#### Abstract

Prognostics health management (PHM) of rotating machinery has become an important process for increasing reliability and reducing machine malfunctions in industry. Bearings are one of the most important equipment parts and are also one of the most common failure points. To assess the degradation of a machine, this paper presents a bearing remaining useful life (RUL) prediction method. The method relies on a novel health indicator and a linear degradation model to predict bearing RUL. The health indicator is extracted by using Hilbert–Huang entropy to process horizontal vibration signals obtained from bearings. We present a linear degradation model to estimate RUL using this health indicator. In the training phase, the degradation detection threshold and the failure threshold of this model are estimated by the distribution of 600 bootstrapped samples. These bootstrapped samples are taken from the six training sets. In the test phase, the health indicator and the model are used to estimate the bearing’s current health state and predict its RUL. This method is suitable for the degradation of bearings. The experimental results show that this method can effectively monitor bearing degradation and predict its RUL.

#### 1. Introduction

Ball bearings are one of the most widely used universal parts in various pieces of mechanical equipment. The operational failure of ball bearings is a major reason for the failure of mechanical equipment. Compared with that of other mechanical parts, the service lives of bearings originating from the same batch can vary greatly in terms of the fault occurrence time, fault type, and severity. In this way, some bearings operate normally over the entire design life, and some do not reach the end of the design life and experience early failure. Therefore, the normal operation of a ball bearing cannot be ensured by timely maintenance based on only the design life. When a bearing fails, it causes noise and reduces the accuracy of the work, causing severe vibration and even damage to the equipment, which stops production and results in workplace hazards. Therefore, condition monitoring and fault diagnosis must be carried out during bearing operations in order to change the traditional timing for predictive maintenance. In this way, we can identify bearing faults and replace bearings prior to the occurrence of failure. Using real-time monitoring, the replacement time can be forecasted so that the bearing can be utilized to its maximum extent and that unexpected equipment issues resulting from degraded bearings are avoided, no mechanical equipment is destroyed, and personal safety is protected. Therefore, research on bearing fault diagnosis and residual life predictions is of great practical significance [1–4].

Technology addressing the remaining useful life prediction of a bearing is known as fault prediction and health management (PHM). PHM consists of two aspects: fault prediction and health management. Fault prediction mainly includes monitoring of the health status of equipment, predicting the remaining life, or predicting the future state. Health management relates to making full use of the forecasted information to make decisions related to the safety and reliability of equipment and to prolong equipment life. In recent years, PHM technology has gradually matured to practical field applications. Usually, for composite objects, it is difficult to establish an accurate failure physical model to predict the RUL. Therefore, the method driving the data is based on a large number of historical failure points that are used to predict the RUL. Data-driven prediction methods mainly use pattern recognition and machine learning techniques combined with changes in the equipment operating conditions to achieve forecasting. Traditional data-driven methods for nonlinear systems mainly include regression-based models [5, 6], Wiener processes, gamma processes, stochastic filtering models, risk-based covariance models, hidden Markov models, and semihidden Markov models [7, 8].

When a ball bearing fails, the vibration signal has a tendency to exhibit strong nonstationary and nonlinear characteristics. Therefore, obtaining fault characteristic information from these nonstationary and nonlinear signals is essential for ball bearing fault diagnosis. Using the Hilbert–Huang transform (HHT), vibration signals can be adaptively decomposed to provide local and time-domain signal information [9, 10].

In this study, horizontal vibration acceleration signals from ball bearings are utilized to extract the health indicator by Hilbert–Huang entropy. This indicator is the input to the linear degradation model. If the indicator reaches the degradation detection threshold, its RUL is predicted using this model.

#### 2. Related Works

The ball bearing-accelerated life test data used throughout this paper are derived from the IEEE PHM 2012 Data Challenge. Under normal circumstances, the RUL of a bearing is within the range of tens of thousands of hours, but such a long running time is impossible to achieve in a laboratory environment. Therefore, an accelerated failure is achieved by imposing an additional load on the bearing or increasing the rotational speed on the PRONOSTIA platform. The purpose of the PRONOSTIA test-bed is to test and verify the bearing fault detection and fault prediction models. Using this platform, a bearing can be rapidly invalidated within a few hours via an accelerated life test so that the bearing failure data can be acquired. PRONOSTIA provides bearing degradation data under various operating conditions. The experimental data in this paper are horizontal acceleration data from bearings from the IEEE PHM 2012 Data Challenge dataset [11–14].

