# Degradation Trend Prediction for Rotating Machinery Using Long-Range Dependence and Particle Filter Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- If the fractional order or LRD property of the health indicator time series (HITS) is too weak (e.g., the Hurst exponent approaching 0.5), especially the time series at the stationary operation stage, the f-ARIMA and FBM models might become unusable and the prediction accuracy reduced dramatically if the classical FOC model lacks the optimal solution.
- (2)
- Similar to the physics-based model, the model parameters of the classical LRD model such as f-ARIMA cannot be adjusted with the evolution of an actual degeneration trend of the HITS, and the prediction accuracies are also greatly decreased.
- (3)
- If some sharp transition points (STPs) are contained in the health indicator time series (HITS), especially the time series at the incipient/severe fault phase, the traditional f-ARIMA and FOC approaches treat all time series values equally, which ignore the fact that the STPs value should be preserved at a larger weight, thus limiting their effectiveness in practical application.

## 2. Fractional Gaussian Noise and LRD Model

_{1}and c

_{2}are constants. Note that the parameter β is the exponent of the LRD. As a matter of fact, the parameter β can be computed by Hurst H, i.e., β = 2 − 2H or H = 1 − β/2.

## 3. Particle Filter Frame

#### 3.1. Bayesian Filter Algorithm

_{k}and y

_{k}represent the state variable value and data measurement at k, respectively. Symbols n

_{k}

_{−1}and u

_{k}denote the process noise representing uncertainty and measurement update noises, and f

_{k}is state transition function. Symbol h

_{k}is measurement function, which denotes the nonlinear relationship between the dynamic system states and the noisy measurements.

- (1)
- Suppose the prior probability distribution $p({x}_{k-1}|{y}_{1:k-1})$ is known, the number of samples is N from the posterior distribution according to Equation (7). Considering the first-order Markov process, the approximation of the posterior distribution can be given by,$$p({x}_{k}|{y}_{1:k-1})={\displaystyle \int p}({x}_{k}|{x}_{k-1})p({x}_{k-1}|{y}_{1:k-1})d{x}_{k-1}$$
- (2)
- The posterior distribution function estimation can be updated by the Bayes formula as follows,$$p({x}_{k}|{y}_{1:k})=\frac{p({y}_{k}|{x}_{k})p({x}_{k}|{y}_{1:k-1})}{p({y}_{k}|{y}_{1:k-1})}$$$$\begin{array}{l}p({x}_{k}|{y}_{1:k-1})\\ ={\displaystyle \int p({x}_{k}|{y}_{1:k-1})}p({y}_{k}|{x}_{1:k-1})d{x}_{k}\end{array}$$

#### 3.2. Particle Filter Frame Algorithm

- (a)
- Initialization.Sampling N particles ${\left\{{x}_{0}^{i}\right\}}_{i=1}^{N}$ from the prior probability distribution function, and the initialization weight is ${\omega}_{0}^{i}=1/N$ accordingly.
- (b)
- Sequential importance sampling (SIS).
- (1)
- Resample independently N times from the above discrete distribution.
- (2)
- The prior weights are used to update the new weights, as follow,$$\begin{array}{l}{\omega}_{k}^{i}={\omega}_{k-1}^{i}p({y}_{k}^{i}|{x}_{k}^{i})\\ ={\omega}_{k-1}^{i}\frac{p({y}_{k}|{x}_{k}^{i})p({x}_{k}^{i}|{x}_{k-1}^{i})}{p({x}_{k}^{i}|{x}_{k-1}^{i},{y}_{1:k})},i=1,2,3,\cdots ,N\end{array}$$
- (3)
- Normalize the importance weights, i.e., ${\stackrel{~}{\omega}}_{k}^{i}={\omega}_{k}^{i}({\displaystyle \sum _{i=1}^{N}{\omega}_{k}^{i}}{)}^{-1}$, where ${\stackrel{~}{\omega}}_{k}^{i}$ is the normalized weight of the i-th particle at time k, and N the number of particles.

- (c)
- Resampling.

