Data Decomposition Techniques with Multi-Scale ...

sensors Article

Data Decomposition Techniques with Multi-Scale Permutation Entropy Calculations for Bearing Fault Diagnosis Muhammad Naveed Yasir and Bong-Hwan Koh * Department of Mechanical, Robotics and Energy Engineering, Dongguk University-Seoul, 30 Pildong-ro 1 gil, Jung-gu, Seoul 04620, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-2260-8591; Fax: +82-2-2263-9379 Received: 8 March 2018; Accepted: 19 April 2018; Published: 21 April 2018







Abstract: This paper presents the local mean decomposition (LMD) integrated with multi-scale permutation entropy (MPE), also known as LMD-MPE, to investigate the rolling element bearing (REB) fault diagnosis from measured vibration signals. First, the LMD decomposed the vibration data or acceleration measurement into separate product functions that are composed of both amplitude and frequency modulation. MPE then calculated the statistical permutation entropy from the product functions to extract the nonlinear features to assess and classify the condition of the healthy and damaged REB system. The comparative experimental results of the conventional LMD-based multi-scale entropy and MPE were presented to verify the authenticity of the proposed technique. The study found that LMD-MPE’s integrated approach provides reliable, damage-sensitive features when analyzing the bearing condition. The results of REB experimental datasets show that the proposed approach yields more vigorous outcomes than existing methods. Keywords: rolling element bearing (REB); fault detection and diagnosis (FDD); local mean decomposition (LMD); multi-scale entropy (MSE); sample entropy; permutation entropy (PE); multi-scale permutation entropy (MPE)

1. Introduction Mechanical defects occur in rolling element bearing (REB) due to wear, fatigue, corrosion, overload, and misalignment. To guarantee the safety of a system exposed to harsh environments, reliable condition monitoring strategies are required. Monitoring and evaluating the current condition of the REB is significant to many researchers for ensuring high operational efficiency. The temperature, vibrational response, and lubrication state are few features of non-destructive condition monitoring. Since the REB system requires a continuous operational state for an indeterminate period, vibration response data can be easily obtained from transducers mounted in the vicinity of the system. Bearing fault detection and diagnosis (FDD) problems have been popular research topics for many decades [1,2]. A variety of techniques exploit vibrational data to assess damage-sensitive features for diagnosis, and many methods use feature extraction and classification tools to monitor the state of the system’s components based on statistical deviation from normal operations [3,4]. Some of the signal-based techniques explored in data analysis processes, such as frequency-spectrum, time-statistical, and time-frequency analysis, provide simultaneous information on the time and frequency of the system response [3]. The time-frequency approach has been employed in different ways for REB fault detection. Defects typically occur in the REB due to micro-cracks and abrasions resulting in strong nonlinear, non-Gaussian, and nonstationary profiles in its vibrational response. Several time-frequency techniques have their own lacunae. Windowed Fourier transform

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(WFT) [5] has a fixed window; Wigner Ville distribution (WVD) [6] causes cross-term interference when dealing with multi-component signals; short-time Fourier transform (STFT) [7] has a fixed analysis window; and Wavelet transform (WT) has different wavelets that should be predefined for each component [8,9]. Wavelet analysis may be applied to FDD problems in rotating machines that can decompose the vibrational signal only if a rectangular time–frequency partition is available, a requirement that would fail to guarantee the instantaneous frequency of each resulting component obtained by WT. Power spectrum analysis and fast Fourier transform (FFT) has been employed to decompose the original signal into its component form for REB faults detection [10]. The time-domain technique exploiting statistical features such as kurtosis, entropy, RMS, skewness and impulse factor are used as benchmark parameters to assess the bearing conditions. The computational intelligence techniques such as machine learning and support vector machine (SVM) have been used to classify the characteristics of the regular healthy and damaged systems [11–13]. Moreover, some of the fault detection problems have been solved by using fuzzy-based techniques [14,15]. Empirical mode decomposition (EMD) has also been an effective tool that provides an advantage over other signal analysis techniques because it tackles non-stationary and non-linear vibrational signals [16,17]. The EMD is inherently combined with the concept of Hilbert transform (HT) to obtain frequency modulation [17]. EMD decomposes the multi-component signal into an amplitude and frequency modulated (AM-FM) signal that produces several intrinsic mode functions (IMFs). In each IMF, HT can calculate the corresponding instantaneous frequency (IF) and instantaneous amplitude (IA). However, there are some deficiencies in EMD, such as mode mixing, envelope line, and end effect, which are still under investigation in recent literature. Furthermore, the impulsive negative instantaneous frequency can appear when computing the instantaneous frequency using the Hilbert transform. A self-adaptive time–frequency analyses technique called Local Mean Decomposition (LMD) was introduced and used to analyze electroencephalogram (EEG) signals [18]. The LMD generates several product functions, each of which is composed of an AM-FM signal. Naturally, this multi-component time–frequency signal can be reconstructed in its original form by summing up all of the instantaneous amplitudes and frequencies of each product function. Hence, the LMD is suitable for analyzing non-stationary signals as well as extracting fault-sensitive signatures from the REB response. Additionally, multi-scale entropy (MSE) was proposed to address the drawbacks found in samples that handle long-term data configuration [19]. The MSE is widely used in a range of data processing applications, such as the analysis of biological signals [20,21], postural control [22], and the vibration of REB [23]. Here, the ingenuity of the MSE is that it is considered a two-step movement: (1) a coarse-grained time series is used to develop illustrations of the system dynamics at variant time scales; and (2) a sample entropy is used to measure the regularity in the time series data [19]. The integrated approach of LMD and MSE has already been introduced in fault detection of REB by Liu et al. [23]. It has also been reported that a single scale method, such as sample entropy or approximate entropy, is unable to provide transparent pictorial information on bearing conditions [13]. Among others, permutation entropy (PE) has recently been used as a damage-sensitive feature after performing the decomposition of its vibrational signal [24].The concept of multi-scale permutation entropy (MPE) is based on the coarse-grained process and calculating the PE at each scale factor. It has been established and implemented for signal processing of human brain activity in applications such as EEG analysis [25], electrocardiogram (ECG) analysis, [26] and REB monitoring [27]. In the present work, a recently developed concept of MPE integrated with LMD has been applied to monitor the condition of an REB system. The obtained time-series data collected from the rotating systems (bearings or gears) have been convoluted and contain the data structure of multiple dimensions. Here, the algorithm of LMD-MPE is projected, and its application to REB shows an enhanced ability to classify the condition of the REB. This paper consists of five sections. In Section 2, the Shannon entropy, sample entropy, PE, MSE, and MPE are explained in detail. The aforementioned methodology

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proposed method is described in Section 3. In addition, theproposed result ofmethod ANOVA [28] and is implemented on experimental data and the validity of the is variance describedtest in Section 3. feature classification performed to discuss cross-verified Finally,are a concluding statement In addition, the result are of ANOVA variance test [28] and featureresults. classification performed to discuss and future research presented.statement and future research direction are presented. cross-verified results.direction Finally, aare concluding 2. 2. Preliminaries Preliminaries The of this this study study is is depicted depicted in in Figure Figure 1. 1. The overall overall schematic schematic of of the the implemented implemented methodology methodology of First, the vibrational vibrationalsignal signalwas wasdecomposed decomposed LMD [18] sample entropy PE were First, the byby LMD [18] andand thethe sample entropy and and PE were then then calculated as damage-sensitive features. The details of the methodology are discussed in calculated as damage-sensitive features. The details of the methodology are discussed in the the following following subsections. subsections.

