Fuzzy feature fusion and multimodal degradation prognosis for mechanical components

Abstract

Reliable degradation prognosis of mechanical components is very important for condition-based maintenance to improve the reliability and reduce the cost of maintenance. This paper reports the development of a fuzzy feature fusion and multimodal regression method for the degradation prognosis of mechanical components. Initially, the raw features from the vibration signals of the mechanical components are extracted. A degradation index is subsequently yielded by merging the obtained features through/using the fuzzy fusion technique. The ensemble empirical mode decomposition is then introduced to decompose the fusion index into several multimodal sub-series to acquire more detailed information. Extreme learning machines are established to predict the sub-series in different modes. The predicted results are obtained by integrating the multimodal sub-results. The reported approach was evaluated with real data from a rolling element bearing. Moreover, two peer models were imported to validate the effectiveness of the proposed method. The experimental results indicate that the reported approach is capable of erecting the degradation index reflecting the bearing degradation and that it had better performance in the remaining useful life prediction than the peer methods.

Keywords

Degradation prognosis fuzzy fusion degradation index ensemble empirical mode decomposition extreme learning machine

1 Introduction

Prognosis and health management (PHM) technology [1] is very important for the healthy service of mechanical components. Effective PHM helps to decrease the production downtime and maintenance costs of mechanical components. It can also reduce or eliminate machine faults while providing an accurate degradation prognosis [2]. Therefore, a useful method for predicting the degradation of mechanical components has to be sought.

Fault prediction includes feature extraction, degradation prognosis, and remaining useful life (RUL) prediction. Among these ones, degradation prognosis is a vital technology of fault prediction [3], which is related to the veracity of fault prediction and the accuracy of RUL or fault probability prediction. Therefore, much research has considered fault diagnosis and degradation prognosis of mechanical component lifetime [4 –6]. There are also different approaches for degradation prognosis, which has been considered. These methods are usually classified into three categories: model-based prognosis models [7], data-driven prognosis models [8], and hybrid prognosis models [9]. Among these ones, data-driven models have been gaining momentum recently and are becoming the mainstream in the degradation prognosis field [10]. Data-driven models collect signals and data from periodic monitoring of the components and predict their future evolution of degradation [2]. Moreover, data-driven models are relatively easier and provide better results, without the assumption of physical parameters.

Mechanical fault diagnosis and degradation prognosis are normally based on running condition indexes, including vibration, temperature, current and voltage, oil, and wear signals [11]. Among these ones indexes, vibration signals can be used to extract information to diagnose abnormal conditions [12], and thus are more effective and suitable than other indexes at reflecting the running condition of mechanical components [11].

For degradation prognosis, feature extraction is the first crucial step therefore a suitable method is utilized for feature fusion. Finally, an effective prognosis model could be selected to prognose the degradation based on the degradation index.

Feature extraction, which effectively reflects the machine running condition using the original vibration signals, is an important prerequisite for degradation prognosis. Typical methods such as time domain, frequency domain, and time-frequency domain analyses have been developed and reported [13]. After extracting the features, it is necessary the fusion of those features.

A variety of feature fusion methods have been adopted in the research. The kernel joint approximate diagonalization of the eigen-matrices approach was applied for feature fusion [14]. The locality preserving projections was used to fuse features [15]. Combining phase space reconstitution (PSR) with between- and within-class scatter evaluation, a feature-weighted fusion method was proposed [9]. A new fusion technique was used based on fuzzy fusion of multiple criteria [16]. This technique was originally used for fusing the time domain and the time–frequency domain of the bearings to establish a degradation index.

Ensemble empirical mode decomposition (EEMD), an improved method of empirical model decomposition (EMD), has been widely adopted for signal decomposition by many researchers. EMD [17], which was proposed by Huang [18], can decompose complicated signals into several intrinsic mode functions (IMFs) self-adaptively. However, there is a mode mixing problem when using EMD. EEMD [18] was thus developed to overcome the mode mixing problem by adding finite white noise into the original signals. The EEMD is used to improve the accuracy of the prognosis result [15 –19].

Data-driven models such as the support vector machine [20], artificial neural network (ANN) [21], and extreme learning machine (ELM) [9] have been used to establish prediction models, such as short-term bus passenger flow forecasting [22] and RUL estimation [23]. Among these ones models, ELM, which was developed by Huang et al. [24], is a novel algorithm of single hidden-layer feed-forward neural networks. With its ability to randomly choose hidden nodes and analytically determine output weights, ELM has an extremely fast learning speed and an advantageous generalization capability. In addition, it has exhibited commendable performance in both classification and regression with a simple structure [9]. Compared with ANN, the training speed of ELM is much faster and achieves better generalization. According the literature review, ELM was selected as the regression model in this work. To validate the availability of the proposed approach, a rolling element bearing, as one representation of a mechanical component, is introduced.