_{eff}, if ${N}_{\mathrm{eff}}=\frac{N}{1+\mathrm{var}({\omega}_{k}^{i})}\approx \frac{1}{{\displaystyle \sum _{i=1}^{N}{({\omega}_{k}^{i})}^{2}}}<{N}_{threshold}$, then generating a new set of particles ${\left\{{x}_{k}^{i,*}\right\}}_{i=1}^{N}$ by resampling N-times from $p({x}_{k}|{y}_{1:k})={\displaystyle \sum _{i=1}^{N}{\stackrel{~}{\omega}}_{k}^{i}\delta ({x}_{k}-{x}_{k}^{i})}$, and set the weights of all the new particles to 1/N accordingly. Finally, for the sampled particles with the associated weights, the state vector ${\stackrel{\wedge}{x}}_{k}$ can be estimated with ${\stackrel{\wedge}{x}}_{k}={\displaystyle \sum _{i=1}^{N}{\stackrel{~}{\omega}}_{k}^{i}{x}_{k}^{i}}$. Figure 1 illustrates the filter evolution procedures of particles in the recursive processes, where the dash line represents the optimal solution.

#### 3.3. LRD Predicting Operator Driven by Particle Filter Frame

**Step 1**.- Particle initialization.
- (a)
- The f-ARIMA degradation models are established by the tested EVI time series or kurtosis time series. In this stage, the initial f-ARIMA model order and coefficients distribution are determined, i.e., $\left\{{x}_{0}^{i}\right\}=[{\varphi}_{0}^{1},{\varphi}_{0}^{2},\cdots ,{\varphi}_{0}^{p},{\theta}_{0}^{1},{\theta}_{0}^{2},\cdots ,{\theta}_{0}^{q},{d}_{0}]$, where p and q are model orders, d
_{0}is initial fractional value, and ϕ(i) and θ(i) are model coefficients. - (b)
- By drifting all of the coefficients set randomly, that is, ${\stackrel{\wedge}{x}}_{k|k-1}^{i}={\stackrel{\wedge}{x}}_{k-1}^{i}+{u}_{k-1}^{i}$, where ${u}_{k-1}^{i}$ is noise sequence, then the particles set $\text{{}{\stackrel{\wedge}{x}}_{k}^{i}\text{}}$ is obtained, the number of particles set $\text{{}{\stackrel{\wedge}{x}}_{k}^{i}\text{}}$ is N, and their weights are initialized as ${\omega}_{0}^{i}=1/N$.

**Step 2**.- For k = 1, 2, …, L (L is step length):
- (a)
- For k = k − 1, predict the time series output ${\stackrel{\wedge}{y}}_{k|k-1}^{i}$ one-step by f-ARIMA model with the parameters ${\stackrel{\wedge}{x}}_{k|k-1}^{i}$.
- (b)
- When the new output time series point y
_{k}arrives, update each particle’s weight: ${\omega}_{k}^{i}={\omega}_{k-1}^{i}p({y}_{k}|{\stackrel{\wedge}{x}}_{k|k-1}^{i})$. - (c)
- Normalize the particles weight, i.e., ${\stackrel{~}{\omega}}_{k}^{i}={\omega}_{k}^{i}({\displaystyle \sum _{i=1}^{N}{\omega}_{k}^{i}}{)}^{-1}$.
- (d)
- Resampling is performed to remove the small weight particles and generate promising new particles with the larger weight, thus the new particles set is $\{{\stackrel{\wedge}{x}}_{k}^{i}\}$.

**Step 3**.- The state vector ${\stackrel{\wedge}{x}}_{k}$ can be estimated as ${\stackrel{\wedge}{x}}_{k}={\displaystyle \sum _{i=1}^{N}{\stackrel{~}{\omega}}_{k}^{i}{x}_{k}^{i}}$.
**Step 4**.- Train the f-ARIMA model using the update state vector ${\stackrel{\wedge}{x}}_{k}$ to get the prediction value of ${\stackrel{\wedge}{y}}_{k+1}$.
**Step 5**.- Algorithm stops when k is equal to step length L, otherwise, go to step (2).

## 4. Experimental Evaluations

#### 4.1. Experimental Setup

#### 4.2. Case 1: The EVI Time Series with Weak LRD Property

_{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.