Figure 1. 1. The The proposed proposed methodology methodology (LMD-MPE) (LMD-MPE) scheme scheme for for REB REB fault fault diagnosis. diagnosis. Figure

2.1. Principle Principle of of LMD LMD 2.1. The LMD LMD method method is is composed composed of of envelope envelope and and frequency frequency modulation modulation of of the the decomposed decomposed form form The of multicomponent signals. The primary aim of the time–frequency LMD method is to decompose the of multicomponent signals. The primary aim of the time–frequency LMD method is to decompose multi-component AM-FM signal into mono-component signals the the multi-component AM-FM signal into mono-component signalsusing usingthe thelocal localmean mean values values of of the signals and envelope components. The LMD is a composite of the product functions that consist of signals and envelope components. The LMD is a composite of the product functions that consist of frequency and amplitude modulation of the signal. The frequency modulation incorporates frequency and amplitude modulation of the signal. The frequency modulation incorporates meaningful meaningful information regarding the vibrational and amplitude produces amplitude modulation. The information regarding the vibrational signals andsignals produces modulation. The product product of the envelope-modulated signal and modulated frequency are called the product function [18]. of the envelope-modulated signal and modulated frequency are called the product function [18]. The summarized summarized formula formula is The is given given as as j

j PFi ( t ) + u j ( t ) x( t ) =  i = 1PF ( t ) + u ( t ) x (t) = ∑ i j i =1

(1) (1)

where PF is the product function and u is the residue, which becomes monotonic or a constant value at the PF endisof process (detailsand of the method can be foundmonotonic in References where thethe product function u is LMD the residue, which becomes or a [18,23]). constant This valueisata suitable for LMD to be appropriate for non-linear and non-stationary in the end ofreason the process (details of considered the LMD method can be found in References [18,23]). This is signals a suitable rotatoryforsignals. LMD appropriate is a self-adaptive method and despite any fixed windowing or the reason LMD toSecondly, be considered for non-linear non-stationary signals in rotatory presenceSecondly, of another fixed so itmethod can cater to theany sudden in theorsignal. The other signals. LMD is parameter, a self-adaptive despite fixedchange windowing the presence of methodfixed combined the LMD MSE to keep the problem solution-efficient. another parameter, so it method can caterand to the sudden change in the signal. The other method combined the LMD method and MSE to keep the problem solution-efficient.

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2.2. Principle of Entropy Entropy is a parameter that assesses the level of disorder in the data. In other words, it represents a lack of information in the signal or data. The basic concept of entropy was first developed by Shannon and thus named the Shannon entropy [29]. Pincus brought forward the methodology of approximate entropy as a measurement tool for a complex system [30]. Richman et al. first introduced sample entropy and compared approximate entropy in the analysis of physiological time-series data [31]. Approximate entropy is implemented for noisy, short, real-time data to predict chaotic oscillation change and sometimes produces an inconsistent result because its performance highly depends on the size of the time-series data. Unlike approximate entropy, sample entropy is not dependent on the length of time-series data. It is based on the calculation of a negative natural logarithm of a distance between two vectors. The ones are maximum norm values where self-matching data points are excluded. This mathematical expression can be written as: H( x ) = −∑ p( xi )logp( xi )

(2)

where p(xi ) represents the density function of unsystematic components in data points and the logarithm is based on two natural functions. The PE can be used to assess chronological information embedded in time-series data. It is simple in computation and its results are consistent. However, the PE of the m dimensions [13] always needs to be appropriately selected to achieve the best result. The more substantial value of m is costly with respect to computational run-time. Given the time series x with delay factor τ and embedded dimension m, the delay vector can be given as [13]: xim = [ x (i ), x (i + τ ), . . . , x (i + (m − 1))τ ]

(3)

Here, the time delay factor is defined as τ and m is an embedded dimension. Each permutation of π and the relative frequency can be obtained as [13]: p(π ) =

Number {t|t ≤ T − (m − 1)τ, xm t has type π } N − ( m − 1) τ

(4)

The PE of the above time series in Equation (3) of m dimension is defined in the arrangement of the Shannon entropy. It can be written as: m!

HPE (m) = − ∑ p(πi ) ln(p(πi ))

(5)

i =1

The normalized PE (NPE) can be described as: HNPE (m) =

HPE (m) ln(m!)

(6)

As before, the PE value depends on the embedded dimensions m and delay factor τ. If m decreases, the algorithm will not work correctly for the detection of anomalies that occur in the time-series data. Specifically, a lower value of m will result in a lower value of PE, and it becomes hard to detect anomalies in different defect conditions. On the contrary, the high value of PE allows for the relatively easy identification of anomalies in regular and irregular time-series data. For calculation, we used the Shannon entropy HPE of Equation (5). 2.3. MPE and MSE Costa first introduced the MSE technique to analyze time-series data [19,21] based on the concept of sample entropy. PE is another entropy-based algorithm, but it has a single value. Thus, PE can

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be integrated with the multi-scale procedure to form a new entropy-based feature called MPE [13]. 5 ofthe 16 MPE employ a coarse-grained procedure that yields new time series data by using mean value in each non-overlapping segment of the equal length of the output variable. Suppose that Suppose we points have Nthat datarepresent points that the time series; coarse-grained points can be we have that N data therepresent time series; coarse-grained data pointsdata can be calculated calculated using(7). Equation (7). using Equation Sensors 2018, and 18, x Both MSE

jτ

1 1 jτ N N (τ ) = x i x, 1i ,1≤ ≤ j ≤j ≤ y j y=(τ)  ∑ j τ τ τ iτ=(i =j(−j -11))ττ++ 1 1

(7) (7)

Here, ττ is and averages thethe data points in is the thenon-overlapping non-overlappingwindow windowlength lengthofofthe thedata datapoint point and averages data points the corresponding window. TheThe MSE calculates thethe complexity of time-series datadata by calculating the in the corresponding window. MSE calculates complexity of time-series by calculating sample entropy of aofparticular setset ofofdata. the sample entropy a particular data.The Thecoarse-grained coarse-grainedtechnique techniquegives givesthe the average average of non-overlapping, consecutivedata datapoints. points.The Thesample sampleentropy entropycan canbebeobtained obtained through Equation non-overlapping, consecutive through Equation (1)(1) at at each coarse-grained factor theof case ofand MSE andThe MPE. The MSE technique wasdesigned initially each coarse-grained scalescale factor in theincase MSE MPE. MSE technique was initially designed the change correlated with the data series multiple time scales. This to measuretothemeasure change correlated with the data series on multiple time on scales. This technique provides technique provides more information in the case of a single-value scale. The schematic diagram of more information in the case of a single-value scale. The schematic diagram of the coarse-grained the coarse-grained process depicted Figure 2 [21]. Both MSE andcoarse-grained MPE methods use the process is depicted in Figureis2 [21]. Both in MSE and MPE methods use the technique coarse-grained technique prior entropy to calculating the sample entropy or PE entropy. prior to calculating the sample or PE entropy.

