Predicting the remaining useful life of rolling element bearings using locally linear fusion regression

Abstract

Predicting the remaining useful life (RUL) of rolling element bearings (REBs) has emerged as a vital technique for guaranteeing the safety, availability, and efficiency of rotating machinery systems. An approach using locally linear fusion regression (LLFR) is developed for the RUL prediction of REBs. The original features, derived from the time domain and time– frequency domain of the vibration signal of the REBs, are extracted first. Utilizing locally linear embedding, the extracted features are then fused into a condition indicator reflecting the entire degradation process. The adaptive network-based fuzzy inference system is then introduced for the RUL prediction. The reported approach is investigated with real REB data. Peer models are employed to validate the performance of the proposed method in this work. The derived experimental results indicate that LLFR has superior prediction ability as compared to the peer models in terms of the introduced performance criteria and that it can obtain more reliable and precise prediction results.

Keywords

Remaining useful life multi-feature fusion regression locally linear embedding rolling element bearings

1. Introduction

Rolling element bearings (REBs) are one of the most vital components in rotating machinery. Studies indicate that almost 40% of motor faults are caused by bearing failure. The performance status of REBs directly affects the health of mechanical systems [1 –3]. Therefore, there is an emerging need to predict the remaining useful life (RUL) of REBs. Accurate RUL prediction can help avoid problems caused by the failure of REBs such as productivity decline, downtime maintenance, and safety accidents, as well as provide a foundation for performance checking schedules and maintenance strategies of mechanical systems [4 –6]. For these reasons, many approaches to RUL prediction have been developed.

Methods currently used in the RUL estimation of REBs can be loosely divided into two categories: model-driven approaches and data-driven approaches. Model-driven approaches depict the rising tendency of a failure mode quantitatively using physical rules and evaluate the RUL by calculating deterministic equations obtained from empirical data [7]. Data-driven approaches are designed to establish a relevant model of the degradation process based on the observed data. It is, however, infeasible to build exact physical models for RUL prediction in practice, especially in an intricate system with various failure mechanisms. Without considering the failure modes, data-driven methods are frequently used because they offer a balance between operability, accuracy, and applicability [8 –11]. Therefore, a data-driven method is adopted to predict the RUL in this work.

Usually, the RUL prediction of REBs can be regarded as a regression issue in which a related model is created via the relationship between sensitive and reliable features and the corresponding RUL throughout the lifecycle. As a result, this approach is composed of two main steps: feature construction and regression model establishment.

There are many technologies that can provide systematic, scientific, and efficient strategies for feature extraction. Most often, such statistical calculation methods as time domain analysis, frequency domain analysis, and time– frequency domain analysis are widely adopted [12 –14]. However, it is challenging to extract the appropriate features from vibration signals because of abundant noise [15, 16]. Reliable, sensitive, and proper features may better reflect the degradation process of REBs and improve the prediction effectiveness [17]. Unfortunately, some features are sensitive to global failure and others are sensitive to local failure. Moreover, an insufficient number of features will result in insufficient information so that the comprehensive and complete degradation process of REBs cannot be reflected. More features may cause overfitting, in which random error is shown instead of the implicit relationship in the model. Thus, a new method is urgently needed for building appropriate indicator fusing multiple original features while retaining sensitivity and variability. In particular, multi-feature fusion methods offer the possibility of addressing this problem [18]. In this approach, the features originating from the vibration signals are fused into a condition indicator that comprehensively reflects the degradation process of the REBs. Wang et al. [19] used the Mahalanobis distance to fuse original features into a new indicator exhibiting the degradation procedure for degradation assessment and RUL prediction. Liu et al. [20] utilized the joints approximate diagonalization of eigen-matrices to merge weighted features into an advanced indicator for RUL prediction. Li et al. [21] adopted the correlation matrix clustering and weight method to establish a new degradation condition indicator. Wei et al. [22] put forward a centralized and distributed information fusion method to obtain a RUL distribution for minimizing prediction uncertainty. However, these approaches have complicated processes and lower fusion efficiencies. They thus may not be able to reduce the evaluation uncertainty for uncertain degradation processes of a system. As an alternative, dimensionality reduction techniques have been introduced as feature fusion approaches. The principal component analysis (PCA) approach, a typical representation of this technique, has been introduced to integrate original features to effectively predict the RUL of bearings [23]. However, it has been found that the PCA approach may be limited by not regarding the nonlinear properties of the original features. Nonlinear dimensionality reduction methods can better address this issue, especially manifold learning, which can derive intrinsic low-dimensional manifold structures implied in high-dimensional space by analyzing topological characteristics of the data distribution and optimizing the fusion strategy. The essential structure can be effectively exhibited from nonlinear data with the manifold learning approach. Such manifolds as locally linear embedding (LLE) [24], isometric feature mapping [25], and diffusion maps (DM) [26] have been developed. In particular, LLE has the advantages of simple calculations, an easily implemented procedure, and intuitive geometric meaning. As a result, in this work LLE is adopted to merge the original features. A preprocessing technique is introduced to improve the prediction effect with consideration of such adverse influences as noise.