#### 4.3. Case 2: The Kurtosis Time Series with STPs

## 5. Conclusions

- (1)
- For engineering applications, a new condition prognostic method is introduced as an effective tool for the long lifetime prediction and assessment of mechanical equipment in the PHM filed, the mechanical running degradation pattern can be predicted well in real-time and the most vital components of the mechanical equipment can be repaired and replaced prior to actual catastrophic failure.
- (2)
- For theoretical analysis, the particle filter frame (PFF) is embedded in the FOC model, an update scheme for online model parameters has been introduced to adapt the initial state model based on the PFF algorithm. Due to the adaptability of the parameters, the performance of our proposed model is remarkably efficient than classical time-series ARMA model and the FOC models, even the initial health status is unknown and the failure degradation behavior is time-varying. That is to say, the main advantage of the proposed approach is that the prognostic model has an ability to generate the reliable probabilistic results despite the uncertainties of the initial model parameters.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Predicting results using different quantitative parameters for case 1 (The EVI time series with weak LRD property).

Real | ARMA | Error | RE | FARIMA | Error | RE | FBM | Error | RE | Proposed | Error | RE |
---|---|---|---|---|---|---|---|---|---|---|---|---|

4.5522 | 4.851179 | 0.298979 | 0.065678 | 4.315315 | −0.23689 | 0.052038 | 3.692463 | −0.85974 | 0.188862 | 4.338669 | −0.21353 | 0.046907 |

3.6649 | 2.928812 | −0.73609 | 0.200848 | 3.116309 | −0.54859 | 0.149688 | 3.829959 | 0.165059 | 0.045038 | 3.754522 | 0.089622 | 0.024454 |

3.036 | 3.122478 | 0.086478 | 0.028484 | 2.500855 | −0.53514 | 0.176266 | 4.138178 | 1.102178 | 0.363036 | 3.1013 | 0.0653 | 0.021508 |

3.3038 | 4.012963 | 0.709163 | 0.214651 | 4.443229 | 1.139429 | 0.344884 | 4.036887 | 0.733087 | 0.221892 | 3.262692 | −0.04111 | 0.012443 |

2.8203 | 3.032396 | 0.212096 | 0.075203 | 3.228642 | 0.408342 | 0.144787 | 3.947579 | 1.127279 | 0.399702 | 2.87064 | 0.05034 | 0.017849 |

3.2525 | 3.090151 | −0.16235 | 0.049915 | 3.305279 | 0.052779 | 0.016227 | 4.043051 | 0.790551 | 0.243059 | 3.19275 | −0.05975 | 0.018371 |

4.1008 | 3.887408 | −0.21339 | 0.052037 | 3.536877 | −0.56392 | 0.137515 | 3.998855 | −0.10194 | 0.02486 | 3.985144 | −0.11566 | 0.028203 |

3.2772 | 4.359959 | 1.082759 | 0.330391 | 4.294689 | 1.017489 | 0.310475 | 4.301419 | 1.024219 | 0.312529 | 3.362803 | 0.085603 | 0.026121 |

2.4798 | 3.038506 | 0.558706 | 0.225303 | 3.073638 | 0.593838 | 0.23947 | 4.407019 | 1.927219 | 0.777167 | 2.569523 | 0.089723 | 0.036181 |

3.9129 | 4.451459 | 0.538559 | 0.137637 | 4.483454 | 0.570554 | 0.145814 | 4.465283 | 0.552383 | 0.14117 | 3.731065 | −0.18184 | 0.046471 |

4.5876 | 3.712715 | −0.87489 | 0.190707 | 4.176524 | −0.41108 | 0.089606 | 4.436422 | −0.15118 | 0.032954 | 4.487611 | −0.09999 | 0.021795 |

2.4631 | 2.732075 | 0.268975 | 0.109202 | 2.190463 | −0.27264 | 0.110689 | 4.635023 | 2.171923 | 0.881784 | 2.706797 | 0.243697 | 0.098939 |

4.55 | 4.025714 | −0.52429 | 0.115228 | 4.073607 | −0.47639 | 0.104702 | 4.550027 | 2.66×10^{−5} | 5.84×10^{−6} | 4.286559 | −0.26344 | 0.057899 |

4.146 | 3.494402 | −0.6516 | 0.157163 | 3.316627 | −0.82937 | 0.200042 | 4.588838 | 0.442838 | 0.106811 | 4.175007 | 0.029007 | 0.006996 |

2.7487 | 2.343199 | −0.4055 | 0.147525 | 2.717725 | −0.03097 | 0.011269 | 4.165975 | 1.417275 | 0.515617 | 2.905444 | 0.156744 | 0.057025 |