Figure 2. 2. The The schematic diagram a coarse-grained procedure for scales two and Figure three. Figure schematic diagram of aof coarse-grained procedure for scales two and three. Figure reproduced from Ref. [21]. reproduced from Ref. [21].

3. Experimental Evaluation 3.1. Characteristic Characteristic Properties Properties of of Bearing Bearing Data Data 3.1. The experimental experimental data data for for this this study study was was accessed accessed from from the the bearing bearing data data center center at at Case Case Western Western The Reserve University (CWRU), USA [32]. The REB model of 6205-2RSJEMSKF was used by Reserve University (CWRU), USA [32]. The REB model of 6205-2RSJEMSKF was used by CWRU CWRU and the the time-series time-series data data were mounted on on the the bearing bearing housing and were recorded recorded through through accelerometers accelerometers mounted housing with magnetic bases. bases.The Theaccelerometers accelerometerswere were located motor-drive fan-drive ends. In with magnetic located at at thethe motor-drive andand fan-drive ends. In this this study, we used the REB data from the motor-drive end. The speed of the motor-drive was 1797 study, we used the REB data from the motor-drive end. The speed of the motor-drive was 1797 rpm rpm varied and varied the change in loading. sampling frequency 12 kHz, thus, a total and with with the change in loading. The The sampling frequency was was 12 kHz, and,and, thus, a total of of 5000, 2048, and 1024 sampled data points were selected to form a set of segments as outlined in 5000, 2048, and 1024 sampled data points were selected to form a set of segments as outlined in Table 1. The REB condition was delineated as being healthy normal or defective in the rolling element, Table 1. The REB condition was delineated as being healthy normal or defective in the rolling inner-race, or outer-race. Furthermore, four different sizes, 0.007, 0.014,0.007, 0.021 and 0.028 inches, element, inner-race, or outer-race. Furthermore, fourfault different fault sizes, 0.014, 0.021 and were created on the inner-race, outer-race, and rolling element. Table 1 also describes cases of REB 0.028 inches, were created on the inner-race, outer-race, and rolling element. Table 1 also describes bearing sorteddamage by the lengths severity of damage in different locations. Table 2 cases of damage REB bearing sorted of bydata the and lengths of data and severity of damage in different illustrates the detail of the inner-race fault size on the REB datasets used in Case 1. locations. Table 2 illustrates the detail of the inner-race fault size on the REB datasets used in Case 1.

The next stage involved the decomposition of the bearing signal using the time-frequency analysis LMD technique. First, the original signal was decomposed into several product functions. The sample entropy and PE values were then quantified from each product function to calculate the MSE and MPE. For verification purposes, the values from the proposed LMD-MPE technique were compared with those from previously published LMD-MSE technique results [23].

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Table 1. REB data cases based on different data lengths and damage types. Case

Healthy

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Inner race Outer-race Rolling element

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LMD MSE PE MPE

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Inner race Outer race Rolling element

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Table 2. REB conditions for Case 1. Set

Damage Condition

Healthy set 1 (H1) Healthy set 2 (H2) Damage set 1 (D1) Damage set 2 (D2) Damage set 3 (D3)

Healthy REB Healthy REB 0.007-inch outer-race 0.014-inch outer-race 0.021-inch outer-race

The next stage involved the decomposition of the bearing signal using the time-frequency analysis LMD technique. First, the original signal was decomposed into several product functions. The sample entropy and PE values were then quantified from each product function to calculate the MSE and MPE. For verification purposes, the values from the proposed LMD-MPE technique were compared with those from previously published LMD-MSE technique results [23]. All three cases were obtained from the drive-end location. The motor speed was 1797 rpm and the motor load were zero horsepower. 3.2. LMD Implementation on the Data from REB Cases 1–3 In this section, we investigated three different damage sizes instigated on the inner-race of the REB in Case 1. The resulting shock impulses by defect were modulated with the normal vibrational signal of the REB, which made it a signal composed of different sources, such as the original and shock-induced impact signals. The vibration signal from REB was decomposed into multiple components and a monotonic residue for feature extraction through LMD. The sampling frequency was 12 kHz and the rotational frequency of the inner-race was 40 Hz. The calculated rotational frequency of REB with respect to the shaft is 85.7 Hz. The time series for all of the datasets (see Table 2) in Case 1 are shown in Figure 3. The total length of the data is 600,000 points and they were further divided into several datasets for the convenience of computation. Each dataset consists of 5000, 2048, and 1024 points in Cases 1, 2, and 3, respectively, as described in Table 1. While eight data segments were used in Case 1, only five segments were used for Cases 2 and 3. Both H1 and H2 were the same types of healthy signals (or the baseline). The only difference was that H1 was the standard baseline signal for all signals acquired in the CWRU dataset at zero motor load (i.e., filename: Normal_0). The H2 was a particular case for a healthy signal and became a baseline for outer-race damage recorded in different instances. The first eight segments of Case 1 were considered to show a general trend in damage-sensitive features when monitoring the bearing condition. The total segments consisted of 40,000 (8 × 5000) data points. The implementation of LMD on Case 1 that yielded product function was integrated with MSE in every health condition of REB to visualize the fault extraction investigation trend. A single damaged area created on the surface of an inner-race had three different severity conditions, as described in Table 2. Furthermore, two independent sets for the normal healthy condition were analyzed.

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(a) (a)

Amplitude Amplitude

(b) (b) (c) (c) (d) (d) (e) (e)

Sample points Sample points Figure 3. Time-series of REB for Case 1 of same segment set: (a) Healthy set 1 (H1); (b) Healthy set 2 (H2); Figure 3.Time-series Time-series REB Case of same segment set:Healthy (a)3Healthy set 1 (b) (H1); (b) Healthy set Figure 3. ofofREB forfor Case 1 of12 same segment set: (a) set 1 (H1); Healthy set 2 (H2); (c) Damage set 1 (D1); (d) Damage set (D2); and (e) Damage set (D3). 2 (H2); (c) Damage set 1 (D1); (d) Damage set 2 (D2); and (e) Damage set 3 (D3). (c) Damage set 1 (D1); (d) Damage set 2 (D2); and (e) Damage set 3 (D3).

