Randomness complexity as a family feature of rolling bearings’ degradation

2.1. Shannon entropy

ShEn is the first proposed complexity, and it quantifies the probability density function (PDF) of the signal as $ShEn=-{\sum }_i{p}_i\mathrm{l}\mathrm{o}\mathrm{g}{p}_i$where $i$ is all amplitude values of the signal and $p_i$ is the probability that amplitude value $a_i$occurs anywhere in the signal [25]. However, in the case of measured signals, the PDF is not known and should be estimated. Also, to consider all amplitude is not reasonable generally. The easy way to evaluate the PDF is to use the histogram where the amplitude range ($N$) of the signal can be divided into $k$ bins linearly so that the ratio $k/N$ is constant. The ratio $k/N$ characterizes the average filling of the histogram. It is worth to notice that there is something difference between the real PDF and the histogram. To reduce the influence, the ratio$k/N$ should be set bigger. However, the bigger ratio will reduce the ability of noise resistance. In this paper, we set it as 50.

2.2. Approximate entropy

The calculation steps of ApEn are as follows [30]:

Step 1: Given an $N$ point time series$u=\left\{u\left(i\right)\right\}$, and form vector sequences $x\left(1\right)$ through $x(N-m+1)$, defined by $x\left(i\right)=\left[u\left(i\right),\cdots ,u\left(i+m-1\right)\right]\text{.}$ These vectors represent $m$ consecutive $u$ values with $i$th point.

Step 2: The distance $d\left[x\right(i),x(j\left)\right]$ between vectors $x\left(i\right)$ and $x\left(j\right)$ can be defined as the maximum difference in their respective scalar components, where:

Step 3: For each vector $x\left(i\right)$, a measure that can describe the similarity between the vector $x\left(i\right)$ and all other vectors $x\left(j\right)$ can be constructed as:

where $\mathrm{\Theta }\left\{x\right\}$is the Heaviside step function, and it is represented as:

The parameter $r$ symbolizes a tolerance value or similarity criterion, which is defined as $r=k\cdot std\left(u\right)$, where $k$ is a positive constant, and $std(·)$ is the standard deviation of the time series. The parameter $r$ is considered as a regularity or frequency of patterns similar to a given pattern of window length $m$.

Step 4: Define ${\varphi }^m\left(r\right)={(N-m+1)}^{-1}{\sum }_{i=1}^{N-m+1}\mathrm{l}\mathrm{o}\mathrm{g}{C}_i^m\left(r\right)\text{,}$ and define $ApEn(m,r)=\underset{N\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\left[{\varphi }^m\right(r)-{\varphi }^{m+1}(r\left)\right]$. Given a finite time series with $N$ data points, the statistic ApEn value is defining the $ApEn(m,r,N)={\varphi }^m\left(r\right)-{\varphi }^{m+1}\left(r\right)$.

The four steps can be described as phase-space reconstruction, distance calculation, similarity calculation and complexity calculation. The concept of ApEn is derived from correlation dimension. When calculating correlation dimension, the correlation integral must be obtained first, defined as:

which is similar to $C_i^m\left(r\right)$. It has been proved that there is a relationship with $\underset{r\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}C\left(r\right)\propto r^D$, where $D$ is the correlation dimension. Since$D=\underset{r\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\left(\mathrm{l}\mathrm{o}\mathrm{g}\right(C\left(r\right))/\mathrm{l}\mathrm{o}\mathrm{g}(r\left)\right)$, $D$ can be estimated by calculating the slope of the $\mathrm{l}\mathrm{o}\mathrm{g}\left(r\right)~\mathrm{l}\mathrm{o}\mathrm{g}\left(C\right(r\left)\right)$ when increasing $r$ from a small value.

2.3. Sample entropy

The calculation procedures of SampEn are as follows [31]:

Step 1 and Step2: Phase-space reconstruction and distance calculation as the same as step 1 and 2 in the calculation of ApEn.