3.0865 | 3.63355 | 0.54705 | 0.17724 | 3.757182 | 0.670682 | 0.217295 | 4.169485 | 1.082985 | 0.350878 | 3.03899 | −0.04751 | 0.015393 |

4.1998 | 3.571195 | −0.62861 | 0.149675 | 3.843438 | −0.35636 | 0.084852 | 4.076447 | −0.12335 | 0.029371 | 4.052326 | −0.14747 | 0.035115 |

3.1988 | 3.510075 | 0.311275 | 0.09731 | 3.290616 | 0.091816 | 0.028703 | 3.958997 | 0.760197 | 0.237651 | 3.30548 | 0.10668 | 0.03335 |

3.0569 | 4.620933 | 1.564033 | 0.51164 | 4.349495 | 1.292595 | 0.422845 | 4.201031 | 1.144131 | 0.374278 | 3.065222 | 0.008322 | 0.002722 |

3.8719 | 4.094481 | 0.222581 | 0.057486 | 4.475258 | 0.603358 | 0.15583 | 4.209925 | 0.338025 | 0.087302 | 3.761908 | −0.10999 | 0.028408 |

3.5429 | 3.102141 | −0.44076 | 0.124406 | 2.865971 | −0.67693 | 0.191066 | 4.244785 | 0.701885 | 0.19811 | 3.568503 | 0.025603 | 0.007227 |

4.4554 | 3.754243 | −0.70116 | 0.157372 | 4.179654 | −0.27575 | 0.06189 | 4.216189 | −0.23921 | 0.05369 | 4.328812 | −0.12659 | 0.028412 |

4.2155 | 3.506033 | −0.70947 | 0.1683 | 3.196617 | −1.01888 | 0.241699 | 3.419768 | −0.79573 | 0.188763 | 4.224678 | 0.009178 | 0.002177 |

2.521 | 2.621368 | 0.100368 | 0.039813 | 2.579941 | 0.058941 | 0.02338 | 3.423397 | 0.902397 | 0.357952 | 2.714081 | 0.193081 | 0.076589 |

3.2362 | 4.057441 | 0.821241 | 0.253767 | 3.836757 | 0.600557 | 0.185575 | 3.377986 | 0.141786 | 0.043813 | 3.143407 | −0.09279 | 0.028673 |

3.1059 | 3.091177 | −0.01472 | 0.00474 | 3.682622 | 0.576722 | 0.185686 | 3.026959 | −0.07894 | 0.025416 | 3.112301 | 0.006401 | 0.002061 |

2.9839 | 2.908971 | −0.07493 | 0.025111 | 2.831969 | −0.15193 | 0.050917 | 2.832535 | −0.15137 | 0.050727 | 2.990418 | 0.006518 | 0.002184 |

2.7785 | 4.240453 | 1.461953 | 0.526166 | 4.188664 | 1.410164 | 0.507527 | 2.850715 | 0.072215 | 0.025991 | 2.796518 | 0.018018 | 0.006485 |

3.3918 | 3.682935 | 0.291135 | 0.085835 | 3.509299 | 0.117499 | 0.034642 | 3.124385 | −0.26741 | 0.078842 | 3.309347 | −0.08245 | 0.02431 |

3.0046 | 3.348826 | 0.344226 | 0.114566 | 3.001548 | −0.00305 | 0.001016 | 2.957211 | −0.04739 | 0.015772 | 3.041626 | 0.037026 | 0.012323 |

3.0101 | 4.347179 | 1.337079 | 0.444197 | 4.64161 | 1.63151 | 0.542012 | 3.099322 | 0.089222 | 0.029641 | 3.001522 | −0.00858 | 0.00285 |

4.226 | 3.36362 | −0.86238 | 0.204065 | 3.548991 | −0.67701 | 0.160201 | 3.639928 | −0.58607 | 0.138682 | 4.066107 | −0.15989 | 0.037836 |

2.9203 | 2.747795 | −0.1725 | 0.059071 | 2.890067 | −0.03023 | 0.010353 | 3.486781 | 0.566481 | 0.19398 | 3.064534 | 0.144234 | 0.04939 |

3.9785 | 4.071772 | 0.093272 | 0.023444 | 3.644438 | −0.33406 | 0.083967 | 3.687806 | −0.29069 | 0.073066 | 3.839084 | −0.13942 | 0.035042 |