The first five segments of the REB—healthy set 1 (H1), healthy set 2 (H2), damage set 1 (D1), The segments of REB—healthy 11 (H1), healthy (H2), damage (D1), damage set five 2 (D2) and damage 3 (D3)—wereset analyzed verify set the 22LMD five The first first five segments of the theset REB—healthy set (H1),to healthy set (H2),methodology. damage set set 11The (D1), damage set 2 (D2) and damage set 3 (D3)—were analyzed to verify the LMD methodology. The five product and a residual generated after LMD implementation are shown inThe Figure damage setfunctions 2 (D2) and damage set 3signal (D3)—were analyzed to verify the LMD methodology. five 4. product functions and residual after implementation shown 4. The first segment of aaH1 of Casesignal 1 wasgenerated used. Next, theLMD multiple componentsare of segment of D3 was product functions and residual signal generated after LMD implementation are shown in in1Figure Figure 4. The first segment of H1 of Case 1 was used. Next, the multiple components of segment 1 of D3 was then decomposed into six product-functions plus one residue, as illustrated in Figure 5. Each plot The first segment of H1 of Case 1 was used. Next, the multiple components of segment 1 of D3 was then decomposed six plus residue, as in 5. plot provided a total into of 5000 data points. It should beone noted that number of product functions depended then decomposed into six product-functions product-functions plus one residue, as illustrated illustrated in Figure Figure 5. Each Each plot provided total of of 5000 5000 datasignal, points.and should be noted noted that number number of product product functions depended on the complexity of the the higher the that signal complexity, the higher thedepended number of provided aa total data points. ItIt should be of functions on the complexity of the signal, and the higher the signal complexity, the higher the number product functions. on the complexity of the signal, and the higher the signal complexity, the higher the number of of product functions. functions. product 0.2

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Figure 4. 4. LMD H1; (b) (b) PF1; PF1;(c) (c)PF2; PF2;(d) (d)PF3; PF3;(e)(e) PF4; PF5; Figure LMDofofnormal normalhealthy healthyCase Case1:1: (a) (a) original H1; PF4; (f)(f) PF5; and and (g) u is the residual signal. Figure of normal healthy Case 1: (a) original H1; (b) PF1; (c) PF2; (d) PF3; (e) PF4; (f) PF5; and (g) u 4. is LMD the residual signal.

(g) u is the residual signal.

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Sample points Figure5.5. Product Product functions functions of of the the damage damage Case Case 11 set: set: (a) (a) original original (D3) (D3) vibrational vibrational signal; signal; (b) (b) PF1; PF1; Figure (c)PF2; PF2;(d) (d)PF3; PF3;(e) (e)PF4; PF4;(f) (f)PF5; PF5;(g) (g)PF6; PF6;and and(h) (h)uuisisthe theresidual residualsignal. signal. (c)

Apparently, the first few product functions inherited most of the features from the original Apparently, the first few product functions inherited most of the features from the original vibrational signal. In Case 1, the first two product functions (PF1 and PF2) were used to demonstrate vibrational signal. In Case 1, the first two product functions (PF1 and PF2) were used to demonstrate feature extraction from the original healthy signal. The basic concept of LMD is to remove the higher feature extraction from the original healthy signal. The basic concept of LMD is to remove the higher frequency with an iterative algorithm and obtain the mean values of all of the new signals. The new frequency with an iterative algorithm and obtain the mean values of all of the new signals. The new envelope signal is subtracted from the original one and the subtracted signal becomes the next input envelope signal is subtracted from the original one and the subtracted signal becomes the next input signal. In the end, the decomposed signal is then formed into a monotonic signal, as is clearly signal. In the end, the decomposed signal is then formed into a monotonic signal, as is clearly visualized visualized by Figures 4g and 5h. by Figures 4g and 5h. 3.3. MSE MSE and and MPE MPE Implementation Implementationon onExperimental ExperimentalData Data 3.3. First introduced introduced by by Costa Costa [17], [17], MSE MSEwas wasdeveloped developed to to estimate estimate the the degree degree of of complexity complexity of of First time-series data data over overmultiple multiple time timescales. scales. Sample Sample entropy entropy is is the the most most common common tool toolused usedin inMSE MSEto to time-series check regularity in data. It is observed that the coarse-grained data sample entropy value reduced in check regularity in data. It is observed that the coarse-grained data sample entropy value reduced accordance with the MPE exhibited exhibited differences differences in in in accordance with theincrease increaseininthe thetime timescale scalefactor. factor. The The MSE MSE and and MPE extracting defect features with different degrees of severity. The values of MSE and MPE vary with extracting defect features with different degrees of severity. The values of MSE and MPE vary with different product product functions, functions, time time scales, scales, and anddamage damageconditions. conditions. Therefore, Therefore, MSE MSE and and MPE MPE can can be be different effectively employed employed as asaaparametric parametric scale scale to to assess assessthe theabnormality abnormality in inmeasurements measurements of of the the signal, signal, effectively such as kurtosis and entropy. such as kurtosis and entropy. 3.3.1. The The Integrated IntegratedApproach Approachof ofLMD LMDand andMSE MSEfor forCases Cases1–3 1–3 3.3.1. Here,MSE MSEwas wasregarded regarded as index an index a feature for characterizing the complexity of the Here, as an and and a feature for characterizing the complexity of the chaotic chaotic time series data the of next stage of faultthe extraction, combined and MSE time series data of REB. Inof theREB. nextIn stage fault extraction, combinedthe LMD and MSELMD approach was approach was used to monitor the condition of an operating REB. The overall process went through used to monitor the condition of an operating REB. The overall process went through the following steps: the following steps: 1. In Cases 1, 2, and 3, the vibrational signal was sampled as shown in Table 1. To establish succinct 1. benchmark In Cases 1, results, 2, and eight, 3, thefive, vibrational sampled shown in Table 1. To establish and five signal sets of was segments wereasused in Cases 1–3, respectively. succinct benchmark results, eight, five, and five sets of segments were used in Cases 1–3, 2. The LMD technique in Equation (1) was used to obtain the product function and residual respectively. signal of the damaged REB conditions. The first five product functions were selected to extract 2. The LMD technique in Equation (1) was used to obtain the product function and residual signal meaningful information. Figures 4 and 5 show the LMD implementation of the data for Case 1. of the damaged REB conditions. The first five product functions were selected to extract meaningful information. Figures 4 and 5 show the LMD implementation of the data for Case 1.

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3.

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16 15 and scale factor was a positive integer, and for Cases 1–3, it was selected as9 of20, 15, respectively. For MSE, the scale factor was a positive integer, and for Cases 1–3, it was selected as 20, 15 and For MSE, the scale factor was a positive integer, and for Cases 1–3, it was selected as 20, 15 and 15, respectively. A 15, graph of the LMD-MSE result was constructed with the help of original REB signal and Product respectively. 3. A graph the LMD-MSE result was constructed withofthe help ofwas original REB Figures signal and Function of of all conditions Case 1. The output the MSE shown 3. A graph1 of the defect LMD-MSE result of was constructed with the help of original REB in signal and 6 and 7 Product Function 1 of all defect conditions of Case 1. The output of the MSE was shown in forProduct original signal and 1, of respectively. Function 1 of Product all defectFunction conditions Case 1. The output of the MSE was shown in Figures 6 and 7 for original signal and Product Function 1, respectively. 6 and 7 forfor original signal and Product Function 1, used respectively. A Figures proper classifier a damage-sensitive feature can be to make the result clearly distinctive 4. A proper classifier for a damage-sensitive feature can be used to make the result clearly 4. from A proper classifier for a damage-sensitive feature can be used to make the result clearly the different damageREB conditions. distinctive fromREB the different damage conditions. distinctive from the different REB damage conditions. In 6 and 7, distinctive differentdamage damage cases cannot be easily separated, 5. Figures In Figures 6 and 7, distinctiveMSE MSEpoints points for different cases cannot be easily separated, 5. In Figures 6 and 7, distinctive MSE points for different damage cases cannot be easily separated, especially in the case andD1. D1.The The other other factor, not allow for for distinction, may may be be especially in the case of of H1H1 and factor,which whichdid did not allow distinction, especially in the case of H1 and D1. The other factor, which did not allow for distinction, may be considered a measurement noise. considered a measurement noise. considered a measurement noise. 6. comparison The comparison between theoriginal originaland and different different faulty conditions showed thatthat the data fromfrom the faulty conditions showed thefrom data 6. The The comparison between between the original and different faulty conditions showed that the data the damaged REB was more separable in the MSE versus Product Function 1 at the damaged REB more separable inMSE the versus MSE versus Product Function at the thethe damaged REB waswas more separable in the Product Function 1 at the1coarse-grained coarse-grained time factor τ = 5 and may be a parameter for measurement factors such as kurtosis coarse-grained = 5aand may be afor parameter for measurement factors as kurtosis time factor τ = 5time andfactor may τbe parameter measurement factors such as such kurtosis and entropy.