Step 3: Given the tolerance value $r$ and count the number of $d_{ij}^m<r$ as $B_i$, and define $B_i^m\left(r\right)={(N-m-1)}^{-1}{B}_i$, where $i=\mathrm{1,2},\cdots ,N-m+1$ and $i\ne j$. Calculate the mean value as$\mathrm{}{B}^m\left(r\right)={(N-m)}^{-1}{\sum }_{i=1}^{N-m}{B}_i^m\left(r\right)$.

Step 4: Calculate$\mathrm{}{B}^{m+1}\left(r\right)$, and define $SampEn(m,r)=\underset{N\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}[-\mathrm{l}\mathrm{o}\mathrm{g}[B^{m+1}\left(r\right)/B^m\left(r\right)\left]\right]$. The statistic SampEn value is estimated by defining the $SampEn(m,r,N)=-\mathrm{l}\mathrm{o}\mathrm{g}\left[B^{m+1}\right(r)/B^m(r\left)\right]$.

2.4. Fuzzy entropy

Based on the ApEn, FuzzyEn expanded the $\mathrm{\Theta }\left\{x\right\}$ in ApEn’s third step. Heaviside step function causes a kind of two-state classifier, which is a crisp one, namely the classifier is one or the other. FuzzyEn combined fuzzy theory with ApEn. By introducing the “membership degree” with a fuzzy function ${\mu }_C\left(x\right)$ which associates each point $x$ with a real number in the range [0, 1], the fuzzy theory gives a property that the higher ${\mu }_C\left(x\right)$is, the higher the membership grade of x in the relevant set. When calculating FuzzyEn, Chen et al. defined the fuzzy function $\mu (d_{ij}^m,n,w)$ as$\mathrm{}\mathrm{e}\mathrm{x}\mathrm{p}(-{\left(d_{ij}^m\right)}^n/w)$, where $w$ determines the width and the gradient of the boundary of the exponential function (It is generally set to be 2), and replace the Heaviside step function in the ApEn [34]. At last, the statistic FuzzyEn is estimated by$FuzzyEn(m,r,w,N)={\varphi }^m(r,w)-{\varphi }^{m+1}(r,w)$. Moreover, FuzzyEn has a difference in distance calculation compared with ApEn. In FuzzyEn, vectors are generalized by removing the baseline of themselves.

2.5. Permutation entropy

The calculation steps of PermEn are as follows [33]:

Step 1: Phase-space reconstruction as same as step 1 in the calculation of ApEn.

Step 2: Arrange $X\left(i\right)$ in an increasing alignment. The $m$ number of values contained in each $X\left(i\right)$ can be arranged in an increasing alignment as:

Accordingly, any vector $X\left(i\right)$ can be mapped onto a set of symbols as $S\left(l\right)=(j_1,j_1,\cdots ,j_m)$, where $l=\mathrm{1,2},\cdots ,k$and $k\le m!$. $S\left(l\right)$ is one of the $m!$ symbol permutations, which is mapped onto the $m$ number symbols $(j_1,j_2,\cdots ,j_m)$ in $m$-dimensional embedding space. If $P_1,P_2,\cdots ,P_k$ are the probability distribution of each sequences where ${\sum }_{l=1}^k{P}_l=1$, then the PermEn for the times series of order $m$ can be defined as the Shannon entropy as for the $k$ symbol sequences:

2.6. Lempel-Ziv complexity

The calculation steps of LZC are as follows [29]:

Step 1: To calculate Lempel-Ziv complexity, the time series should be conducted “coarse-graining” operation first. In general, the time series would change to a sequence that only contains two symbols. The sequence $S_N$ is reconstructed by comparing the value of each sample of the previous sequence with the median value $m$ (or mean value, we take the median value as default). If the sample’s value is large than $m$, it will change to 1, otherwise as 0.

Step 2: By obtained the two-symbol $N$ point sequence $S_N=\{s_1{s}_2\cdots s_N\}$, initialize $S_{v,0}=\left\{\right\}$, $Q_0=\left\{\right\}$, $C_N\left(0\right)=0$ and $r=0$. Set $Q_r=\left\{Q_{r-1}{S}_r\right\}$. Due to the $Q_r$ does not belong to $S_{v,r-1}$, so set $C_N\left(r\right)=C_N(r-1)+1$, $Q_r=\left\{\right\}$ and $r=r+1$.