The RUL of a machine is the expected life or usage time remaining before the machine requires repair or replacement. Predicting RUL from system data is a central goal of predictive-maintenance algorithms. To estimate the RUL of a system, a model that can perform an estimation based on the time evolution or statistical properties of condition indicator values needs to be developed.

Predictions from such models are statistical estimates with the associated uncertainty. They provide a probability distribution of the RUL of the test machine. Developing a model for RUL prediction is the next step in the algorithm-design process after identifying promising condition indicators. Because the developed model uses the time evolution of condition indicator values to predict RUL, this step is often iterative with the step of identifying condition indicators.

There are three families of RUL estimation models: similarity model, degenerate model, and survival model.

##### 2.1. Similarity Model

A similarity model is based on a historical database that combines the RUL prediction of a test machine with the behavior of a known machine. Such a model compares the trend of test data or conditional indicator values with the same information extracted from other similar systems.

Many scholars have proposed various similarity models for RUL. A similarity-based prognostic approach is developed for estimating the RUL of a valve asset using full-scope high-fidelity simulators to generate the run-to-failure data [15]. A Bayesian hierarchical model- (BHM-) based prognostics approach is applied to analyze and predict the discharge behavior of Li-ion batteries with variable load profiles and variable amounts of available discharge data [16]. A sparse representation model is used to extract the inherent relationships of training samples and measure the similarities between testing samples and training samples, and then a weight is given to every training sample to note its similarity to the testing sample [17]. A framework for RUL estimation of an aircraft engine is proposed by using the whole lifecycle data and performance-deteriorated parameter data without failures based on the theory of similarity and supporting vector machine (SVM) [18]. Another proposal involves the combined use of a fuzzy similarity method for RUL prediction and the belief function theory for uncertainty treatment [19]. A prognostics-based qualification method using an efficient relevance vector machine (RVM) regression model to predict the RUL of an LED by calculating the accumulated sum of products of similarity weights and historical LED RUL values at the 210th hour has been proposed [20]. A general data-driven, similarity-based approach for residual useful life estimation is proposed. Distance score and weight are computed by calculating the fuzzy similarity between test trajectory pattern and reference trajectory patterns [21].

Using a similar model requires running fault data similar to systems (components). “Running to the fault” data are the data that begin during normal operation and end when the machine is in a near-failure or maintenance state. “Run to the failure” data show similar degradation behavior. In other words, as the system degrades, the data change in a particular way. IEEE PHM 2012 data support the entire six-training-run failed dataset, which does not exhibit very similar degradation behavior. It is not easy to choose an appropriate model based on similarity.

##### 2.2. Degenerate Model

This model infers the behavior of the past to predict future conditions. This type of RUL calculates the condition indicator for a degenerate indicator that fits the linear or exponential model, giving a degenerate contour in your ensemble. It then uses the degraded contour of the test component to count the remaining time until the indicator reaches a specified threshold. These models are most useful when there is a status indicator of a known value indicating failure. The two available degradation model types are the linear degenerate model and the exponential degenerate model.

The linear degenerate model describes the degenerate behavior as a linear stochastic process with an offset. Linear degenerate models are useful when the system does not experience cumulative degradation. A nonparametric estimator is proposed for percentiles of the time-to-failure distribution obtained from a linear degradation model using the kernel density method in [22]. A parametric method is described to estimate the survival function of a general linear degradation model with one soft failure and several hard failures in [23]. The stochastic stress is developed by incorporating the interaction between basic variables and the time variable into the quadratic response surface method, and the stochastic strength is characterized by the linear degradation model in [24].

The exponential degenerate model describes the degenerate behavior as an exponential stochastic process with an offset. The exponential degradation model is useful when the test component undergoes cumulative degradation. This model has been presented in many literatures with specific applications. The variation of the ON-state resistance is identified as the failure precursor, and an exponential degradation model that fits successfully with the experimental data is developed [25]. Based on the experimental data, a nonlinear dual-exponential degradation model for MOSFETs is obtained [26]. An exponential degradation model is developed to describe the degradation characteristics of devices for residual life estimation, and this model is based on a gamma-prior Bayesian updating approach and an acceptance-rejection algorithm [27]. The proposed method is applied to the degradation data of plasma display panels (PDPs), following a biexponential degradation model [28]. Using a base case exponential degradation model, we identify conditions necessary to establish stochastic ordering among the RLDs of similar components [29]. An uncertainty RUL prediction method is proposed based on the exponential stochastic degradation model that considers the multiple uncertainty sources of oscillator stochastic degradation processes simultaneously [30].

The linear and exponential degradation models are intuitive and clear, and the method of parameter calculation is easy. These models are therefore preferred.