In past years, many data-driven approaches, such as auto-regressive models [27], hidden Markov models [28], artificial neural networks (ANN) [29 –31], Bayesian networks [32], support vector regression (SVM) [33 –37], the fuzzy system model [38], and stochastic processes [40 –42] have been applied to various disciplines, including mechanical fault diagnosis and prognosis. Especially, adaptive network-based fuzzy inference systems (ANFISs), in which fuzzy membership functions and fuzzy rules are derived via the learning of abundant historical data instead of via human experience or intuition, is an efficient way of handling the regression problem [43]. Combining the superiority of both ANNs and fuzzy inference systems, ANFIS presents significantly better self-learning ability, robustness, and adaptability. Soualhi et al. [44] developed an approach that incorporates ANFIS for the prognosis and health management of REBs. Ma et al. [45] introduced degradation process prediction into condition monitoring and employed ANFIS to effectively estimate the RUL of bearings. Chen et al. [46] adopted ANFIS and high-order particle filtering to form a prognostic method of machine condition. In view of these advantages, ANFIS is used as the regression model in this study.

Based on the above-mentioned analysis, locally linear fusion regression (LLFR) based on LLE and ANFIS is proposed for the RUL prediction of REBs in this study. Initially, features showing the operating status are extracted from the vibration signals of the REBs. Second, LLE is adopted to fuse the obtained features into a condition indicator. Finally, ANFIS is employed to predict the RUL. Furthermore, two benchmark models with different fusion methods are used to evaluate the performance of the proposed model. RUL prediction is composed of a degradation estimation and time prediction of the RUL.

The rest of the paper proceeds as follows. Section 2 systematically presents the proposed method. The reported approach is investigated using real data and compared with peer models in Section 3. Section 4 presents the conclusions of the paper.

2 Methodology

In this section, the systematic methodology of the proposed approach is given. The implementation process of the reported approach, including feature extraction, LLE fusion, and ANFIS, is demonstrated in detail. Moreover, the motivation behind using the proposed method is also explained.

2.1 Feature extraction

The vibration signals of REBs consist of random noise and periodical fluctuations. The features from the time domain and time– frequency domain, which have clear physical meaning, efficiently denoise the original signals [47]. To better grasp the degradation characteristics, 15 time domain features are first extracted. Furthermore, the energy values of sub-frequency bands obtained by a wavelet packet transform (WPT) are considered as the timeߝfrequency domain features.

The raw signal, which is composed of M segments of signal with length N, is denoted as $x_{n}^{i}$ (i ∈ [1, M], n ∈ [1, N]). The features $f_{m}^{i}$ (m is number of features categories) are obtained as shown in Table 1. In Table 1, the band energy is denoted as E and $D_{b}^{a, h}$ is the wavelet packet coefficients of the hth frequency band at the ath level.

Table 1
The specification of the features

Features Formula

Maximum f₁ = max{ |x_n| } (1)

Minimum f₂ = min{ x_n } (2)

Average absolute amplitude $f_{3} = \frac{1}{N} \sum_{n = 1}^{N} | x_{n} |$ (3)

Mean value $f_{4} = \frac{1}{N} \sum_{n = 1}^{N} x_{n}$ (4)

Root mean square $f_{5} = \sqrt{\frac{1}{N} \sum_{n = 1}^{N} x_{n}^{2}}$ (5)

Peak-to-peak value f₆ = f₁ - f₂ (6)

Variance $f_{7} = \frac{1}{N - 1} \sum_{n = 1}^{N} (x_{n} - f_{4})^{2}$ (7)

Standard deviation $f_{8} = \sqrt{f_{7}}$ (8)

Kurtosis $f_{9} = \frac{1}{N} \sum_{n = 1}^{N} x_{n}^{4}$ (9)

Skewness $f_{10} = \frac{1}{N} \sum_{n = 1}^{N} x_{n}^{3}$ (10)

Kurtosis factor $f_{11} = \frac{f_{9}}{(f_{5})^{4}}$ (11)

Crest factor $f_{12} = \frac{f_{1}}{f_{5}}$ (12)

Waveform factor $f_{13} = \frac{f_{5}}{f_{3}}$ (13)

Impulse factor $f_{14} = \frac{f_{1}}{f_{3}}$ (14)

Margin factor $f_{15} = \frac{f_{1}}{{(\frac{1}{N} \sum_{n = 1}^{N} \sqrt{| x_{n} |})}^{2}}$ (15)

Energy values $f_{15 + h} = E_{a, h} = \sum_{b} {| D_{b}^{a, h} |}^{2}$ (16)