2.8849 | 3.696822 | 0.811922 | 0.281438 | 3.812164 | 0.927264 | 0.32142 | 3.740511 | 0.855611 | 0.296583 | 3.004821 | 0.119921 | 0.041569 |

2.781 | 2.821017 | 0.040017 | 0.014389 | 2.481329 | −0.29967 | 0.107757 | 4.339559 | 1.558559 | 0.560431 | 2.787078 | 0.006078 | 0.002185 |

3.8017 | 4.081894 | 0.280194 | 0.073702 | 4.310372 | 0.508672 | 0.133801 | 4.144882 | 0.343182 | 0.090271 | 3.668168 | −0.13353 | 0.035124 |

4.6639 | 3.529533 | −1.13437 | 0.243223 | 3.672363 | −0.99154 | 0.212598 | 4.21923 | −0.44467 | 0.095343 | 4.540967 | −0.12293 | 0.026358 |

3.8834 | 2.812674 | −1.07073 | 0.275719 | 2.667587 | −1.21581 | 0.313079 | 4.152104 | 0.268704 | 0.069193 | 3.957954 | 0.074554 | 0.019198 |

3.3754 | 4.384567 | 1.009167 | 0.298977 | 4.223374 | 0.847974 | 0.251222 | 4.010072 | 0.634672 | 0.188029 | 3.422911 | 0.047511 | 0.014076 |

**Table A2.**Predicting results using different quantitative parameters for case 2 (The Kurtosis time series with STPs).

Real | ARMA | Error | RE | FARIMA | Error | RE | FBM | Error | RE | Proposed | Error | RE |
---|---|---|---|---|---|---|---|---|---|---|---|---|

3.643158 | 3.583676 | −0.05948 | 0.016327 | 3.573756 | −0.0694 | 0.01905 | 4.032532 | 0.389374 | 0.106878 | 3.658809 | 0.015651 | 0.004296 |

4.458712 | 3.72162 | −0.73709 | 0.165315 | 3.698368 | −0.76034 | 0.17053 | 3.767629 | −0.69108 | 0.154996 | 4.125354 | −0.33336 | 0.074765 |

3.885578 | 3.867426 | −0.01815 | 0.004672 | 3.848826 | −0.03675 | 0.009459 | 3.486229 | −0.39935 | 0.102777 | 4.097066 | 0.211488 | 0.054429 |

3.26448 | 4.058685 | 0.794206 | 0.243287 | 4.109618 | 0.845139 | 0.258889 | 3.156036 | −0.10844 | 0.033219 | 3.507751 | 0.243271 | 0.074521 |

3.326908 | 4.248171 | 0.921263 | 0.276913 | 4.428144 | 1.101236 | 0.331009 | 4.602094 | 1.275186 | 0.383295 | 3.306727 | −0.02018 | 0.006066 |

3.596278 | 4.40949 | 0.813212 | 0.226126 | 4.669793 | 1.073515 | 0.298507 | 3.575741 | −0.02054 | 0.005711 | 3.490924 | −0.10535 | 0.029295 |

3.911841 | 4.51446 | 0.602619 | 0.15405 | 4.809623 | 0.897783 | 0.229504 | 3.624969 | −0.28687 | 0.073334 | 3.781957 | −0.12988 | 0.033203 |

4.374046 | 4.545397 | 0.171351 | 0.039175 | 4.892967 | 0.518921 | 0.118636 | 3.181498 | −1.19255 | 0.272642 | 4.17795 | −0.1961 | 0.044832 |

4.252138 | 4.496206 | 0.244068 | 0.057399 | 4.834369 | 0.582232 | 0.136927 | 2.674354 | −1.57778 | 0.371057 | 4.28264 | 0.030502 | 0.007173 |

3.393007 | 4.374221 | 0.981215 | 0.289187 | 4.699951 | 1.306944 | 0.385188 | 2.841052 | −0.55195 | 0.162674 | 3.724723 | 0.331716 | 0.097765 |

3.612475 | 4.199098 | 0.586623 | 0.162388 | 4.481938 | 0.869463 | 0.240683 | 3.447842 | −0.16463 | 0.045574 | 3.52586 | −0.08662 | 0.023977 |