The rest of the paper is organized as follows. The theoretical background of the proposed degradation prognosis method is introduced in Section 2. The degradation prognosis procedure is illustrated in Section 3. Experimental results and comparison with peer models are presented in Section 4. Finally, the conclusions drawn from the presented work are presented in Section 5.

2 Methodologies

2.1 Feature extraction in different domains

Feature extraction has an important influence on the degradation prognosis method in terms of its accuracy and efficiency. In practice, fault diagnosis of mechanical components often uses time domain and time–frequency domain parameters from vibration signals [25 –28]. The features extracted from the original vibration signals show accurate information about the initially fault in mechanical components.

In all, 16 features were used in this study. Time-frequency domain can be obtained using several tools [29]. In this paper, eight features were extracted from the time–frequency domain using the wavelet transform package. The remaining eight time domain features (standard deviation, root mean square, variance, maximum, peak-to-peak value, absolute mean value, waveform index, and kurtosis) were extracted using the formulas in Table 1. These features represent the degradation condition of mechanical components from a healthy condition to collapse.

According to Table 1, P is the feature, q is the extracted vibration signal, and W is the sample number for μ = 1, 2, …, W.

Table 1
List of the original extracted feature

Features Formula

Standard deviation $P_{1} = \sqrt{\frac{1}{W - 1} \sum_{μ = 1}^{W} {(q_{i} - \frac{1}{W} \sum_{μ = 1}^{W} q_{μ})}^{2}}$ (1)

Root mean square $P_{2} = \sqrt{\frac{1}{W} \sum_{μ = 1}^{W} q_{μ}^{2}}$ (2)

Variance $P_{3} = \frac{1}{W - 1} \sum_{μ = 1}^{W} {(q_{i} - \frac{1}{W} \sum_{μ = 1}^{W} q_{μ})}^{2}$ (3)

Maximum P₄ = max{ |q_μ| } (4)

Peak to peak value P₅ = q_max - q_min (5)

Absolute mean value $P_{6} = \frac{1}{W} \sum_{μ = 1}^{W} | q_{μ} |$ (6)

Waveform index $P_{7} = \frac{max (q)}{(1 / W) \sum_{μ = 1}^{W} | q_{μ} |}$ (7)

Kurtosis $P_{8} = \frac{(1 / W) \sum_{μ = 1}^{W} x_{μ}^{4}}{{(\sqrt{(1 / W) \sum_{μ = 1}^{W} q_{μ}^{2}})}^{4}}$ (8)

Energy values P_9-16 = E_ς, ς = 1, 2, …, 8

Features	Formula
Standard deviation	$P_{1} = \sqrt{\frac{1}{W - 1} \sum_{μ = 1}^{W} {(q_{i} - \frac{1}{W} \sum_{μ = 1}^{W} q_{μ})}^{2}}$ (1)
Root mean square	$P_{2} = \sqrt{\frac{1}{W} \sum_{μ = 1}^{W} q_{μ}^{2}}$ (2)
Variance	$P_{3} = \frac{1}{W - 1} \sum_{μ = 1}^{W} {(q_{i} - \frac{1}{W} \sum_{μ = 1}^{W} q_{μ})}^{2}$ (3)
Maximum	P₄ = max{ \|q_μ\| } (4)
Peak to peak value	P₅ = q_max - q_min (5)
Absolute mean value	$P_{6} = \frac{1}{W} \sum_{μ = 1}^{W} \| q_{μ} \|$ (6)
Waveform index	$P_{7} = \frac{max (q)}{(1 / W) \sum_{μ = 1}^{W} \| q_{μ} \|}$ (7)
Kurtosis	$P_{8} = \frac{(1 / W) \sum_{μ = 1}^{W} x_{μ}^{4}}{{(\sqrt{(1 / W) \sum_{μ = 1}^{W} q_{μ}^{2}})}^{4}}$ (8)
Energy values	P_9-16 = E_ς, ς = 1, 2, …, 8

where E_ζ is the energy values of sub-frequency bands obtained by a three-layer wavelet packet transform.