Multi-scale entropy Multi-scale entropy

and entropy. and entropy.

Scale factor Scale factor

Figure 6. The original REB signal for Case 1. An average of eight segments were in each set. Scale Figure 6. The original REB signal for Case 1. An average of eight segments were in each set. Scale

Figure 6. The REB signal for (sample Case 1. An average of eight in eachofset. factor 20 original vs. multi-scale values entropy) were used segments to obtain were the results theScale MSEfactor factor 20 vs. multi-scale values (sample entropy) were used to obtain the results of the MSE 20 vs.technique. multi-scale values (sample entropy) were used to obtain the results of the MSE technique.

Multi-scale entropy Multi-scale entropy

technique.

Scale factor Scale factor

Figure 7. REB Case 1 and Product Function11with with an average average ofofeight segments perper set.set. ScaleScale factor Figure 7. 7. REB Function eight segments factor Figure REBCase Case11and and Product Product Function 1 with anan average of eight segments per set. Scale factor 20 vs. multi-scale values (sample entropy) were used to obtain the results of the MSE technique. 20 20 vs.vs. multi-scale entropy)were were used to obtain the results the technique. MSE technique. multi-scalevalues values (sample (sample entropy) used to obtain the results of theofMSE

Figures 6 and 7 show the MSE method applied to both healthy and different damage severities

Figures 6 and 7 show the MSE method applied to both healthy and different damage severities Figures 6 anddamage 7 showtype thecases MSEtomethod applied to bothbetween healthythem. and different damage severities and the same illustrate the differences Curves from the original and the same damage type cases to illustrate the differences between them. Curves from the original and the same damage type cases to illustrate the differences between them. Curves from the original vibrational signal and Product Function 1 after applying LMD are depicted in Figures 6 and 7,

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respectively. In these figures, the data points are given as an average of the eight segments in each case scenario. In Figure 6, when the scale factor1was than five, curveinofFigures all health condition vibrational signal and Product Function afterless applying LMDthe areMSE depicted 6 and 7, casesrespectively. exhibited ambiguity in the the sample entropy and the same or a higher value of In these figures, data points arevalues given as anhad average of thevalues eight segments in each D1 than The few instances where values D1 less andthan D2 were not MSE clearly discernable were at caseH1. scenario. In Figure 6, when thethe scale factorofwas five, the curve of all health condition cases exhibited ambiguity the sample entropy values andtechniques, had the same a 13 and 15, as shown in Figure 7. After in decomposition through LMD thevalues three or different higher value of D1 than H1. The few instances where the values of D1 and D2 were not clearly damage severities (D1, D2, and D3) could be easily distinguished from the normal healthy (H1) data, as discernable were7.atAt 13 this and 15, as shown in Figurethe 7. After decomposition through LMD techniques, that illustrated in Figure stage, we discussed conventional techniques and demonstrated the three different damage severities (D1, D2, and D3) could be easily distinguished from the normal the proposed method or LMD-MPE properly performs with relatively small computational time and healthy (H1) data, as illustrated in Figure 7. At this stage, we discussed the conventional techniques has a more robust result. In the next section, we will establish a comparison between Case 1 and Case 3 and demonstrated that the proposed method or LMD-MPE properly performs with relatively small for PEcomputational calculation of the validity of the REB data. This comparison is based on two criteria: (i) the same time and has a more robust result. In the next section, we will establish a comparison damage type Case but different damage intensity, such 0.014, andREB 0.021 inches 1); or is (ii) the between 1 and Case 3 for PE calculation ofas the0.007, validity of the data. This (Case comparison samebased intensity of damage but different damage types, such as the 0.021 inches damage intensity on two criteria: (i) the same damage type but different damage intensity, such as 0.007, 0.014,(inner race, and outer race, or rolling element, similar Cases of 2 and 3). In the next section, 0.021 inches (Case 1); or (ii) the same to intensity damage but different damagesince types,Cases such 1–3 as are the 0.021 inches damage intensity (inner race, outer rolling element, similar to Cases 2 and 3). scenarios based on different damage severities and race, dataor lengths, we developed a reliable solution in Inwhatever the next section, 1–3 are scenarios based onworked different damage severities and data which scenariosince wasCases presented, the PE calculation successfully. lengths, we developed a reliable solution in which whatever scenario was presented, the PE worked 3.3.2.calculation PE for Cases 1–3 successfully.

PE originally 3.3.2.was PE for Cases 1–3proposed to examine the complexity of the natural time series data [33]. PE offers several advantages, such as less computational complexity and stability in the results, over PE was originally proposed to examine the complexity of the natural time series data [33]. PE otheroffers entropy calculation techniques. The calculation of PE values depends on the τ delay factor and several advantages, such as less computational complexity and stability in the results, over m dimensions. Incalculation Figures 8techniques. and 9, theThe embedded dimension m depends varies from 8 infactor obtaining other entropy calculation of PE values on the2 τtodelay and the optimal value of the permutation order. All the healthy condition sets of Cases 1–3 have variations in m dimensions. In Figures 8 and 9, the embedded dimension m varies from 2 to 8 in obtaining the the PE value.value Theof theory also statesorder. that the smallest of the embedded dimension m produces optimal the permutation All the healthyvalue condition sets of Cases 1–3 have variations in the PE value. TheHowever, theory also that thevalue smallest value embedded dimension m produces inconsistent results. forstates the higher of m, the of PEthe yields remarkably independent metrics inconsistent However,the forhealthy the higher value of m, theREB PE yields remarkably independent metrics that can be usedresults. to distinguish and damaged datasets, as depicted in Figures 8 and 9. that can be used to distinguish the healthy and damaged REB datasets, as depicted in Figures 8 and All 9. the The figures also show the implementation of PE in the decomposed REB vibrational signals. The figures also show the implementation of PE in the decomposed REB vibrational signals. All the REB conditions manifest recognizable separation from each other in the healthy and damaged cases REB conditions manifest recognizable separation from each other in the healthy and damaged cases (i.e., inner, outer race, and rolling element), as shown in Figures 8 and 9. Note that the “H1” and (i.e., inner, outer race, and rolling element), as shown in Figures 8 and 9. Note that the “H1” and “Healthy” of both graphs have approximately equal values. The case of “Outer race” in Figure 9 might “Healthy” of both graphs have approximately equal values. The case of “Outer race” in Figure 9 might be compared withwith D3 in 8. Moreover, it has been shown that thethe values are approximately be compared D3Figure in Figure 8. Moreover, it has been shown that values are approximatelyequal in theequal graphs. The results show that, when have data sizes or different damage in the graphs. The results show that,we when weeither have different either different data sizes or different types, the PEtypes, values the differences among REB conditions (i.e., healthy, inner-race, damage thecan PEdistinguish values can distinguish the differences among REB conditions (i.e., healthy, inner-race, and rolling element) by changing m in the permutation outer-race, andouter-race, rolling element) by changing the value ofthe m value in theofpermutation order. order.