3.561138 | 3.999812 | 0.438674 | 0.123184 | 4.222849 | 0.661711 | 0.185815 | 3.387714 | −0.17342 | 0.048699 | 3.579461 | 0.018323 | 0.005145 |

3.465567 | 3.809977 | 0.34441 | 0.099381 | 3.963394 | 0.497827 | 0.14365 | 5.315971 | 1.850404 | 0.53394 | 3.502904 | 0.037337 | 0.010774 |

3.168745 | 3.662308 | 0.493563 | 0.15576 | 3.700369 | 0.531624 | 0.167771 | 4.683602 | 1.514857 | 0.478062 | 3.289646 | 0.120902 | 0.038154 |

3.819911 | 3.583102 | −0.23681 | 0.061993 | 3.562851 | −0.25706 | 0.067295 | 2.746134 | −1.07378 | 0.2811 | 3.56325 | −0.25666 | 0.06719 |

3.500932 | 3.587702 | 0.08677 | 0.024785 | 3.53361 | 0.032678 | 0.009334 | 2.37261 | −1.12832 | 0.322292 | 3.623144 | 0.122212 | 0.034908 |

3.570946 | 3.677685 | 0.106739 | 0.029891 | 3.616021 | 0.045075 | 0.012623 | 2.29524 | −1.27571 | 0.357246 | 3.54248 | −0.02847 | 0.007972 |

3.168281 | 3.840316 | 0.672035 | 0.212113 | 3.826746 | 0.658465 | 0.20783 | 2.838346 | −0.32994 | 0.104137 | 3.329741 | 0.16146 | 0.050961 |

3.293096 | 4.05039 | 0.757294 | 0.229964 | 4.099097 | 0.806001 | 0.244755 | 2.776671 | −0.51643 | 0.156821 | 3.249593 | −0.0435 | 0.01321 |

3.446315 | 4.274201 | 0.827886 | 0.240224 | 4.429192 | 0.982877 | 0.285197 | 2.604462 | −0.84185 | 0.244276 | 3.388605 | −0.05771 | 0.016745 |

3.162353 | 4.475022 | 1.312669 | 0.415092 | 4.722749 | 1.560396 | 0.493429 | 2.233966 | −0.92839 | 0.293575 | 3.278397 | 0.116044 | 0.036696 |

3.466872 | 4.619211 | 1.152339 | 0.332386 | 4.955867 | 1.488995 | 0.429492 | 3.825701 | 0.358829 | 0.103502 | 3.350685 | −0.11619 | 0.033513 |

3.173705 | 4.68192 | 1.508215 | 0.475222 | 5.108123 | 1.934418 | 0.609514 | 2.313129 | −0.86058 | 0.271158 | 3.293016 | 0.119311 | 0.037594 |

2.735459 | 4.651447 | 1.915989 | 0.700427 | 5.101645 | 2.366186 | 0.865005 | 1.647156 | −1.0883 | 0.39785 | 2.919846 | 0.184387 | 0.067406 |

2.741126 | 4.531477 | 1.790351 | 0.653145 | 4.975918 | 2.234792 | 0.815283 | 1.354652 | −1.38647 | 0.505804 | 2.755165 | 0.014039 | 0.005121 |

4.596144 | 4.340782 | −0.25536 | 0.05556 | 4.725386 | 0.129242 | 0.02812 | 1.409942 | −3.1862 | 0.693234 | 3.86102 | −0.73512 | 0.159944 |

5.237255 | 4.110381 | −1.12687 | 0.215165 | 4.39778 | −0.83947 | 0.160289 | 1.939538 | −3.29772 | 0.629665 | 4.953759 | −0.2835 | 0.054131 |

9.022867 | 3.878593 | −5.14427 | 0.570137 | 4.052994 | −4.96987 | 0.550809 | 2.780149 | −6.24272 | 0.691877 | 7.451932 | −1.57093 | 0.174106 |

5.710835 | 3.684756 | −2.02608 | 0.354778 | 3.721233 | −1.9896 | 0.348391 | 4.30398 | −1.40685 | 0.246348 | 6.932928 | 1.222093 | 0.213995 |

11.78736 | 3.562644 | −8.22472 | 0.697757 | 3.501299 | −8.28607 | 0.702962 | 3.285498 | −8.50187 | 0.72127 | 9.278318 | −2.50905 | 0.212859 |