Features	Formula
Maximum	f₁ = max{ \|x_n\| } (1)
Minimum	f₂ = min{ x_n } (2)
Average absolute amplitude	$f_{3} = \frac{1}{N} \sum_{n = 1}^{N} \| x_{n} \|$ (3)
Mean value	$f_{4} = \frac{1}{N} \sum_{n = 1}^{N} x_{n}$ (4)
Root mean square	$f_{5} = \sqrt{\frac{1}{N} \sum_{n = 1}^{N} x_{n}^{2}}$ (5)
Peak-to-peak value	f₆ = f₁ - f₂ (6)
Variance	$f_{7} = \frac{1}{N - 1} \sum_{n = 1}^{N} (x_{n} - f_{4})^{2}$ (7)
Standard deviation	$f_{8} = \sqrt{f_{7}}$ (8)
Kurtosis	$f_{9} = \frac{1}{N} \sum_{n = 1}^{N} x_{n}^{4}$ (9)
Skewness	$f_{10} = \frac{1}{N} \sum_{n = 1}^{N} x_{n}^{3}$ (10)
Kurtosis factor	$f_{11} = \frac{f_{9}}{(f_{5})^{4}}$ (11)
Crest factor	$f_{12} = \frac{f_{1}}{f_{5}}$ (12)
Waveform factor	$f_{13} = \frac{f_{5}}{f_{3}}$ (13)
Impulse factor	$f_{14} = \frac{f_{1}}{f_{3}}$ (14)
Margin factor	$f_{15} = \frac{f_{1}}{{(\frac{1}{N} \sum_{n = 1}^{N} \sqrt{\| x_{n} \|})}^{2}}$ (15)
Energy values	$f_{15 + h} = E_{a, h} = \sum_{b} {\| D_{b}^{a, h} \|}^{2}$ (16)

2.2 LLE for feature fusion

Some of the obtained original features contain similar properties and some have partial information of the degradation process. To reflect the degradation process accurately, all original features are fused into a condition indicator for condition estimation and RUL prediction. In this work, feature fusion is achieved by the LLE algorithm.

As a representation of manifold learning, LLE is an efficient tool of dimensionality reduction based on unsupervised learning, especially for nonlinear systems [24]. Mapping data points of high-dimensional space into low-dimensional space, LLE achieves a low-dimensional approximation for the raw data. The steps in feature fusion using LLE are outlined as follows:

Step 1. Seek the k nearest neighbors for each features sample point f_i using the Euclidean distance. Letting f_i and f_j be neighbors, calculate the distances D_ij based on the following equation: $D_{ij} = | f_{i} - f_{j} | .$ (1)

Step 2. Calculate weights ω_ij, which can reconstruct f_i with its k neighbors. Reconstruction error α is minimized by the following cost function: $α (ω) = \sum_{i = 1}^{M} {| f_{i} - \sum_{j = 1}^{k} ω_{ij} f_{j} |}^{2},$ (2) subject to two conditions, ${\begin{matrix} \sum_{j = 1}^{k} ω_{ij} = 1 \\ ω_{ij} = 0 \end{matrix} .$ (3)

Step 3. Calculate the low-dimensional embedding y_i using ω_ij. The value of y_i is obtained by minimizing $θ (y) = \sum_{i = 1}^{M} {| y_{i} - \sum_{j = 1}^{k} ω_{ij} y_{j} |}^{2},$ (4) subject to two conditions, ${\begin{matrix} \sum_{i = 1}^{M} y_{i} = 0 \\ \frac{1}{M} \sum_{i = 1}^{M} y_{i} y_{i}^{T} = I_{d \times d} \end{matrix},$ (5) where I_d×d is the identity matrix. To seek y under these conditions, a matrix W is established based on matrix ω. $W = (I - ω)^{T} (I - ω) .$ (6)

As a result, the condition indicator y is the d eigenvector, which is equal to the d smallest non-zero eigenvalues of W.

2.3 ANFIS for estimation and prediction

As mentioned above, a nonlinear regression model is required for degradation evaluation and RUL prediction based on the obtained condition indicator. Integrating neural networks and fuzzy logic rules, ANFIS is a multi-layer adaptive system capable of handling the complexity and nonlinearity [43]. The difference between ANFIS and fuzzy inference systems is that ANFIS adopts a back-propagation method to control errors. In a sense, ANFIS is a combination of ANN and fuzzy logic. Compared with ANN, ANFIS has better self-learning capability, stability, and self-adaptability. Furthermore, high efficiency, a fast convergence rate, and good prediction performance are also reflected in ANFIS. Therefore, ANFIS is used as the regression model in this work.

ANFIS can find optimal solutions with the back-propagation method or with a combination of the least squares method and the back-propagation method to estimate the membership function parameters by combining the two systems [48]. Thus, the degradation estimation y_i+1 and RUL prediction R_i can be expressed as $y_{i + 1} = ANFIS$ (7) $R_{i} = ANFIS [y_{i}, y_{i - 1}, \dots, y_{i - g}]$ (8)

Two inputs, y₁ and y₂, and one output, Y, in this system are assumed, as illustrated in Fig. 1. Moreover, a common set of fuzzy if– then rules in the Sugeno fuzzy model is depicted as $〈 \begin{matrix} If y_{1} is A_{1} and y_{2} is B_{1}, then F_{1} = p_{1} y_{1} + q_{1} y_{2} + r_{1} \\ If y_{1} is A_{2} and y_{2} is B_{2}, then F_{2} = p_{2} y_{1} + q_{2} y_{2} + r_{2} \end{matrix} .$ (9) ∥

Fig.1

The structure of ANFIS.

Figure 1 shows that ANFIS consists of five layers. Let O_u,s represent the output at the uth node of the sth layer. The information of each layer is described as follows:

Layer 1. Nodes in this layer are adaptive nodes. O_1,s, which determines the degree to which a given input meets the fuzzy sets, is the membership function of A_s and B_s. It is described as $O_{1, s} = {\begin{matrix} μ_{A_{s}} (y_{1}), s = 1, 2 \\ μ_{B_{s}} (y_{2}), s = 3, 4 \end{matrix},$ (10) where μ_As and μ_Bs are the membership functions used. A Gaussian function is selected as the membership function, which is denoted as $μ A_{s} (y) = e^{- \frac{(y - c)^{2}}{2 σ^{2}}},$ (11) where {c, σ} is the parameter set that can transform the function shapes.