2.2 Fuzzy fusion of multiple features

In the fuzzy fusion of multiple criteria method, the cost function is composed of a fuzzy nearness data fusion of kurtosis, smoothness factor, and kurtosis coefficient. It improves the robustness that would identify the optimal resonance band. Therefore, the cost function constructed by the multiple indicator fusion results is adopted in this paper. The mathematical expression is as follows [30]: $A_{n} = F (B_{m} (x_{n} (t))),$ (1) where A_n represents the nth index, F(.) is the function of data fusion, B_m is the mth index and x_n(t) is the time domain component.

The realization of the data fusion function F(.) uses the fuzzy nearness, which is a measurement of the proximity degree of fuzzy sets. A collection of initially minimum spectral subsets is X (f) = {C₁, C₂, …, C_i, …, C_l}. The corresponding set of time domain components is {x₁ (t), x₂ (t) , …, x_i (t) , …, x_l (t)}. Every index sequence is normalized over the interval from 0 to 1. ${NB}_{m} (x_{n} (t)) = \frac{B_{m} (x_{i} (t)) - {min}_{t = 1}^{l} B_{m} (x_{i} (t))}{{max}_{t = 1}^{l} (B_{m} (x_{i} (t))) - \min_{i = 1}^{l} (B_{m} {(x}_{i} (t)))}$ (2)

Then, to calculate the mean value $\bar{{NB}_{m}}$ and the standard deviation σ_m of each normalized index sequence: $\bar{{NB}_{m}} = \frac{1}{l} \sum_{i = 1}^{l} {NB}_{m} (x_{i} (t)),$ (3) $σ_{m} = \frac{1}{l - 1} \sqrt{\sum_{i = 1}^{l} {({NB}_{m} - \bar{{NB}_{m}})}^{2}} .$ (4)

To calculate the grand mean $\bar{{NB}_{0}}$ and the total standard deviation σ₀ of the 16-index sequence: $\bar{{NB}_{0}} = \frac{1}{16} \sum_{m = 1}^{16} {\bar{B}}_{m}$ (5) $σ_{0} = \frac{1}{15} \sqrt{\sum_{m = 1}^{16} {({\bar{NB}}_{m} - \bar{{NB}_{0}})}^{2}}$ (6)

In this paper, the triangle fuzzy membership function is chosen. The fuzzy quantity of each indicator is expressed as follows:

$\begin{matrix} {\tilde{B}}_{m} & = & (a_{m 1}, a_{m 2}, a_{m 3}) \\ = & (\bar{N B_{m}} 2 σ_{m}, \bar{N B_{m}}, \bar{N B_{m}} + 2 σ_{m}) \end{matrix}$ (7)

So, the blur quantity of the data fusion cost function is as follows:

$\begin{matrix} {\tilde{B}}_{0} & = & (a_{01}, a_{02}, a_{03}) \\ = & (\bar{{NN}_{0}} 2 σ_{0}, \bar{{NB}_{0}}, \bar{{NB}_{0}} + 2 σ_{0}) \end{matrix}$ (8)

The nearness of the post of the mth index fuzzy quantity and the fuzzy quantity of data fusion is: $D_{m} = \frac{1}{1 + \frac{a_{m 1} + 4 a_{m 2} + a_{m 3} a_{01} 4 a_{02} a_{03}}{6}}$ (9)

If the fuzzy quantity of the mth index is closer to the fuzzy amount of the data fusion, the nearness D_m is bigger. This means that the better the reliability and stability, the greater the weight. Based on this principle, Equation (18) can be obtained by using Equations (9) and (10). $A_{n} = \frac{D_{m}}{\sum_{m = 1}^{16} D_{m}} {NB}_{m} (x_{n} (t))$ (10)

A fusion degradation index is finally obtained: $y (i) = A_{1} + A_{2} + \dots + A_{n}$ (11) where y(i) is the final degradation index.

2.3 Feature decomposition using EEMD

Huang et al. [31] introduced the EMD method, which allows for adaptive decomposition of non-linear and non-stationary signals. This method can decompose the signal into several narrow-banded IMF components. It is a widely used approach. However, it may incorrectly reveal an intermittent signal’s characteristic information as a result of mode mixing, which may cause the decomposition to render the physical meaning of the IMFs unclear. To overcome this problem, Wu and Huang proposed EEMD [32].

EEMD is an improved version of EMD, and it can improve the prediction accuracy [33]. It solves the mode-mixing problem [34] by adding white noise of finite amplitude fairly. It also uses statistical characterization of the Gaussian white noise to improve the distribution of extreme points in the original signal. The process of EEMD is as follows [35].