Figure 8. Product FunctionofofCase Case1,1,damage damage size: and 0.021 inches; damage type: type: Figure 8. Product Function size: 0.007, 0.007,0.014, 0.014, and 0.021 inches; damage outer-race; data points: 1024; PE calculation with different m permutation order. outer-race; data points: 1024; PE calculation with different m permutation order.

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Figure 9. 9. Product 0.021 inches; different damage locations; PE Figure ProductFunction FunctionofofCase Case3,3,damage damagesize: size: 0.021 inches; different damage locations; calculation with different m permutation order. PE calculation with different m permutation order.

3.3.3. MPE MPE for for Cases Cases 22 and and 33 3.3.3. The MPE MPE algorithm algorithm was was implemented implemented in in the the REB REB data, data, and and the the preprocessing preprocessing for for the the The decomposition of the signal remained the same as in the LMD method. The length of data samples decomposition of the signal remained the same as in the LMD method. The length of data samples N, N, delay factor τ, and embedding dimension m are effective parameters in the calculation of MPE. If delay factor τ, and embedding dimension m are effective parameters in the calculation of MPE. If m m is a small value, then few characteristics will emerge that will produce a few dynamic changes, is a small value, then few characteristics will emerge that will produce a few dynamic changes, and andchanges the changes happen intime the time series cannot be determined precisely. On other the other hand, if the that that happen in the series cannot be determined precisely. On the hand, if the the value m increases, then the segment space of the timewill series will be standardized, which value of m of increases, then the segment space of the time series be standardized, which requires requires significant computing time and can be interpreted inaccurately. Therefore, the selection significant computing time and can be interpreted inaccurately. Therefore, the selection range of m is range of m is 4–7 in general [13,27]. chose m = 4 factor and the factor τ =PE 1 to PE in aInlater 4–7 in general [13,27]. We chose m = We 4 and the delay τ =delay 1 to calculate incalculate a later context. the context. In the stage, the integrated approach of LMD MPE was used to detectof thedamaged features next stage, the next integrated approach of LMD and MPE wasand used to detect the features of damaged REB. Again, the MPE theofcomplexity of athe signal scale at a different scale factor REB. Again, the MPE measured themeasured complexity the signal at different factor and different and different severity of bearing damage. The whole process included the following for both severity of bearing damage. The whole process included the following steps for bothsteps the MSE and the MSE and MPE methods: MPE methods: 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6.

Only five segment sets of each health condition were investigated among the bearing data of Only five segment sets of each health condition were investigated among the bearing data of Cases 1–3, as shown in Table 1 (i.e., five segments of all cases and their subclasses, such as Cases 1–3, as shown in Table 1 (i.e., five segments of all cases and their subclasses, such as healthy, healthy, inner race defect, outer race fault, and rolling element defect, were investigated.) inner race defect, outer race fault, and rolling element defect, were investigated.) The LMD method was used to calculate the product function of each damaged condition. The The LMD method was used to calculate the product function of each damaged condition. The first first five product functions were selected to obtain meaningful information regarding the five product functions were selected to obtain meaningful information regarding the bearing bearing condition. Figures 4 and 5 show the LMD implementation from Equation (1). condition. Figures 4 and 5 show the LMD implementation from Equation (1). The scale factor was an integer number and selected as 15 for both Cases 2 and 3. The The scale factor was an integer number and selected as 15 for both Cases 2 and 3. The coarse-grained coarse-grained calculation was performed through Equation (7). calculation was performed through Equation (7). The PE values were calculated for each coarse-grained signal using Equation (5). The sample The PE values for the each coarse-grained signal using Equation (5). The sample entropy values were were calculated calculated for MSE graph. entropy values forwas the MSE graph. Then, the MSEwere and calculated MPE graph created with every cross-grained point. The respective Then, the MSE and MPE graph was created with every cross-grained point. The respective sample entropy and PE values are depicted in Figures 10 and 11. sample10 entropy PE values depicted in Figures by 10 and 11.of the original REB signal. The Figure showsand a spider plot are of MSE preprocessed LMD Figure 10 shows a spider plot of MSE preprocessed by LMD of original REB signal. The same same original dataset of REB was used to produce Figure 11the after performing the LMD-MPE original dataset of REB was used tolocation produceofFigure 11 after performing LMD-MPE algorithm. The different damage the same damage severitythe could not be algorithm. identified The different location theThe samesame damage severity could not be identified precisely, as precisely, as damage shown in Figureof10. original REB condition scenario dataset after shown in Figure 10. The same original REB condition scenario dataset after processing using processing using the LMD-MPE approach presented in Figure 11 shows a clear separation among the damage locations of the same damage severity.

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the LMD-MPE approach presented in Figure 11 shows a clear separation among the damage 12 of 16 the same damage severity.

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Figure10. 10.Spider Spiderplot plotfor forMSE MSEof ofthe thePF4 PF4(Case (Case2): 2):damage damagesize sizeisis0.021 0.021inches. inches. Figure Figure 10. Spider plot for MSE of the PF4 (Case 2): damage size is 0.021 inches. Healthy Inner race Healthy Scale Factor 5

Scale Factor 4

1.77

1.91 Factor 5 Scale Factor 6 Scale 1.77 Scale Factor 6 1.91 2.07 1.56 1.56

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Scale Factor 8

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Scale Factor 8 2.28

Scale Factor 9 2.28 Scale Factor 9 2.31 2.31 Scale Factor 10

Scale Factor 10

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Scale Factor 7 2.07 Scale Factor 7

1.61 Factor 4 Scale

1.43 0.95

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Rolling element Outer race

Rolling element Sacle Factor 3 Sacle Factor 3 1.45

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Outer Inner race race

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0.88 1.09 0.95 0.81 0.73 Scale Factor 2 1.25 1.04 0.52 0.48 0.44 0.93 0.4 1.08 1.25 0.54 0.73 1.62 0.52 0.48 0.44 0.62 0.93 0.40.36 1.08 0.57 0.54 Scale Factor 1 1.14 0.62 1.71 0.31 0.36 0.58 0.57 0.25 0.51 0.76 1.01 Scale Factor 1 1.71 1.14 0.31 0.6 0.58 1.16 0.51 0.76 1.01 0.65 0.25 0.62 0.6 0.65 1.31 0.64 1.16 1.73 0.65 0.62 1.19 0.65 1.96 0.64 1.28 1.31 Scale Factor 15 1.73 1.19 1.24 1.3 1.96 2.62 1.79 Scale Factor 15 1.28 1.24 1.92 1.3 2.62 1.79 1.86 1.92 1.95 1.04

1.62

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Scale Factor 2.39 11

1.86 2.48

Scale Factor 11 2.48 Scale Factor 12 Scale Factor 12

1.95 2.6 2.6

2.56 2.56

Scale Factor 14 Scale Factor 14

Scale Factor 13 Scale Factor 13

Figure Figure11. 11.Spider Spiderplot plotfor forMPE MPEof ofPF4 PF4(Case (Case2): 2):damage damagesize sizeisis0.021 0.021inches. inches. Figure 11. Spider plot for MPE of PF4 (Case 2): damage size is 0.021 inches.