Step 3: Set $Q_r=\left\{Q_{r-1}{S}_r\right\}$ and judge whether the $Q_r$ belongs to $S_{v,r-1}$, if true, set $C_N\left(r\right)=C_N(r-1)+1$ and $r=r+1$; if not, then set $C_N\left(r\right)=C_N(r-1)+1$, $Q_r=\left\{\right\}$ and $r=r+1$. Repeat step 3 to the end of the sequence. Finally, we have the $C_N\left(r\right)$.

Step 4: The length of the sequence has obvious influence on the $C_N\left(N\right)$. So, Lempel and Ziv gave a normalized LZC, which is defined as $C_{normalizedN}=C_N\left(N\right)\times \mathrm{l}\mathrm{o}\mathrm{g}N/N$.

2.7. ${\mathit{C}}_0$ complexity

The calculation steps of $C_{0}C$ are as follows [36]:

Step 1: Implement Fast Fourier Transform (FFT) for the time series $s\left(t\right)$ and gain the spectrum where $x\left(k\right)=FFT\left(s\right(t\left)\right)$.

Step 2: Obtain $G_N$, where it stands for the mean square value of the amplitude spectrum. Introduce a variable $r$ where$\mathrm{}r(r>1)$. Where the part that its value is bigger than ${rG}_N$ is considered as regular component. Where the opposite part is considered as irregular part:

Step 3: Transform the regular part $\tilde{x}\left(t\right)$ into $\tilde{s}\left(t\right)$ through the inverse FFT, and $\tilde{s}\left(t\right)$ is the regular part of the original signal.

Step 4: Define $C_{0}C$ as the ratio of the component of $\left|s\right(t)-\tilde{s}(t\left)\right|$ to the $s\left(t\right)$ where it stands for the ratio between the irregular part to the original signal:

Ref [36] suggests that $r$ should be 5 to 10, and we will discuss the parameter as below.

Compute the $C_{0}C$ of white noise with different $r$ as shown in Fig. 1. When $r>6$ the value of $C_{0}C$ is close to 1. Actually, the $C_{0}C$ of a random time series should be equal to 1. By using the group simulation signals in Section 3.1, we have four $C_0$ complexities curves with $r=$ 6, 10, 100 and 200, as shown in Fig. 2. Too higher $r$ will lead to the fact that the $C_0$ complexity of pseudo-periodical signal does not close to 0. In this paper, we set $r=\text{10}$ as default.

Fig. 1The C0C value with different r

Fig. 2The curves of the C0C with different SNRs and r

From the procedures of the sevens, we can see that ApEn derives from correlation dimension. They have many similarities in their forms and calculation. SampEn and FuzzyEn are two modifications of ApEn in different aspects. Although, PermEn have somewhat similar in form to ApEn, e.g. phase-space reconstruction, but it calculates complexity in ShEn form. Thus, we can classify the complexities into two categories: one is the ApEn, SampEn and FuzzyEn, the other is PermEn and ShEn. As to the rest of the sevens, LZC and $C_{0}C$, we put them into another category, for they have threshold parameters for coarse-graining or as the boundary of periodicity and randomness. To compare the performance of the sevens, we need to reduce the influence of parameters, so the communal parameters should be consistent. The setting of parameters is within the recommended values of their original references. It is suggested that the tolerance of ApEn, SampEn and FuzzyEn should be set 0.1-0.25 std. The embedding dimension is suggested to set as 2 or 3. We set them as 0.2 std and 2. Particularly, though PermEn has the phase-space reconstruction procedure which has the same form as ApEn, FuzzyEn and PermEn, it is essentially different. For PermEn, vectors come from the phase-space reconstruction are not to be compared. Comparisons exist within the vectors. For the other two, comparisons are implemented between vectors to calculate the distance. So, the embedding dimension $m$ of PermEn is different from the others. Large $m$ will extremely increase the calculation time, and we set $m=\text{6}$.