Step 1. Obtain y(i) as a degradation index and add white noise c_j(i) to y(i): $y_{j} (i) = y (i) + c_{j} (i)$ (12) where c_j(i) indicates the jth added white noise series, y_j(n) represents the noise-added signal of the jth trial, and j = 1,2, … z; z is the number of IMFs.

Step 2. Decompose y_j(i) into different IMFs b_k,j utilizing the EMD algorithm. $y_{j} (i) = \sum_{j = 1}^{N} b_{k, j} + r_{N}$ (13) where b_k_,j denotes the kth IMF of the jth trial, r_N denotes the residue of the jth trial, and N is the number of IMFs of the jth trial.

Step 3. If j < z, then repeat steps 1 and 2 z times and add different white noise each time.

Step 4. The final result of the IMFs is obtained by calculating the ensemble means of the corresponding components. $b_{k} = (\sum_{j = 1}^{z} b_{k, j}) / z$ (14) where b_k is the final IMF decomposed by EEMD.

In this paper, the EEMD method was applied to perform with an ensemble number of 500 and a white noise amplitude that is 0.2 times the standard deviation of the bearing fuzzy fusion degradation index.

Fig.1

Schematic of mechanical components degradation prognostic model.

2.4 Degradation prognosis using ELM

Huang presented a simple learning algorithm for single hidden-layer feed-forward neural networks called ELM [36]. Its learning speed was thousands of times faster than traditional feed-forward network learning algorithms such as BP algorithms, and it obtained better generalization performance [25]. The structure of ELM can show that it can randomly assign the weights of the input layers. Moreover, it computes and analyzes the weights of the output layers by a simple generalized inverse operation [37].

Suppose there are k samples (c_r, d_r), r = 1, 2, …, k, where c_r = (c_r1, c_r2, …, c_rv) ^T and d_r = (d_r1, d_r2, …, d_rw) are the rth input feature and its target value, respectively. Then, the model of single hidden layer feedforward neural networks with R hidden nodes can be described as follows:

$\begin{matrix} \sum_{r = 1}^{R} β_{r} γ_{r} (c_{p}) & = & \sum_{r = 1}^{R} β_{r} γ_{r} (ω_{r} . c_{p} + g_{r}) = o_{p}, \\ p = 1, \dots, k \end{matrix}$ (15) where β_r = [β_r1, β_r2, …, β_rw] ^T is the weight vector of the connectors between the rth hidden and output nodes, γ (.) is the activation function, and ω_r = [ω_r1, ω_r2, …, ω_rv] ^T is the weight vector that connects the input node to the rth hidden node. The operation ω_r. c_p is the inner product of ω_r and c_p. The variable g_r is the rth bias of the hidden nodes, which are randomly assigned.

Standard SLFNs with R hidden nodes with activation function g_r can approximate these k samples between output o_p and target s_p with zero error means. It is represented as: $\sum_{j = 1}^{N} ∥ o_{p} - s_{p} ∥ = 0$ (16)

There is an existing optimal value of β_i such that $\sum_{r = 1}^{R} β_{r} γ_{r} (ω_{r} . c_{p} + g_{r}) = s_{p}, p = 1, \dots, k .$ (17)

By using the following substitutions:

$\begin{matrix} H (ω_{r}, \dots, ω_{R}, g_{r}, \dots, g_{R}, c_{1}, \dots, c_{k}) \\ = {[\begin{matrix} γ_{r} (ω_{1}, g_{1}, c_{1}) & \dots & γ_{r} (ω_{R}, g_{R}, c_{1}) \\ ⋮ & ⋮ & ⋮ \\ γ_{r} (ω_{1}, g_{1}, c_{k}) & \dots & γ_{r} (ω_{R}, g_{R}, c_{K}) \end{matrix}]}_{k \times R} \end{matrix}$ (18) the preceding equations can be written as the following substitutions: $H β = T$ (19) where H is the hidden-layer output matrix of the neural network, $β = {[β_{1}^{T}, \dots β_{R}^{T}]}_{R \times v}^{T}$ and $T = {[T_{1}^{T}, \dots T_{R}^{T}]}_{R \times v}^{T}$ .

The output weight β can be written as: $β = H^{+} T,$ (20) where H⁺ = (H^TH) ^-1T^T.

As ELM does not need to iteratively adjust input weights and threshold in training, it decreases the training complexity and improves the training speed. However, its accuracy is still related to feature extraction.

2.5 Overall description of the proposed approach

The main procedure of the proposed approach is demonstrated in Fig. 1.