##### 2.3. Survival Model

The survival model is a statistical method for statistical time event data. This is useful when the user does not have a complete history of running failures. The survival model is mainly used in the food and finance industry and is not applied much in the machinery industry. A survival model is introduced to contemplate two simultaneous accelerating factors affecting a food product’s shelf life: temperature and illumination [31]. The empirical analysis is based on an extension of the discrete-time survival analysis model, which allows for a structural break in its baseline hazard function and a unique set of individual loan accounts in [32]. IEEE PHM 2012 Data supported six complete training run-to-failure datasets, and this survival model is not suitable.

The model-based method presented in this paper is shown in Figure 1. The method consists of two modules of training stage and test stage, i.e., health indicator construction and RUL prediction. In the health indicator construction module, the Hilbert–Huang entropy is extracted from the horizontal vibration signal of the bearing. The degradation trend of the health indicator was evaluated. In the RUL prediction module, the input linear degenerate model’s health indicator input and model parameters are initialized in the training stage. Finally, according to the estimation parameters of the test phase, the degradation trend and mechanical RUL are predicted [2].

#### 3. Bearing Health Indicator

The construction of health indicators is the key to predicting the remaining service life. Bearings are the most common mechanical part in rotating machinery, and its health indicators have attracted much attention in practice. Given that, in many cases, it is difficult to measure and quantify the health indicators of a bearing, many vibration-based methods have been proposed to construct a bearing health indicator. Although many bearing health indicators have been proposed for a constant operating environment in the past few years, most bearing health indicators do not have a theoretical lower limit, so theoretical baselines do not exist. More theoretical studies should be conducted. In addition, performance indicators based on monotonicity, variance, trend, predictability, early fault detection, and calculation time should be proposed [33].

Ball bearing fault diagnosis signals include vibration, temperature, chemical analysis of the lubricant, and sound intensity. Vibration detection is the most effective method for bearing fault diagnosis. There are three analysis methods used for vibration detection.

##### 3.1. Time-Domain Analysis

Time-domain analysis is the first detection method in the vibration detection method, and it determines if the mechanical equipment has failed by calculating time-domain statistical characteristic parameters (mean value, variance, kurtosis, etc.) of the vibration signal. For the mechanical equipment vibration signal calculated and analyzed only in the time domain, it can only determine if the overall operational status of the equipment is normal and cannot determine the severity of a fault and predict any RUL [34–37].

##### 3.2. Frequency-Domain Analysis

Frequency-domain analysis is based on Fourier analysis to observe the operation state of the device by examining the frequency of fault-based characteristics over the spectrum diagram and the corresponding amplitude of the related frequencies. Commonly used frequency-domain vibration signal detection methods include fast Fourier transforms, power spectra, and filtering. However, each of these methods has flaws and loses some characteristics of nonlinear vibrations, meaning that the monitoring frequency band has a decisive impact on the analysis results [38–41].

##### 3.3. Time-Frequency Analyses

When a ball bearing fails, its vibration signal often has both nonstationary and nonlinear characteristics. The time- and frequency-domain analyses can only describe the signal as a whole and not the local characteristics of a signal for a certain time or a specified frequency range. Time-frequency analyses can be used to analyze the time and frequency of signals simultaneously. The time-frequency analysis method is the most effective method for the analysis of nonstationary and nonlinear vibration signals associated with ball bearings. The time-frequency analysis method can accurately describe the time and frequency characteristics of fault signals and has more advantages. Conventional time-frequency analysis methods include short-time Fourier transform (STFT), Wigner–Ville distribution (WVD), wavelet transform (WT), and EMD [42–45].

In this study, a novel Hilbert–Huang entropy as a new bearing health indicator is addressed. In a ball bearing vibration test, the time-domain signal describes the ball bearing vibration amplitude with time. The vibration acceleration signal is collected at a time scale of 0.1 s using PRONOSTIA. Therefore, the bearing health indicator is also conducted in time slices on signal slice . The proposed approach uses Hilbert–Huang entropy and its upper peak envelope to extract new heath indicators from the horizontal acceleration signals able to track the degradation of the critical components of bearings. The degradation states are detected by a supervised fault diagnostic given by analyzing the extracted health indicators. The proposed approach is decomposed into four parts (Figure 2).