Layer 2. Nodes of this layer are fixed nodes, labeled by Π. The output O_2,s, which denotes the firing strength of a rule, is a product of the incoming signals. $O_{2, s} = w_{s} = μ A_{s} (y_{1}) \times μ B_{s} (y_{2}), s = 1, 2 .$ (12)

Layer 3. Nodes of this layer, labeled N, are fixed nodes. The ratio of the sth rule’s firing strength to the sum of the firing strengths of all rules is calculated by this layer. The output of this layer is the normalized firing strength, expressed as $O_{3, s} = {\bar{w}}_{t} = \frac{w_{t}}{w_{1} + w_{2}}, t = 1, 2 .$ (13)

Layer 4. Nodes of this layer are adaptive nodes. The contribution of each rule is calculated as the output of this layer, which is given by $O_{4, s} = {\bar{w}}_{s} F_{i} = {\bar{w}}_{s} (p_{s} + q_{s} + r_{s}),$ (14) where p_s, q_s, and r_s are consequent parameters.

Layer 5. The sum of all the incoming signals in this layer is calculated by the single node as the final output. Meanwhile, the fuzzy results of each rule are converted into crisp output by defuzzification. $O_{5, s} = \sum_{s} w_{s} F_{s} = \frac{\sum_{s = 1} w_{s} F}{\sum_{s = 1} w_{s}} .$ (15)

2.4 Overview of the proposed approach

Having separately described the abovementioned constituents, the LLFR process can be implemented, as illustrated in Fig. 2.

Fig.2

The flow chart of the proposed approach.

Step 1. Acquire the vibration signal of the REBs throughout their life cycle.

Step 2. Extract the original features from the time domain and timeߝfrequency domain of the vibration signal.

Step 3. Employ the LLE method to fuse the extracted features into a condition indicator that can adequately reflect the degradation process of the REBs.

Step 4. Adopt equal interval sampling and smoothing to denoise the indicator.

Step 5. Apply ANFIS to the degradation estimation and RUL prediction.

3 Experimental results and analysis

3.1 Experimental data and performance criteria

The reported approach was investigated using real REB data originating from the IEEE PHM 2012 Prognostic Challenge [49]. To depict the degradation process of the tested REB significantly, an experimental platform was used to provide real data throughout the lifecycle [50]. The experimental platform consisted of three modules: rotating module, degradation generation module, and measurement module (Fig. 3). With a sampling frequency 25.6 kHz and operating conditions of 1800 rpm and 4000 N loads, the vibration signal of the REB was acquired every 10 s. The collected vibration signal is demonstrated in Fig. 4.

Fig.3

The structure of the experimental platform [49].

Fig.4

Raw signal of the REB to be tested.

As seen in Fig. 4, the amplitude changed from stationarity to a gradual increase. The raw vibration signal roughly reflects the whole life process, from normal conditions to complete failure.

In this work, the mean absolute error (MAE), mean absolute relative (MARE), root mean square error (RMSE), correlation coefficient (CC), and equal coefficient (EC) are introduced to assess the performance of the reported approach. $MAE = \frac{1}{L} \sum_{i = 1}^{L} | y_{i} - {\hat{y}}_{i} |,$ (16) $MARE = \frac{1}{L} \sum_{i = 1}^{L} | \frac{y_{i} - {\hat{y}}_{i}}{y_{i}} |,$ (17) $RMSE = \sqrt{\frac{1}{L} \sum_{i = 1}^{L} {(y_{i} - {\hat{y}}_{i})}^{2}},$ (18) $CC = \frac{\sum_{i = 1}^{L} (y_{i} - \bar{y}) ({\hat{y}}_{i} - \bar{\hat{y}})}{\sqrt{\sum_{i = 1}^{L} {(y_{i} - \bar{y})}^{2}} \sqrt{\sum_{i = 1}^{L} {({\hat{y}}_{i} - \bar{\hat{y}})}^{2}}},$ (19) $EC = 1 - \frac{\sqrt{\sum_{i = 1}^{L} (y_{i} - {\hat{y}}_{i})^{2}}}{\sqrt{\sum_{i = 1}^{L} y_{i}^{2}} + \sqrt{\sum_{i = 1}^{L} {\hat{y}}_{i}^{2}}},$ (20) where y_i represents the observed values, $\bar{y}$ is the average of the observed values, ${\hat{y}}_{i}$ represents the prediction values, is the average of prediction values, and L is the length of the output data.

3.2 Feature extraction and fusion

According to the above-mentioned procedure, multiple features derived from time domain and timeߝfrequency domain of the vibration signal were extracted for building a condition indicator. Fourteen time domain features were obtained first. Additionally, the energy values of sub-frequency bands derived by the WPT as the timeߝfrequency features were adopted. The decomposition level of three layer was obtained using the empirical formula [51]. Meanwhile, “db4,” one of most widely used and effective mother wavelets, was introduced for decomposing the raw signal. Eight energy values were acquired. Thus, a total of twenty-three features were obtained in this work, which are given in Fig. 5.