Step 1. Feature extraction. Multifarious features from time domains and time–frequency domains are extracted from the rolling element bearing. The 16 original features selected for this study, which include 8 time domain features and 8 time–frequency domain features, are shown in Table 1. Features P1–P8 are the standard deviation, variance, kurtosis, maximum, peak-to-peak value, absolute mean value, root mean square, and wave form factor, respectively. Furthermore, the eight energy values P9–P16 are obtained by three-layer decomposition of the wavelet packet transform.

Step 2. Feature fuzzy fusion. The features extracted in Step 1 are considered the raw features. The fuzz fusion of multiple features is used to build a degradation index using the 16 features.

Step 3. Degradation index decomposition. The fused degradation index is decomposed by EEMD. After decomposition, the degradation index transforms to a multimodal sub-series, including seven IMFs and one residue component.

Step 4. Degradation prognosis. Based on Steps 1–3, multimodal ELM is erected using the sub-series of each mode. The sub-results of the different modes are summarized into a total result.

3 Experimental results and analysis

Rolling element bearings are among the most vital and easily destroyed components in rotating machinery because of their delicate structures, heavy loads, and harsh running environment [30], which makes monitoring them essential. In this section, the reported degradation prognosis method is applied to the bearing. In the aforementioned methods, fuzzy fusion has a capacity preferable to that of feature fusion. EEMD focuses on characteristic decomposition. ELM is an efficient learning algorithm with a fast training speed. An approach validation scheme, which is combined with the preceding three methods, was implemented in this work. The schematic procedure of the degradation mode can be seen in Fig. 1 and it will be explain in the following sections.

Fig.2

Bearing experiment rig and sensors.

3.1 Experimental data description

With the development of information technologies, the test data for experimentally determining the operating status are easy to obtain. In this work, experimental data from the Prognostics Center of Excellence were used to verify the validity of the degradation prognosis proposed in this paper. They were generated by the NSF I/UCR Center for Intelligent Maintenance Systems with support from Rexnord Corp. in Milwaukee, WI [38]. The bearing experiment rig and sensor placement are demonstrated in Fig. 2.

As shown in the Fig. 2, four bearings were installed on a shaft. A radial load of 6000 lbs was applied to the shaft and bearing by a spring mechanism. The vibration data from each bearing were collected every 10 min with the sampling rate set at 20 kHz. A data collection length of 20480 points was collected by an NI DAQCard-6062E. The rotational speed was set at 2000 rpm by using an AC motor associated with the shaft through rub belts [39]. The raw vibration signals are shown in Fig. 3.

Fig.3

Acceleration of bearing ending with failure roller.

Fig.4

The extracted time-domain and time-frequency domain features.

3.2 Performance criteria

The mean absolute error (MAE), root mean square error (RMSE), and correlation coefficient (CC) are introduced to estimate the performance of the model. $MAE = \frac{1}{L} \sum_{θ = 1}^{L} | y_{θ} - {\hat{y}}_{θ} |$ (21) $RMSE = \sqrt{\frac{1}{L} \sum_{θ = 1}^{L} {(y_{θ} - {\hat{y}}_{θ})}^{2}}$ (22) $CC = \frac{\sum_{θ = 1}^{L} (y_{θ} - \bar{y}) ({\hat{y}}_{θ} - \bar{\hat{y}})}{\sqrt{\sum_{θ = 1}^{L} (y_{θ} - \bar{y})^{2}} \sqrt{\sum_{θ = 1}^{L} ({\hat{y}}_{θ} - \bar{\hat{y}})^{2}}}$ (23) where L is the length of the output data, y_θ is the observed values, ${\hat{y}}_{θ}$ represents the prediction values, $\bar{y}$ represents the average of the observed values, and is the average of prediction values.

3.3 Feature extraction

Only the vibration signals of the bearing were used in this study. Features can represent the change of bearing vibration signals. In this work, a total of 16 features including eight time domain features and eight time–frequency domain features were extracted from the raw vibration signals of the bearing, as illustrated in Fig. 4.

The Fig. 4 shows the 16 original features derived from the raw signals. Undoubtedly, feature extraction is an extremely critical procedure [40]. As observed in Fig. 4, the original features were sensitive in either early or severe failure. The fusion result shows that it is much better than many other indexes in performance of degradation prognosis. Thus, it is necessary to adopt a method for feature fusion.

3.4 Feature fuzzy fusion

Feature fusion will increase the amount of useful information available for degradation prognosis and remove some of the inevitable redundant information. Redundant features may reduce the accuracy and efficiency of degradation prognosis. To improve the performance of degradation prognosis, feature fusion is necessary.