##### 3.4. Hilbert–Huang Transform

Hilbert–Huang transform is a time-frequency method widely used in speech recognition and seismic signal analysis. The Hilbert–Huang transform consists of empirical mode decomposition (EMD) and Hilbert transform. EMD can decompose the signal into a small amount of intrinsic mode function (IMF) components, which is very adaptive and efficient. The decomposition method is suitable for nonlinear and nonstationary signal analysis and time-frequency energy representation because the base of the expansion is adaptive. The method has a powerful ability to reveal the real physical significance of data checking because of its complete empirical characteristics [46].(1)*EMD*. The time slice signal contains the above frequency information, and separating the frequency information is the key to feature extraction. EMD is a signal decomposition method originally proposed by Norden E. Huang in the 1990s. In this method, the signal can be adaptively decomposed according to the time-scale characteristics of the signal itself, and it is not necessary to preset the kernel function, such as in wavelet analysis. Therefore, an analysis of the vibration signal containing nonlinear noise has obvious advantages. For any one-dimensional time-series signals, the EMD can decompose the signal components over different frequency ranges, resulting in a series of intrinsic mode functions (IMFs) with their own feature scales and explicit physical meanings [47–49]. After the EMD decomposition process, the expression of the original signal is generated as (1)where is the residual function and represents a series of IMFs with different characteristic time scales and is arranged from small to large. Different frequency components in the original time series are also separated and included in each IMF component from high to low frequencies. Each IMF contains information for a frequency segment in the original time series and varies with time. EMD decomposition can determine ball bearing vibration feature signals, which contain fault information. These serve as the criteria for ball bearing fault diagnosis and identification.(2)*Hilbert transform*. According to Hilbert transform, the original signal could be expressed bywhere is the amplitude of the corresponding IMF and represents the quantity of IMFs. The summation of the real part indicated the Hilbert spectrum, which could be expressed as The marginal Hilbert spectrum was calculated as follows:

##### 3.5. Hilbert–Huang Entropy

To assess the health status of the bearings, a feature representing the degraded condition of the bearing needs to be identified. This section employs a new feature, known as Hilbert–Huang entropy (HHE), which is measured by the Shannon entropy based on the horizontal and vertical acceleration spectra [50, 51]. For signal processing, the Shannon entropy reveals the irregular changes in signal properties, such as spectra and amplitude distribution. Therefore, HHE can determine the irregular acceleration distribution levels in a 0.1 s time slice, which is selected as the feature for bearing degradation. Correspondingly, the Hilbert–Huang entropy is expressed according to the definition of information entropy aswhere and is the probability of occurrence of an event, and here, it refers to the probability density of the spectrum and the occurrence probability of amplitude corresponding to the *i*th frequency.

The HHE of the horizontal acceleration over six training sets is presented in Figure 3. The HHE shows the degraded condition of the bearings.

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##### 3.6. Health Indicator

For comparing the health indicator of the standard model with the HHE of the training set 1, we need to transform the HHE into a standard format (Figure 4).

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We now consider signal-amplitude transformations. Amplitude transformations follow the rules as follows:

The new form of Hilbert–Huang entropy in all training sets can be seen in Figure 5.

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Envelope analysis is one of the most successful and widely used methods in early fault diagnosis of rolling bearings. These technologies have been comprehensively described and analyzed in the field of rotating machinery, especially in the diagnosis of bearings. In this study, we use the peak envelopes and use spline interpolation with not-a-knot conditions over local maxima separated by at least samples [52].

Cubic spline interpolation smooths curves without adding irregularities to the signal, and therefore, it was selected as an appropriate technique to apply to the data. Assuming that there are points on an interval ,

Let be a cubic polynomial on each of the intervals so that we have different polynomials of this type in total. According to the definition of the cubic spline interpolation, can be represented by the following form [46, 53, 54]:where , and are undetermined coefficients. is the health indicator (HI) in the time slice .

#### 4. Degradation Model

##### 4.1. RUL Estimation Model

The fault of the machine is the life expectancy or time remaining before the machine needs to be repaired or replaced. Predicting residual service life from system data is the core goal of the predictive maintenance algorithm. A model that fits the time evolution of a health indicator and predicts how long it will last before a health indicator crosses the threshold that indicates a failure threshold, and one that compares the time evolution of a health indicator from a system that runs to fault, can calculate the most likely failure time for the current system. In this study, we consider the following simple linear degradation model. The linear degradation model object implements the following continuous-time linear degradation model:where is the model intercept, which is constant. is the model slope and is modeled as a random variable with a normal distribution with mean and variance. is the model additive noise and is modeled as a normal distribution with zero mean and variance.

Based on the degradation paths described in Figure 6, it is clear that the simple linear degradation model with zero origin is the logical assumption to fit the data.