Fig.5

The extracted features for raw signal.

As can be seen in Fig. 5, the results reflected by these features have large differences. Some features, e.g., f₃, f₁₈, and f₁₉ were sensitive only to severe failure, but they were unresponsive for the beginning and development of failure. On the other hand, some features, e.g., f₄ and f₅, were sensitive to the occurrence and development of failure but unresponsive at the complete failure stage. From features f₆ and f₈, a rough degradation trend can be observed. However, the overall information is noisy. Moreover, irregular trends are reflected in the lesser features such as f₁ and f₁₃. Of course, there also are accepted features that can better exhibit the failure process, e.g., f₂, f₉, and f₁₆. To exhibit the overall condition of these obtained features more specifically, a sensitive degree statistics of these features in different stages is shown in Table 2.

Table 2

Sensitive degree of the features

	Beginning	Development	Failure
f₁, f₁₃	0	0	0
f₁₁, f₁₂, f₁₄, f₁₅	0	0	1
f₃, f₁₈, f₁₉, f₂₂, f₂₃	0	0	2
f₂₀, f₂₁	0	1	2
f₄, f₅,	1	1	0
f₆, f₇, f₈, f₁₇	1	2	2
f₂, f₉, f₁₀, f₁₆	2	2	2

Note: 0 = insensitive, 1 = mid-sensitive, 2 = sensitive.

Based on the above analysis, it is shown that none of the features could exhibit all of the information on the degradation process. An ideal condition indicator is desired that contains complete failure information of the different features and maintains consistent sensitivity over changing stages of the degradation process. Consequently, LLE was introduced to integrate the original features scientifically and systematically. In this work, LLE, with the target dimension of one, fused the twenty-three features to be a condition indicator, as shown in Fig. 6.

Fig.6

The condition indicator of multi-feature fusion for REB.

As shown in Fig. 6, the obtained condition indicator had much noise and drastic fluctuations, which affected the evaluation effect and prediction performance. Thus, a denoising technique was introduced to address the issue. In this work, the condition indicator was processed using equal interval sampling of 1 min and median filtering with a window size of 10. The derived result is given in Fig. 7.

Fig.7

The denoised condition indicator.

Figure 7 exhibits a multi-feature indicator, which synthetically reflected the degradation process of the REB. An obvious staggered trend is observed. The running state of the REB can be considered as having three periods: the normal stage (Stage 1), the failure growth stage (Stage 2), and the serious stage (Stage 3).

The curve condition of each stage reflects the current situation. In the Stage 1, the relatively stationary volatility of the curve shows that the REB was under regular working conditions. Accompanying the occurrence of early fault, the REB then began to step into the Stage 2. The gradually increased curve indicates that the failure was continuously developing. The failure growth stage occupies a large proportion of the failure process. This is owing to the fact that the failure behavior takes time to progress from an early fault to complete failure. A sharp increase in the curve shows that the REB entered into the Stage 3, the serious stage. It is quite clear that a complete and comprehensive process is exhibited by the condition indicator. What is more, LLE completed the fusion task excellently.

Before establishing the regression model, a sample set, including training and testing, must be created. A failure process composed of the Stage 2 and the Stage 3 was utilized for degradation estimation and RUL prediction owing to the minimal contribution of the Stage 1 to the failure. Around 60% of the sample was utilized for training and 40% for testing.

3.3 Degradation estimation

To highlight the performance of the proposed model, peer models in which LLE was replaced by either DM or PCA were introduced. The comparison models can be called DM-ANFIS and PCA-ANFIS, respectively. The selection of the input structures of the peer models was the same as that for the proposed method. The optimal structure, which is expressed as y_i-G = [y_i-1, y_i-2, …, y_i-g] was determined by trial and error. The results of the degradation estimation are displayed in Fig. 8 for the different models.

Fig.8

The estimation results of different models: (a) LLFR, g = 5; (b) DM-ANFIS, g = 5; (c) PCA-ANFIS, g = 5.

As is observed in Fig. 8(a) and (b), the predicted curve of LLFR and DM-ANFIS better coincided with the variation of the observed curve. This reflects the better fitting performance of the manifold learning. More specifically, LLFR exceeded DM-ANFIS in local capture. The worse results for PCA-ANFIS are in sharp contrast to the results for the other models in Fig. 8(c), and suggest shows that manifold learning is superior to PCA in multiple feature integration, and that the effect of LLE was the best. The ability to estimate extreme values is a crucial criterion for measuring model performance. In contrast to Figs. 5(a), 8(b) and (c) show that the predicted extreme values of the reported model more closely approach the observed values. LLFR has the ability to capture the extremum. However, the benchmark models show obviously deviation, especially in the evaluation of the serious stage. To all appearances, the reported approach is superior to the peer models.

To further exhibit the performance of the models in detail, the absolute relative errors yielded by each model are given in Fig. 9. It is clear that the LLFR model has the best stability among these models. The others show obvious drastic fluctuations.

Fig.9

Comparisons of absolute relative error using different models.

Based on the adopted performance criteria, a quantitative analysis is additionally given in Table 3.