The above features are not insensitive to identifying the degradation process, but they are not comprehensive enough to describe the degenerative process. Therefore, it is very important to rebuild the features from the original features to estimate the degradation of the bearing. In this study, the fuzzy fusion method was used to execute the fusion task. The result, a degradation index, was obtained using Equation (19). Figure 5 presents the overall degradation trend of the bearing more clearly. To better describe the failure, the degradation process can be divided into three phases. At the beginning, the curve remains stable, which indicates normal working conditions; this is called the normal phase (Phase 1). Over time, the tendency toward degradation increases gradually. This phase, where the fault starts to occur, is regarded as the failure development phase (Phase 2). At the last phase, the curve sharply increases, which is denoted as the severe phase (Phase 3). Relying on the preceding analysis, the conclusion can be drawn that the fusion index is able to demonstrate more detailed information about the failure process. Additionally, it was found that the normal phase contained less information. It can even be said that Phase 1 made little contribution to the degradation. Phase 1 was therefore removed from the experiment. Phases 2 and 3 (700–984) of the degradation indexes were utilized to model the degradation estimation. In this study, about 65% of the sample sets were used for training; the rest were used for testing.

Fig.5

Degradation index for bearing prognosis.

3.5 Degradation index decomposition

According to the aforementioned procedure, EEMD, deriving more details and it was subsequently used for the decomposition of the character index. was Seven IMFs and one residual component were ultimately gained; they are exhibited in Fig. 6.

As shown in the figure, the seven individual elements of b₁–b₇ were acquired with EEMD. In particular, one independent component, r, described the overall tendency of the degradation index in the form of a monotonic function. Overall, eight components were derived after decomposition. As seen in these components, they became stationary and visually regular. The obtained sub-series are exhibited from high to low frequency so that more details are presented. The EEMD technique attenuated the coupling and interruption implied in the data to some extent. Subsequently, multimodal ELM was established in each mode.

Fig.6

The decomposition results of the characteristics index.

3.6 Prediction results

To compare the performance of the reported approach, mono ELM was introduced to compare the influence of the multimodal prediction. Furthermore, support vector regression (SVR), which is commonly used as an effective tool in many disciplines [41], was also adopted for comparison. The final results of all of the models are displayed in Fig. 7.

Figure 7(a)–(c) show the respective prediction results. As can be seen, the values from the proposed method and ELM are much closer to the observed values as a whole. Their fitting effect exceeds that of SVR, which is obviously worse among these ones models. SVR conforms to the beginning stage, but not to the whole. In particular, the differences in their results are quite distinct in the prediction of extreme values. It is apparent that the proposed method is superior to the others and exhibits outstanding performance. The preceding comparison reveals that the proposed method is more stationary and has a better generalization capacity.

Fig.7

The prediction results for different models: (a) the proposed method; (b) ELM; (c) SVR.

Fig.8

Comparisons of boxplot of absolute errors using different models.

As shown by the median displayed in Fig. 8, the absolute error mean for the proposed method is the lowest, and the stationarity is outstanding. The results yielded through the proposed method and ELM are shorter than those yielded through SVR when comparing the length of every box entity. This demonstrates that the distributions of the absolute error for the proposed method and ELM are relatively concentrated. Furthermore, the position of the entity for the proposed method is lower than that for SVR. Most of the absolute error of the proposed method is thus within a lower range. Compared with the distance between the median and quartiles, the condition of the proposed method is relatively symmetrical and shows basically normal distribution. Furthermore, none of the benchmarks are balanced clearly. None of the benchmarks are clearly balanced, which indicates that the proposed method is better in terms of the distribution of absolute error. All of the models have some outliers reflecting significant error values. However, SVR has the most outliers among these ones models; the other two have relatively few. This shows the advantage of ELM from another aspect. Based on the preceding analyses, one may experience the advantages of the proposed method as compared with other methods.

Table 2 shows the results of MAE, RMSE, and CC. The proposed method has lower values than those of the other two models. This means that, among the three models, the proposed method has better accuracy and performance than the other two in terms of the various performance criteria. These differences also illustrate that multi-feature fuzzy fusion and multimodal decomposition can better extract essential features of the degradation process.