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##### 4.2. Degradation Detected Threshold and Failure Threshold

It is difficult to determine degradation detected threshold (DDT) and failure threshold (FT) because at failure, values of different machines generally have a large variation range. For data-driven methods, RUL is obtained when a health indicator exceeds a predefined DDT, and the machine should stop when a health indicator exceeds a predefined FT. However, the DDT and FT are usually determined experimentally because the health indicator values of different bearings at a failure time are generally different [55]. In this study, there are six training datasets to get DDT and FT. We manually labeled beginnings of the wear-out periods of six training sets DDT values. Training FT values are the maximum HI in each training sets (shown in Figure 7). We use bootstrap sampling to get the best elevate value of DDT and FT. Bootstrap is a nonparametric method for estimating population values from small samples. The idea of bootstrap is to generate a series of bootstrap pseudosamples, each of which is the initial data with playback sampling. Through the calculation of pseudosamples, the distribution of statistics is obtained. In this study, to bootstrap 600 times and find the confidence interval of the average, the average for each pseudosample is calculated. We get 600 averages. The confidence intervals can be obtained by calculating the quantiles of 600 averages. It has been proved that bootstrap sampling can be unbiased close to the overall distribution when the initial sample is large enough. The bootstrap sample is applied to construct a DDT and FT value. Distribution and parameter estimation are listed in Table 1, and we get 2.97 as the best estimated value of DDT and 5.88 for FT.

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#### 5. Experiment, Discussion, and Conclusion

##### 5.1. Experiment

To verify the model prediction results in this paper, we used eleven test sets and determined the RUL of these test sets using the health indicator and linear degradation model. According to the 2012 IEEE PHM Data Challenge criterion, we computed the scores and mean errors for these eleven RULs using the model and its parameters. All the results are listed in Table 2, including those from Tianyi Wang and Edwin Sutrisno (2012 IEEE PHM Data Challenge winners). The score of our method is in between those of the two winners and very close to the result of Edwin Sutrisno’s. The mean error rate was lower than that of the two winners. Therefore, this health indicator algorithm and this model are effective.

##### 5.2. Discussion

###### 5.2.1. Effects of Different FT Values and DDT Values in Linear Degradation Model

In Section 4, the 95% DDT confidence interval is between the lower endpoint, 2.9561, and the upper endpoint, 2.99284, while FT confidence interval is between the lower endpoint, 5.85236, and the upper endpoint, 5.90777. In Table 3, scores are computed with different DDT and FT values. There is little effect when different values of DDT and FT are selected in the confidence interval. This health indicator algorithm and the way to compute DDT and FT of linear degradation model are effective.

The estimation of FT is based on the 6 training sets, and the optimal value and confidence interval of FT is obtained by bootstrap sampling. The experiment results show that when the FT value is greater than the upper limit of the confidence interval, the score of this algorithm can be improved. When FT ≥ 5.93, the scores of our method are higher than those of the two winners. In future research, we will analyze how many training sets are needed to get the optimal FT value.

In Table 3, we found values of FT are selected in the confidence interval. This health indicator algorithm and the way to compute DDT and FT of linear degradation model are effective.

###### 5.2.2. Comparison between the Linear Degradation Model and the Exponential Degradation Model

Theoretically, the RUL can be predicted by the exponential degradation model. In our research, a comparison is conducted using the same values of DDT and FT in the linear degradation model and the exponential degradation model. We used the method in Section 4 to estimate RUL of the test sets and determine the score listed in Table 4. No matter what values were utilized, the score was lower than that in Table 3, but the scores are in between those of the two winners. The linear degradation model is accurate enough to estimate the RUL of the bearing.

##### 5.3. Conclusion

In this study, a novel health indicator algorithm and a method to compute the degradation detected threshold and failure threshold of the linear degradation model for a bearing RUL prediction are developed. Horizontal vibration signals collected from accelerated degradation bearings tests are utilized to demonstrate the effectiveness of the method. This health indicator is processed by Hilbert–Huang entropy and amplitude transformations. The DDT and FT values are estimated by bootstrapped sampling and parameter estimation of these samples distribution. The RUL prediction model uses a linear degradation model. The results demonstrate that the proposed model-based method’s performance is close to the best winners’ methods in the 2012 IEEE PHM Data Challenge. In the future, we will use this proposed method to test more open experimental datasets and to establish more robust wear-out period detection and RUL prediction models, which will be able to describe anomalies in the degradation trends.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was partially supported by the Appropriative Researching Fund for Guangdong Provincial Key Laboratory of Precision Equipment and Manufacturing Technology under Grant PEM201604 and Bureau of Education of Guangzhou Municipality under Grant 2017192201.