Table 3

Comparison of the estimation performance

Model	CC	MARE	EC
LLFR	0.9003	0.1348	0.8744
DM-ANFIS	0.8949	0.2122	0.8718
PCA-ANFIS	0.8405	0.2598	0.8237

In the table, the superiority of the reported model is clearly exhibited in terms of the CC, MARE, and EC. It is also shown that LLFR has a better effect and accuracy compared with the benchmark models. Moreover, observing the CC and EC, the preponderance of manifold learning is displayed once again. Therefore, it is concluded that LLFR can significantly merge inherent characteristics of the data, remove the interference of random elements, and obtain a good result.

3.4 RUL prediction

After the degradation estimation of the REB, RUL prediction was performed using each model. The results are shown in Fig. 10.

Fig.10

The RUL prediction results of different models: (a) LLFR, g = 5; (b) DM-ANFIS, g = 2; (c) PCA-ANFIS, g = 2.

Figure 10(a-c) demonstrate the results of the RUL prediction using LLFR, the DM-ANFIS model, and the PCA model, respectively. At the beginning of the RUL prediction, the LLFR results coincide with the observed RUL. The comparison models have larger fluctuations from the start. Hence, the reported model has the best representation in the beginning stage, whereas the peers yield large errors. As time goes on, the comparison models gradually deviate from the observed values. In contrast, the LLFR model traces the RUL trend continuously, and the obtained predicted values converge to the observed RUL. Thus, among these models, LLFR furnishes more precise prediction results and exhibits the best performance. Furthermore, the results of DM-ANFIS are slightly better than those of PCA-ANFIS. However, these models fail to provide ideal results in the serious stage. Thus, the performance of the LLFR model is superior to that of the peer models in this stage.

Based on the above analysis, the fusion ability of LLE outperforms the comparison fusion methods, such that random errors of the stochastic process implied in the raw features are weakened greatly. A statistical analysis of absolute error is demonstrated in Fig. 11 using a boxplot.

Fig.11

The boxplot of absolute error using different models.

Overall, the boxes corresponding to the proposed model are in the low range of absolute error, whereas the competing models are in relatively high ranges. This indicates that the reported model has the best comprehensive performance among these models. Comparing the heights of the boxes, which reflect the distributions of the absolute errors, it is shown that LLFR has a relatively concentrated distribution of errors, as shown by its shortest box height. Observing the position of the median lines, the reported model is the best stationary in that the mean values of the absolute errors are the minimum. Thus, LLFR outperforms its comparison models.

Visual results are shown with the above quality analysis. Naturally, a quantitative comparison was performed for reflecting more specific information. The prediction performance comparison of the three competing models is given in Table 4.

Table 4

Comparison of the prediction performance

Model	RMSE	MAE	EC
LLFR	13.419	9.5732	0.8863
DM-ANFIS	30.059	25.203	0.7845
PCA-ANFIS	33.578	28.233	0.7569

Table 4 compares the performance for the different models. Comparing these criteria, the advantages of the LLFR model are displayed based on the RMSE, MAE, and EC. It is found that the reported model is superior to the others in terms of prediction accuracy. The related RMSE, MPE, and EC differences between LLFR and DM-ANFIS are 55.36%, 62.02%, and 12.98%, respectively, and between LLFR and PCA-ANFIS the differences are 60.04%, 66.09%, and 17.10%, respectively. These results also demonstrate the effectiveness of the proposed method.

4 Conclusion

The condition of REBs is extremely important in rotating machinery. A reliable RUL prediction can effectively improve the implementation of condition-based maintenance for REBs. A locally linear fusion fuzzy regression approach is herein developed to predict the RUL of REBs. In the proposed method, the LLE technique is adopted to merge original features obtained from the vibration signal of the REBs to create a condition indicator. The condition indicator is then used by ANFIS to estimate the RUL. The reported method was applied to experimental data from an REB. The experimental results revealed that the reported approach demonstrated better performance than competing approaches, and can provide more reliable and accurate prediction results. It was able to accurately predict the RUL of the REB. The primary innovation of this work is the proposed predictor following a philosophy that the comprehensive information can be obtained by using multiple feature fusion. In particular, the fusion method can be replaced with other methods in the implementation process. Thus, the approach may be generalized in some extent. Further work to improving the prediction accuracy should be undertaken in future.

Footnotes

Acknowledgments

This work is supported in part by the National Key Research & Development Program of China (2016YFE0132200), the Project of Chongqing Science & Technology Commission (cstc2015zdcy-ztzx70012), Fundamental Research Funds for the Central Universities (106112014CDJZR095501), and Chongqing Research Program of Basic Science & Frontier Technology with Grant (cstc2017jcyjB0305). The valuable comments and suggestions from the editor and the two reviewers are very much appreciated.

References

Lei

, Lin

, Zuo

M.J.

and He

, Condition monitoring and faultdiagnosis of planetary gearboxes: A review, Measurement48 (2014), 292–305.

Ghods

and Lee

H.H.

, Probabilistic frequency-domain discretewavelet transform for better detection of bearing faults ininduction motors, Neurocomputing188 (2016), 206–216.

Cerrada

, Sánchez

R.V.

, Li

, Pacheco

, Cabrera

Valente de Oliveira

and Vásquez, R.E.A review on data-drivenfault severity assessment in rolling bearings, MechanicalSystem and Signal Processing99 (2018), 169–196.