Table 2

The comparisons of three models

Model	MAE	RMSE	CC
The proposed	0.381	0.875	0.960
ELM	0.620	1.444	0.890
SVR	0.861	2.023	0.813

4 Conclusions

In this paper, a data-driven prognostics method based on fuzzy fusion and multimodal degradation prognosis is proposed. The objective of this work was to develop a method to estimate the degradation process of mechanical components. Initially, the raw features were collected from the time domain and time–frequency domain of vibration signals for the bearings. Then, the acquired original features were fused into a degradation index using the fuzzy fusion technique. Subsequently, EEMD was adopted to decompose the fusion index into multimodal sub-series. A multimodal ELM was established to estimate the sub-series. Finally, the sub-results of each mode were added into the prognosis results. The adopted method is applied to actual bearing data, and the degradation prognosis results of the bearing are shown and discussed. The proposed approach can achieve a prognosis with better accuracy, and it provides a good method for making a prognosis of the RUL of mechanical components.

Footnotes

Acknowledgments

This work is supported in part by the National Key Research & Development Program of China (2016YFE0132200), and the Chongqing Science & Technology Commission (cstc2015zdcy-ztzx70012).

References

Medjaher

, Tobon-Mejia

D.A.

and Zerhouni

, Remaining useful lifeestimation of critical components with application to bearingsreliability, IEEE Transactions on Reliability61(2) (2012), 292–302.

Laayouj

and Jamouli

, Prognostic of degradation using remaininguseful life estimation, Mediterranean Conference on Control and Automation2015.

Cerrada

, Li

, Sánchez

R.V.

, Pacheco

, Cabrera

and

Valente de Oliveira , A fuzzy transition based approach for faultseverity prediction in helical gearboxes, Fuzzy Sets &Systems (2016), 30458–4.

Cong

F.Y.

, Chen

and Pan

, Kolmogorov-Smirnov test for rollingbearing performance degradation assessment and prognosis,&(9), Control17 (2011), 1337–1347.

Dong

, Yin

and Tang

, Bearing degradation process predictionbased on the support vector machine and markov model,(1-2), Shock andVibration2014 (2014), 1–15.

Wang

, Miao

, Zhou

Q.H.

and Zhou

G.W.

, An intelligent prognosticsystem for gear performance degradation assessment and remaininguseful life estimation, ASME Journal of Vibration and Acoustics137(2) (2015), 021004.

C.J.

and Lee

, Gear fatigue crack prognosis using embeddedmodel, gear dynamic model and fracture mechanics,(4), Mechanical System & Signal Processing19 (2005), 836–846.

Cerrada

, Sánchez

R.V.

, Li

, Pacheco

and Cabrera

, J.Valente de Oliveira and R.E. Vásquez, A review on data-drivenfault severity assessment in rolling bearings,, Mechanical System & Signal Processing99 (2018), 169–196.

Liu

, He

, Liu

, Lu

S.L.

, Zhao

Y.L.

and Zhao

J.W.

, Remaining useful life prediction of rolling bearings using PSR, JADE, and Extreme Learning Machine, Mathematical Problems in Engineering2016 (2016), 1–13.

10.

Song

Y.C.

, Liu

D.T.

, Yang

and Peng

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

11.

Hong

and Zhou

, Remaining useful life prognosis of bearing based on Gauss process regression, International Conference on Biomedical Engineering and Informatics (2012), 1575–1579.

12.

, Cabrera

, Valente de Oliveira

, Sánchez

R.V.

, Cerrada

and Zurita

, Extracting repetitive transients for rotating machinery diagnosis using multiscale clustered grey infogram, Mechanical Systems & Signal Processing76-77 (2016), 157–173.

13.

, A hybrid feature selection scheme and self-organizing mapmodel for machine health assessment,(5), Applied Soft ComputingJournal33 (2011), 4041–4054.

14.

Liu

, He

, Liu

, Lu

and Zhao

, Feature fusion using kernel joint approximate diagonalization of eigen-matrices forrolling bearing fault identification,, Journal of Sound &Vibration385 (2016), 389–401.

15.

Liu

, Li

and Ye

, A method for rolling bearing fault diagnosis based on sensitive feature selection and nonlinear feature fusion, International Conference on Intelligent ComputationTechnology and Automation (2015), 30–35.

16.

, Valente de Oliveira

, Sánchez

R.V.

and Cerrada

, Fuzzy determination of informative frequency band for bearing faultdetection, Journal of Intelligent & Fuzzy Systems30(6) (2016), 3513–3525.

17.

Huang

N.E.

, Shen

, Long

S.R.

, Wu

M.C.

, Shih

H.H.

, Zheng

, Yen

N.C.

, Tung

C.C.

and Liu

H.H.

, The empirical mode decomposition andthe Hilbert for nonlinear and non-stationary time series analysis,, Proceedings Mathematical Physical & Engineering Sciences454 (1971), 903–995spectrum.

18.

Wang

C.J.

, Li

H.Y.