3.1. Performance in sinusoidal signals with additive noise

To figure out the performance of different complexities in rolling bearings, it is the primary and paramount to test in periodical signals with different intensity noise, where the sine wave is the simplest periodical signal. Now, we set a class of simulation signals defined as $S\left(t\right)=X\left(t\right)+e\left(t\right)$, where $X\left(t\right)=\mathrm{s}\mathrm{i}\mathrm{n}(2\pi \times 10t)$ and $e\left(t\right)$ represents the additive noise. The sampling frequency is 10000 Hz with 1 s duration. Fig. 3 shows the complexities with different SNRs. To be more vivid, all the complexities have been normalized.

Fig. 3The curve of seven complexities versus SNRs

It is obvious to see that all the complexities tend to descend with the decreasing of the additive noise. The complexities should decrease with the decreasing of the additive noise. Among the sevens, ShEn and PermEn are the worst, since they do not have a good monotonous tendency. The rests of complexities present good monotonicity. Among them, ApEn the rightmost is the most sensitive to the noise, and $C_{0}C$ the leftmost is the most insensitive to the noise.

3.2. Performance in logistic map

In Section 3.1, we have discussed the performance of seven complexities in a specific group of simulation signals. In this part, we will use more general simulation signals. The logistic map can be taken as an easy platform for it can generate periodical and chaotic signals. As many references shown, bearings are nonlinear components and many researches have reported nonlinear phenomena such as chaos, bifurcations and quasi-periodicity in bearings [38-40]. The logistic map is an easy nonlinear system, where it can be simple mathematical written as $x_{n+1}=\mu x_n(1-x_n)$. Fig. 4 shows the process of the logistic map where $2.5<\mu <4$, meanwhile, it shows the largest Lyapunov exponent (LLE) of the logistic map. Where $LLE<0$ means the system is periodical. $LLE=0$ means bifurcation occurs, and $LLE>0$ means chaotic behaviors occur. In the logistic map, the period 2 bifurcation happens at $\mu =3$; the 4 bifurcation happens at $\mu =\text{3.449}$. After $\mu =\text{3.83}$, there is a short period 3.

Analogically, we have calculated the seven complexities of the logistic map, and the results are shown in Fig. 5. The LLE is a reference of the sevens. The trend of the sevens should be similar to the LLE When $LLE>0$. Let’s define a term of $LL{E}_1$ which satisfies that $LL{E}_1=LLE$when $LLE>0$, $LL{E}_1=$0 when $LLE<0.$ It is observed that PermEn has quite a few fluctuations before $\mu =\text{3.449}$. ShEn has some platforms before $\mu =\text{3.449}$. FuzzyEn has something wrong around $\mu =\text{3.5}$ and before $\mu =\text{3}$. The reason of that must lie in the fuzzy membership of the FuzzyEn. LZC and $C_{0}C$ have some wrong value about $\mu =\text{3.6}$, where there exist chaotic behaviors, but both have values close to zero. All the sevens exhibit the short period 3 after $\mu =\text{3.83}$. As it shows, ApEn and SampEn almost coincide with $LL{E}_1$. In this part, ApEn and SampEn have the best performance. It should be notice that the application scope of the $LLE$ is narrow. It only can be calculated in the determined chaos systems. It cannot be computed with an arbitrary time series.

Fig. 5The seven complexities of the logistic map

a) ShEn, PermEn and SampEn versus $LL{E}_1$

b) ApEn, FuzzyEn, LZC and $C_{0}C$ versus $LL{E}_1$

3.3. Performance in rate of convergence

The length of data can affect the complexities value too. By using the data where the complexities are not convergent is inaccurate. In this part, we will make certain the performance in rate of convergence of the seven complexities, and it is a supplementary performance of complexities. In order to study the influence of data length on the sevens, we use the simulation signal in Section 3.1 with SNR = –10 dB as an example. The length is from 100 to 4000, as illustrated in Fig. 6.