Sikorska

, Hodkiewicz

and Ma

, Prognostic modelling optionsfor remaining useful life estimation by industry, Me-chanicalSystems and Signal Processing25 (2011),1803–1836.

Wang

and Shen

, An equivalent cyclic energy indicator forbearing performance degradation assessment, Journal of Vibration and Control22 (2016), 2380–2388.

, Valente de Oliveira

Sánchez

R.V.

, Cerrada

, Zurita

and Cabrera

, Fuzzy determination of informative frequencyband for bearing fault detection, Journal of Intelligent & Fuzzy Systems,30(6) (2016), 3513–3525.

, Kurfess

and Liang

, Stochastic prognostics for rollingelement bearings, Mechanical Systems and Signal Processing14(5) (2000), 747–762.

Singleton

R.K.

II , Strangas

E.G.

and Aviyente

, Extended Kalmanfiltering for remaining-useful-life estimation of bearings, IEEE Transactions on Industrial Electronics62 (2015), 1781–1790.

Chen

and Tsui

K.L.

, Condition monitoring and remaining useful life prediction using degradation signals: Revisited, IIE Transactions45(9) (2013), 939–952.

10.

Yin

, Ding

S.X.

, Xie

and Luo

, A review on basic data-drivenapproaches for industrial process monitoring, IEEE Transactionson Industrial Electronics61(11) (2014) 6418–6428.

11.

Mosallam

, Medjaher

and Zerhouni

, Data-driven prognosticmethod based on Bayesian approaches for direct remaining useful lifeprediction, Journal of Intelligent Manufacturing27(5) (2016), 1–12.

12.

Wei

, Wang

, He

and Bao

, A novel intelligent method forbearing fault diagnosis based on affinity propagation clustering andadaptive feature selection, Knowledge-Based Systems116 (2017), 1–12.

13.

and Liang

, Time-frequency signal analysis for gearbox faultdiagnosis using a generalized synchrosqueezing transform, Mechanical Systems and Signal Processing26(1) (2012), 205–217.

14.

, Sánchez

R.V.

, Zurita

, Cerrada

, Cabrera

and Vásquez

R.E.

, Multimodal deep support vector classification withhomologous features and its application to gearbox fault diagnosis, Neurocomputing168 (2015), 119–127.

15.

Zhang

and Li

, Bearing condition recognition and degradationassessment under varying running conditions using NPE and SOM, Mathematical Problems in Engineering (2014), 781583.

16.

Cabrera

, Sancho

, Li

, Cerrada

, Sánchez

R.V.

Pacheco

and Valente de Oliveira

, Automatic feature extraction oftime-series applied to fault severity assessment of helical gearboxin stationary and non-stationary speed operation, Applied SoftComputing (2017), RUL: doi.org/10.1016/j.asoc.2017.04.016.

17.

Jammu

N.S.

and Kankar

P.K.

, A review on prognosis of rolling element bearings, International Journal of Engineering Science & Technology3(10) (2011) 7497–7503.

18.

, Liang

and Wang

, Criterion fusion for spectral segmentation and its application to optimal demodulation of bearingvibration signals, Mechanical Systems and Signal Processing64-65 (2015), 132–148.

19.

Wang

, Peng

, Zi

, Jin

and Tsui

K.L.

, A two-stagedata-driven-based prognostic approach for bearing degradation problem, IEEE Transactions on Industrial Informatics12(3) (2016), 924–932.

20.

Liu

, He

, Liu

, Lu

, Zhao

and Zhao

, Remaining useful life prediction of rolling bearings using PSR, JADE, and extremelearning machine, Mathematical Problems in Engineering (2016), 8623530.

21.

, Lei

, Liu

and Lin

, A particle filtering-basedapproach for remaining useful life predication of rolling elementbearings, International Conference on Prognostics and HealthManagement, IEEE, (2015), pp. 1–8.

22.

Wei

, Chen

, Zhou

and Wang

, Remaining useful lifeprediction using a stochastic filtering model with multi-sensorinformation fusion, pp, Prognostics and System Health Management Conference, IEEE, (2011), 1–6.

23.

Dong

and Luo

, Bearing degradation process prediction based onthe PCA and optimized LS-SVM mode, Measurement46(9) (2013), 3143–3152.

24.

Roweis

S.T.

and Saul

L.K.

, Nonlinear dimensionality reduction bylocally linear embedding, Science290 (2000), 2323–2326.

25.

Tenenbaum

J.B.

, Silva

and Langford

J.C.

, A global geometric framework for nonlinear dimensionality reduction, Science290 (2000), 2319–2323.

26.

Coifman

R.R.

and Lafon

, Diffusion maps, Computational Harmonic Analysis & Computational Harmonic Analysis21(1) (2006), 5–30.

27.

Song

, Liu

, Yang

and Peng

, Data-driven hybrid remaining useful life estimation approach for spacecraft lithium-ion battery, Microelectronics Reliability75 (2017), 142–153.

28.

Dong

and He

, A segmental hidden semi-Markov model (HSMM)-based diagnostics and prognostics framework and methodology, Mechanical Systems and Signal Processing21(5) (2007) 2248–2266.

29.

Ali

J.B.