The spider plot is a graphical representation of multivariate data projected onto a The spider is a graphical representation of multivariate data projecteddata onto projected a two-dimensional The spiderplot plot isof a more graphical representation of variables. multivariate onto a two-dimensional graph than three quantifiable These variables represent graph of more than three of quantifiable variables. These variables represent multiple axes, represent which all two-dimensional graph more than three quantifiable variables. These variables multiple axes, which all start from a single central point. Here, the spider plot is used to start from axes, a single centralallpoint. Here, spider is used to demonstrate comparison healthy multiple start fromthe acondition singleplot central Here, the the spider plot outer isofused to demonstrate the which comparison of healthy cases ofpoint. REB (i.e., healthy, inner race, race, condition cases of REB (i.e., healthy, inner race, outer race, and rolling element) at different scale demonstrate the comparison of healthy condition cases of REB (i.e., healthy, inner race, outer race, and rolling element) at different scale factors. It also explains the differences among the values of factors. It also explains the differences among the values of each healthy condition at separate axes and rolling It also explains differences among values of each healthyelement) conditionatatdifferent separatescale axesfactors. of the scale factor. The the value of the scale factorthe starts from of the scale factor. The value of the scale factor starts from the center point (origin) and ends at the each healthy condition at separate axes of the scale factor. The value of the scale factor starts from the center point (origin) and ends at the outer circle ring. In Figure 10, the healthy case has a outer circlepoint ring. In Figureand 10, the healthy has a maximum whichthe shows the outer circle the center ends thecase outer ring. In value, Figure case has a maximum value, (origin) which shows theatouter circlecircle of the spider plot. 10, All the healthy faulty conditions of the spider plot. All the faulty conditions (inner-race, outer-race, and rolling element) are shown maximum value, which outer circle of thetoward spiderthe plot. All ofthe (inner-race, outer-race, andshows rollingthe element) are shown center thefaulty circle.conditions In a few toward the center of the and circle. In a few instances, the outertoward race sample entropy value is higher (inner-race, outer-race, rolling element) are shown the center of the circle. a than few instances, the outer race sample entropy value is higher than the rolling element sample In entropy the rollingthe element sample entropy value value (i.e., atisscale factors 9,the 11 and 15).element In Figure 11, defects on instances, outer race sample entropy higher than rolling sample entropy value (i.e., at scale factors 9, 11 and 15). In Figure 11, defects on the rolling element and inner-race the rolling element and inner-race have the same entropy value only at scale factor 15, which shows valuethe (i.e., at scale factors 9, 11 and Figure defects on the element and inner-race have same entropy value only at 15). scaleInfactor 15,11, which shows the rolling robustness of MPE. Note that have the same entropy value only at scale factor 15, which shows the robustness of MPE. Note that the MSE and MPE results have the same damage size and product function, and the entropy values the MSE andare MPE have damage product function, and entropy values represented theresults averages of the fivesame segment sets.size Theand same damage severity of the different damage represented are the averages of five segment sets. The same damage severity of different damage locations shows that the bearing defects are more easily separable in the MPE than MSE toward the locations shows that the bearing defects are more same original REB signals, as shown in Figure 11. easily separable in the MPE than MSE toward the same original REB signals, as shown in Figure 11.

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the robustness of MPE. Note that the MSE and MPE results have the same damage size and product function, and the entropy values represented are the averages of five segment sets. The same damage severity of 18, different damage locations shows that the bearing defects are more easily separable 13 in ofthe Sensors 2018, x 16 MPE than MSE toward the same original REB signals, as shown in Figure 11. It isisshown shownthat that healthy defective REB signals be accurately discerned, but it It thethe healthy andand defective REB signals can becan accurately discerned, but it becomes becomes very hard to identify the types or locations of the fault occurrences in Figure 10. Moreover, very hard to identify the types or locations of the fault occurrences in Figure 10. Moreover, Figure 11 Figure the 11 LMD-MPE shows the method LMD-MPE method using spidercan plot, which can all easily identify all health shows using a spider plot,awhich easily identify health condition cases condition cases of REB. Here, in this spider plot, the healthy case signal has been positioned near the of REB. Here, in this spider plot, the healthy case signal has been positioned near the center of the center All of the circle.cases All damage cases (inner-race, outer-race, and rolling element) canclearly be identified circle. damage (inner-race, outer-race, and rolling element) can be identified except clearly except and the outer-race and rolling at The scaleMSE factor Thealgorithms MSE and MPE the outer-race rolling element at scaleelement factor 15. and15. MPE were algorithms applied to were applied to the same PF4results signal,are andshown the results are shown in11. Figures 10 and 11. The factor the same PF4 signal, and the in Figures 10 and The scale factor is a scale significant is a significant of thethat MSE and MPEillustrated that was clearly using a spider plot. parameter of theparameter MSE and MPE was clearly using aillustrated spider plot. Figures 10 and 11 Figures and 11values showcorresponding the entropy values corresponding to every scale from 1 to 15. The show the10 entropy to every scale factor from 1 to 15. Thefactor function mapped with a function mapped with a total 15 scale factors and MPE shows separable entropy values compared total 15 scale factors and MPE shows separable entropy values compared with that of the MSE case, with that in of these the MSE case, as shown in these figures. as shown figures. 3.3.4. 3.3.4. Comparison of MSE and and MPE MPE Results Results of of Case Case 22 Using Using Box Box Plot Plot To validateand andcompare compare proposed method, a boxplot five segments is presented in To validate thethe proposed method, a boxplot of fiveof segments is presented in Figure 12. Figure 12. five-scale-factors The first five-scale-factors boxplottheindicates the distribution data by The first standardizedstandardized boxplot indicates distribution of data by of minimum, minimum, median, first,quartile and third quartile values, shown Figure 12. Thebetween separation maximum, maximum, median, first, and third values, as shown inas Figure 12.inThe separation the between thethe median the normal healthy and cases,atparticularly at all mentioned scale median of normalofhealthy and damage cases,damage particularly all mentioned scale factors 1–5, is factors 1–5, is remarkable. The calculated value of the median of the different defect condition cases remarkable. The calculated value of the median of the different defect condition cases shows their shows their differences. of the interquartile range (IQR/variation box)(Figure of MPE12b) (Figure 12b) differences. The size of The the size interquartile range (IQR/variation box) of MPE is more is more compact than of the(Figure MSE (Figure 12a), which that the LMD-MPE method has compact than that of that the MSE 12a), which shows shows that the LMD-MPE method has more more certainty the PE value calculation the sample entropy calculation method, LMD-MSE. certainty in theinPE value calculation than than the sample entropy calculation method, LMD-MSE. The The whisker of scale the scale factor of MSE the MSE technique is longer the MPE shown in scale factors whisker of the factor of the technique is longer thanthan the MPE shown in scale factors 2–5. 2–5. The data outlier (positive red mark) in scales 2 and 4 (shown inLMD-MSE the LMD-MSE technique) shows The data outlier (positive red mark) in scales 2 and 4 (shown in the technique) shows that that inconsistent values appear during calculation thesample sampleentropy. entropy. However, However, in the somesome inconsistent values appear during thethe calculation of ofthe the LMD-MPE did notnot exist in all factors, as shown in Figure 12b. These LMD-MPE technique techniquethe theoutlier outlier did exist in scale all scale factors, as shown in Figure 12b. points These validated that the LMD-MPE technique technique shows superior distinction the between damage conditions of points validated that the LMD-MPE shows superior between distinction the damage REB compared with compared the prior technique. offers a new prospective conditions of REB with the The priorLMD-MPE technique.technique The LMD-MPE technique offersmethod a new for tackling the REB fault detection prospective method for tackling theproblem. REB fault detection problem.

