained by performing the prediction process in Bayesian inference. In this experiment, the proposed model is applied to predict 40 points vibration intensity ahead; i.e., 400 min in advance.  
The EVI prediction results and errors generated by the proposed method and the ARMA and FOC methods are shown in Figure 6, using the 400 EVI points (from the 801 st point to 1200 th point) as the historical-based data. The results indicate that the ARMA, \(f\)-ARIMA and FBM models cannot generally track the EVI fluctuation trend well under the time series with weak LRD property. As shown in Figure 6d,e, the predicted EVI time series fit the actual time series better than the ARMA, \(f\)-ARIMA, and FBM models. It can be seen from Figure 6d,e that, compared with the ARMA and FOC methods, the EVI predictions by the proposed method are closer to the actual values, the prediction results having smaller error values. The specific parameters for quantitative assessment can be found in Table A1 in Appendix A.  
Figure 5: Weak long-range dependence (LRD) characteristic time series and its Hurst curve. (**a**) Weak LRD characteristic time series; (**b**) Hurst curve.