, Saidi

, Mouelhi

, Chebel-Morello

and Fnaiech

, Linear feature selection and classification using PNN and SFAM neural networks for a nearly online diagnosis of bearing naturally progressing degradations, Engineering Applications of Artificial Intelligence42 (2015), 67–81.

30.

Vijay

G.S.

, Kumar

H.S.

, Srinivasa

P.P.

, Sriram

N.S.

and Rao

R.B.K.N.

, Evaluation of effectiveness of wavelet based denoising schemesusing ANN and SVM for bearing condition classification, Computational Intelligence and Neuroscience (2012),582453.

31.

Bai

, Sun

Z.Z.

, Zeng

, Deng

and Li

, A multi-pattern deepfusion model for short-term bus passenger flow forecasting, Applied Soft Computing58 (2017), 669–680.

32.

Mosallam

, Medjaher

and Zerhouni

, Data-driven prognosticmethod based on Bayesian approaches for direct remaining useful lifeprediction, Journal of Intelligent Manufacturing27(5) (2016), 1037–1048.

33.

Miao

, Zhang

, Liu

Z.W.

and Zhang

, Conditionmulti-classification and evaluation of system degradation processusing an improved support vector machine, Microelectronics Reliability75 (2017), 223–232.

34.

and Zuo

M.J.

, An LSSVR-based algorithm for online systemcondition prognostics, Expert Systems with Applications39(5) (2012), 6089–6102.

35.

Loutas

T.H.

, Roulias

and Georgoulas

, Remaining useful life estimation in rolling bearings utilizing data-driven probabilistic E-support vectors regression, IEEE Transactions on Reliability62(4) (2013), 821–832.

36.

Bai

, Wang

, Li

, Xie

and Wang

, Dynamic forecast of daily urban water consumption using a variable-structure supportvector regression model, Journal of Water Resources Planning & Management141(3) (2015), 04014058.

37.

, Cheng

, Yu

, Bai

and Li

, Water-quality predictionusing multimodal support vector regression: Case study of JialingRiver, China, Journal of Environmental Engineering143(10) (2017), 04017070.

38.

, Ledo

, Delgado

, Cerrada

, Pacheco

, Cabrera

Sánchez

R.V.

and Valente de Oliveira

, A Bayesian approach toconsequent parameter estimation in probabilistic fuzzy systems and its application to bearing fault classification, Knowledge-Based Systems129 (2017), 39–60.

39.

Bai

, Wang

, Xie

, Li

and Li

, Additive model for monthly reservoir inflow forecast. Journal of Hydrologic Engineering20(7) (2014), 04014079.

40.

Ali

J.B.

, Chebel-Morello

, Saidi

, Malinowski

and Fnaiech

, Accurate bearing remaining useful life prediction based on Weibull distribution and artificial neural network, Mechanical Systemsand Signal Processing56-57 (2015), 150–172.

41.

Wang

, Youn

B.D.

and Hu

, A generic probabilistic framework forstructural health prognostics and uncertainty management, Mechanical Systems and Signal Processing28 (2012), 622–637.

42.

Son

K.L.

, Fouladirad

, Barros

, Levrat

and Iung

, Remaininguseful life estimation based on stochastic deterioration models: Acomparative study, Reliability Engineering & System Safety112(4) (2013), 165–175.

43.

Jang

J.S.R.

, ANFIS: Adaptive-network-based fuzzy inference system, IEEE Transactions on Systems Man and Cybernetics23(3) (2010), 665–685.

44.

Soualhi

, Razik

, Clerc

and Doan

D.D.

, Prognosis of bearing failures using hidden Markov models and the adaptive neuro-fuzzy inference system, IEEE Transactions on Industrial Electronics61(6) 2864–2874.

45.

, Kang

J.S.

and Zhao

C.Y.

, Research on condition monitoring of bearing health using vibration data, Applied Mechanics & Materials226-228 (2012), 340–344.

46.

Chen

, Zhang

, Vachtsevanos

and Orchard

, Machine condition prediction based on adaptive neuro-fuzzy and high-order particle filtering, IEEE Transactions on Industrial Electronics58 (2011), 4353–4364.

47.

Lei

, He

, Zi

and Chen

, New clustering algorithm-basedfault diagnosis using compensation distance evaluation technique, Mechanical Systems and Signal Processing22(2) (2008), 419–435.

48.

Jang

J.S.R.

, Sun

C.T.

and Mizutani

, Neuro-fuzzy and soft computing: A computational approach to learning and machine intelligence, IEEE Transactions on Automatic Control42(10) (1997), 1482–1484.

49.

FEMTO-ST, IEEE PHM Data Challenge, online website, lastaccessed on Jan 20, (2014) http://www.femto-st.fr/en/Researchdepartments/AS2M/Research-groups/PHM/IEEE-PHM-2012-Datachallenge.php.

50.

Nectoux

, Gouriveau

, Medjaher

, Ramasso

, Morello

Zerhouni

and Varnier

, PRONOSTIA: Anexperimental platform for bearings accelerated life test,pp, IEEE International Conference on Prognostics and Health Management (2012), 1–8.

51.

Seo

and Kim

, O, Kisi, V.P. Singh and K. Parasuraman, River stageforecasting using wavelet packet decomposition and machine learningmodels, Water Resources Management30(11) (2016), 4011–4035.