Figure Figure 12. 12. Boxplot Boxplot for for the the MSE MSE and and MPE MPE of of the the first first five five scale scale factors: factors: (a) MSE (multi-scale entropy); and and (b) (b) MPE, MPE, where where II is is aa healthy healthy signal, signal,IIIIisisinner innerrace, race,III IIIisisouter outerrace, race,and andIV IVisisrolling rollingelement. element.

4. Discussion The ANOVA statistical test [28,34] was also implemented in each scale factor case to verify that all the healthy conditions are expressively diversified among each healthy condition of REB (i.e., healthy, inner-race, outer-race, and rolling element defects). The ANOVA test was also employed to

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4. Discussion The ANOVA statistical test [28,34] was also implemented in each scale factor case to verify that all the healthy conditions are expressively diversified among each healthy condition of REB (i.e., healthy, inner-race, outer-race, and rolling element defects). The ANOVA test was also employed to determine which method (LMD-MSE or LMD-MPE) provides better performance in the bearing monitoring problem. In the results of the ANOVA tests, the sum of squares between the two groups had lower MSE values (see Table 3) than MPE (see Table 4). The p-values were less than 0.05 in both the LMD-MSE and LMD-MPE techniques, which showed significant results at P. The F-test value was the ratio of variation between the group means to the variance within the damage condition groups means. The low F-test value shows that the group means were close to each other, and the high F-test shows a condition where the variability of the group means was large relative to the group’s variability. This means that the F-test could be a measure of separation among damage group means, as shown in Table 4. It is shown that, in both scale factor cases, the F-test (LMD-MSE = 27.49, LMD-MPE = 102.55) were higher than F-crit (3.24), as shown in Tables 3 and 4, respectively. These values indicate that the four REB health conditions comprised of the healthy signal, inner race, outer race, and rolling elements were statistically and significantly different from one another. However, the F-test of LMD-MPE (102.55) was much higher than the F-test score of LMD-MSE (27.49). Therefore, it can be concluded that the LMD-MPE method is better than LMD-MSE. Moreover, in the datasets of LMD-MSE, all values were very close to the mean, resulting in a small variance (SS = 0.15) between the four REB healthy conditions. However, the datasets of the LMD-MPE values were spread further away from the mean, leading to a more substantial variance (SS = 0.25) between the four signal features. The total variance of MPE was 0.257, which was higher than that of the MSE (0.179). Table 3. One-way ANOVA test for sample entropy at scale 2 of MSE. Source of Variation

Sum of Squares (SS)

Degree of Freedom (DF)

Mean Square (MS)

Between Groups Within Groups Total

0.1503 0.0292 0.1795

3 16 19

0.0501 0.0018

p-Value

F-Test

F-Crit

1.5 × 10−6

27.4866

3.2389

Table 4. One-way ANOVA test for PE at scale 2 of MPE. Source of Variation

Sum of Squares (SS)

Degree of Freedom (DF)

Mean Square (MS)

Between Groups Within Groups Total

0.2446 0.0127 0.2574

3 16 19

0.0816 0.0008

p-Value

F-Test

F-Crit

1.16 × 10−10

102.5448

3.2389

Additionally, classification has been performed using the features from LMD-MSE and LMD-MPE. For comparison, MATLAB [35] has been employed for classification of over 15 datasets of REB Case 2. Here, the time delay factor and embedding dimension is set to unity and four, respectively. The label of healthy and damage condition is applied as an input for supervised learning and five-folds cross-validation are implemented for the training and testing of classifiers. Table 5 shows the result for average classification accuracy of various classifiers and area under the curve (AUC) of the receiver operator characteristics (ROC). The case of LMD-MPE provides relatively better accuracy in classification compared to LMD-MSE using all three classifiers. Considering the sensitivity and specificity, the classification result becomes successful as the AUC approaches to unity. Note that the result from the method of LMD-MPE gives more consistent value of AUC close to 1.00.

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Table 5. Comparison of average of classifiers on REB data Case 2.

Classifier Fine Gaussian SVM Medium Gaussian SVM Fine KNN

LMD-MSE

LMD-MPE

Accuracy (%)

AUC

Accuracy (%)

AUC

66.3 88.0 84.2

0.94 1.00 0.96

87.1 90.4 86.3

1.00 0.99 0.98

5. Conclusions This research introduced LMD, a self-adaptive technique that can be integrated with MPE to tackle the problem of REB fault diagnosis using the test data from CWRU. The results of the proposed LMD-MPE method were compared with that of the previously used LMD-MSE technique. We found that LMD-MPE showed improvement and robustness in detecting an identical damage size (0.021 inches) with different damage locations (i.e., inner-race, rolling element, and outer-race) of REB compared with LMD-MSE. Specifically, the performances of LMD-MSE and LMD-MPE were reviewed and discussed through the graphical representation of spider plots. The results were also verified by two statistical analysis tests: boxplot and ANOVA. The boxplot showed that the LMD-MPE variation performance of the REB condition was higher than that of the LMD-MSE method. The ANOVA test also demonstrated that the variance between the groups was 0.25 in the LMD-MPE case, which can be compared with 0.15 for the LMD-MSE case. Moreover, the LMD-MPE and LMD-MSE results were compared using the SVM and KNN classifiers. The classifier accuracy and AUC values for LMD-MPE showed an acceptable performance in fault classification of REB. This anticipated technique can be readily extended to incorporate an online monitoring system for REB diagnosis. Future work can be directed toward fully self-automated or self-adaptive methods for fault diagnosis by calculating the MPE and optimizing the dimensions of PE for further development of a robust early-warning system in REB fault diagnosis using noisy and complex gearbox data in practice. Acknowledgments: This work was supported by a research program (2015R1D1A1A01059701) of the Basic Science Research Program through the National Research Foundation of Korea (NRF) and funded by the Ministry of Education, Science and Technology. This work was also supported by the research program of Dongguk University (2017). Author Contributions: Muhammad Naveed Yasir provided the idea of algorithms and carried out data analysis. Bong-Hwan Koh supervised the work and provided major direction of the research. Conflicts of Interest: The authors declare no conflict of interest.

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