6.161553 | 3.534682 | −2.62687 | 0.426333 | 3.397958 | −2.7636 | 0.448522 | 4.699429 | −1.46212 | 0.237298 | 8.261117 | 2.099564 | 0.340752 |

17.11001 | 3.607951 | −13.5021 | 0.789132 | 3.439155 | −13.6709 | 0.798997 | 9.469801 | −7.64021 | 0.446534 | 12.61797 | −4.49204 | 0.262538 |

12.79646 | 3.772676 | −9.02378 | 0.705178 | 3.639572 | −9.15689 | 0.71558 | 4.514397 | −8.28206 | 0.647215 | 14.24967 | 1.453207 | 0.113563 |

4.624278 | 4.003519 | −0.62076 | 0.134239 | 3.945269 | −0.67901 | 0.146836 | 6.213077 | 1.588799 | 0.343578 | 7.733383 | 3.109106 | 0.672344 |

3.468857 | 4.263509 | 0.794651 | 0.229082 | 4.336045 | 0.867188 | 0.249992 | 8.735766 | 5.266909 | 1.518341 | 3.912321 | 0.443463 | 0.127841 |

15.5777 | 4.510005 | −11.0677 | 0.710483 | 4.723523 | −10.8542 | 0.696777 | 4.796129 | −10.7816 | 0.692116 | 10.67374 | −4.90396 | 0.314806 |

6.759972 | 4.701735 | −2.05824 | 0.304474 | 5.057507 | −1.70246 | 0.251845 | 6.469648 | −0.29032 | 0.042947 | 10.07051 | 3.310534 | 0.489726 |

7.891755 | 4.805755 | −3.086 | 0.391041 | 5.288081 | −2.60367 | 0.329923 | 3.409635 | −4.48212 | 0.56795 | 7.36274 | −0.52901 | 0.067034 |

6.637513 | 4.803148 | −1.83437 | 0.276363 | 5.347143 | −1.29037 | 0.194406 | 4.985756 | −1.65176 | 0.248852 | 7.050467 | 0.412954 | 0.062215 |

1.390226 | 4.692493 | 3.302267 | 2.375345 | 5.250833 | 3.860607 | 2.776964 | 9.331603 | 7.941377 | 5.712292 | 3.44751 | 2.057284 | 1.47982 |

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**Figure 2.**The flow chart of the proposed method for degeneration trend prognostics of rotating machinery.

**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.

**Figure 5.**Weak long-range dependence (LRD) characteristic time series and its Hurst curve. (

**a**) Weak LRD characteristic time series; (

**b**) Hurst curve.

**Figure 6.**The EVI prediction results by the proposed method and the other state-of-the-art methods. (

**a**) Prediction result using the autoregressive–moving-average (ARMA) model; (

**b**) Prediction result using the fractionally autoregressive integrated moving average (f-ARIMA) model; (

**c**) Prediction result using the fractional Brownian motion (FBM) model; (

**d**) Prediction result using the proposed model; (

**e**) Prognostic errors.

**Figure 7.**Kurtosis time series and its Hurst curve. (

**a**) kurtosis time series with sharp transition points; (

**b**) Hurst curve.

**Figure 8.**The kurtosis prediction results by the proposed method and the other state-of-the-art methods. (

**a**) Prediction result using the ARMA model; (

**b**) Prediction result using the f-ARIMA model; (

**c**) Prediction result using the FBM model; (

**d**) Prediction result using the proposed model; (

**e**) Prognostic errors.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Q.; Liang, S.Y.
Degradation Trend Prediction for Rotating Machinery Using Long-Range Dependence and Particle Filter Approach. *Algorithms* **2018**, *11*, 89.
https://doi.org/10.3390/a11070089

**AMA Style**

Li Q, Liang SY.
Degradation Trend Prediction for Rotating Machinery Using Long-Range Dependence and Particle Filter Approach. *Algorithms*. 2018; 11(7):89.
https://doi.org/10.3390/a11070089

**Chicago/Turabian Style**

Li, Qing, and Steven Y. Liang.
2018. "Degradation Trend Prediction for Rotating Machinery Using Long-Range Dependence and Particle Filter Approach" *Algorithms* 11, no. 7: 89.
https://doi.org/10.3390/a11070089