, Xiang

and Zhao

, A new signal classification method based on EEMD and FCM and its application inbearing fault diagnosis,(1), Applied Mechanics & Materials602-605 (2014), 1803–1806.

19.

Lei

, He

and Zi

, Application of the EEMD method to rotorfault diagnosis of rotating machinery,(4), Mechanical Systems &Signal Processing23 (2009), 1327–1338.

20.

Aroussi

M.E.

, Bearings prognostic using mixture of gaussians hidden Markov model and support vector machine,(3), International Journalof Network Security & Its Application5 (2013), 1–4.

21.

Bangalore

and Tjernberg

L.B.

, An artificial neural network approach for early fault detection of gearbox bearings,(2), IEEE Transactions on Smart Grid6 (2015), 980–987.

22.

Bai

, Sun

, Zeng

, Jun

and Li

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

23.

Loutas

T.H.

, Roulias

and Georgoulas

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

24.

Huang

G.B.

, Zhu

Q.Y.

and Siew

C.K.

, Extreme learning machine: A new learning scheme of feedforward neural networks, IEEE International Joint Conference on Neural Networks2 (2005), 985–990.

25.

Hong

and Dhupia

J.S.

, A time domain approach to diagnose gearbox fault based on measured vibration signals,(7), Journal of Sound &Vibration333 (2014), 2164–2180.

26.

Pacheco

, Cerrada

, Sánchez

R.V.

, Cabrera

, Li

, and Valente de Oliveira

, Attribute clustering using rough set theory for feature selection in fault severity classification of rotating machinery,, Expert Systems with Applications71 (2017), 69–86.

27.

and Liang

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

28.

Yuan

, Zhao

, Fei

, Zhao

and Wang

, Study on fault diagnosis of rolling bearing based on time-frequency generalized dimension,(3), Shock and Vibration2015 (2015), 1–11.

29.

Cabrera

, Sancho

, Li

, Cerrada

and Sánchez

R.V.

, F.Pacheco and J. Valente de Oliveira, Automatic feature extraction oftime-series applied to fault severity assessment of helical gearboxin stationary and non-stationary speed operation, Applied SoftComputing (2016), 30188–6.

30.

, Liang

and Criterion

, Criterion fusion for spectral segmentation and its application to optimal demodulation of bearing vibration signals,(3), Mechanical Systems & Signal Processing64-65 (2015), 132–148.

31.

Liu

, Huang

and Gill

E.W.

, Wind direction estimation fromrain-contaminated marine radar data using the enble empiricalmode decomposition method,, IEEE Transaction on Geoscience &Remote Sensing99 (2016), 1–9sem.

32.

Z.H.

and Huang

N.E.

, Enble empirical mode decomposition: Anoise-assisted data analysis method,(01), Advances in Adaptive DataAnalysis1 (2011), 0900004–sem.

33.

X.J.

, Cheng

Z.W.

, Yu

Q.B.

, Bai

and Li

, Water-qualityprediction using multimodal support vector regression: Case study ofJialing River, China,(10), Journal of Environmental Engineering143 (2017), 04017070.

34.

Lei

and Zi

, EEMD method and WNN for fault diagnosis oflocomotive roller bearings, Inc;, 2011–Pergamon Press.

35.

Lei

, He

and Zi

, Application of EEMD method to rotor faultdiagnosis of rotating machinery,(4), Mechanical System & Signal Processing23 (2009), 1327–1338.

36.

M.B.

, Huang

G.B.

, Saratchandran

and Sundararajan

, Fully complex extreme learning machine,(1), Neurocomputing68 (2005), 306–314.

37.

Castaño

, Fernández-Navarro

, and Hervás-Martínez

, Pca-elm: A robust and pruned extreme learning machine approach based on principal component analysis,(3), Neural Processing Letters37 (2013), 377–392.

38.

Qiu

, Lee

, Lin

and Yu

, Robust performance degradation assessment methods for enhanced rolling element bearing prognostics,(3-4), Advanced Engineering Informatics17 (2003), 127–140.

39.

Qiu

, Lee

, Lin

and Yu

, Wavelet filter-based weak signature detection method and its application on rolling elementbearing prognostics,(4-5), Journal of Sound & Vibration289 (2006), 1066–1090.

40.

Wang

, Chen

and Dong

, Feature extraction of rolling bearing’s early weak fault based on EEMD and tunable Q-factorwavelet transform,(1-2), Mechanical Systems & Signal Processing48 (2014), 103–119.

41.

, Sánchez

R.V.

, Zurita

, Cerrada

, Cabrera

, and Vasquezl

R.E.

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