Fig. 6The curve of seven complexities versus SNRs

As we can see, PermEn and ShEn have similar increasing convergence trend, and the complexities value are convergent after 2000 data points. SampEn, FuzzyEn, LZC and $C_{0}C$ have similar convergence trend. The complexities valued have a fluctuant decreasing trend. ApEn is different from the above, and it has a fluctuant increasing trend. From the comparison of convergence rate, we can confirm that $C_{0}C$ is the best for it can be convergent about 1000 data points. ShEn and PermEn are no doubt the worst. To be more accurate and explicable, the length of each data should beyond 2000.

3.4. Brief summary of comparisons of simulation signals

From the three comparisons of simulation signals above, we can have some brief conclusions. Primarily, every complexity conforms to the basic principle i.e. the higher the regularity is, the lower the complexities value. Among them, the performances of ShEn and PermEn are dissatisfactory. Take ShEn as an example, the parameter of average filling of the histogram ($k/N$) must be set. Though, to be a certain extent, such a method can improve anti-noise performance, the complexity still has its inherent problems. For instance, it is relatively insensitive to the changes in the tails of the distribution and slow convergence. Similarly, LZC and $C_{0}C$ have their inherent problems too. Coarse-graining may change the dynamic properties of the original time series. As to $C_{0}C$, it is lack of rigor to define a threshold as the boundary of periodicity and randomness. Relatively, ApEn and SampEn perform better than the others. As to a modification of ApEn, we consider that SampEn has the best performance among them. Thus, we will take SampEn as the benchmarking of the seven complexities.

4.1. Example 1 (inner race fault)

The Example 1 is from the IEEE PHM 2012 Prognostics Challenge data and the data were provided by FEMTO-ST Institute [41]. The purpose of the challenge was to estimate the remaining useful life of bearings. FEMTO-ST Institute has made an experimentation platform which is named PRONOSTIA as shown in Fig. 7. In this challenge, the data were monitored with 3 different loads. There were 6 complete run-to-failure data for training and 11 truncated data for predicting the remaining useful life. Tests were stopped when the vibration signal reached 20 g. However, we have no idea of the failure type of the bearings. In this paper, the first dataset is taken as the Example 1 within 2803 files. The parameters of the test and bearings are listed in Table 2.

Fig. 7Overview of PRONOSTIA

As a fact that we don’t know the failure type of the Example 1, we can study the envelop spectrum of it. The envelope spectrum of the last file of the Example 1 is shown in Fig. 8. By calculation of characteristics frequencies of the test, we can obtain the ball pass frequency on inner race is 221.66 Hz, the ball pass frequency on outer race is 168.34 Hz and fundamental train frequency is 12.95 Hz. In Fig. 8, there is a peak at 218.8 Hz. So, we can infer that the final failure type of Example 1 is inner race fault.

Fig. 8The envelope spectrum of e Example 1’s last file data

The seven complexities of the Example 1 are calculated as shown in Fig. 9. The seven complexities present a downtrend. They show consistent results, however, the ShEn seems to have more fluctuation. Among them, ApEn, SampEn and LZC seem to be more similar. Take SampEn as an example, before #500 (where # means the number file), the complexity increases to a peak and then decreases for a long time. As the figure shows, there is a local peak appeared about #2400 to #2600. Before #500, oil film is not fully formed is the reason of the arise of the complexities. When formed, the complexities’ decreasing indicate that the fault of the bearing is deepening. The reason for the peak around about #2400 to #2600 is probably the pitting of the surface are planished by the rotating. So, the signal may present not so periodical. And then, the new cracks appear making the signal more periodical, so the complexities are then dropped.

The complexities present a kind of similarity and can be explained with Fig. 3. Assume that the curves in Fig. 3 are seven functions ($y_i\left(x\right)=f_i\left(x\right)$, $i=\mathrm{1,2},\cdots ,7$) that reflect monotonic decreasing, they have a similar trend. Take each file data as x ($x=\mathrm{1,2},\cdots ,2803$) and put them to $y_i\left(x\right)=f_i\left(x\right)$. If the functions $f_i\left(x\right)$ are similar, then the results are similar too. To give quantitative similarity results, in this paper, we will propose a trend similarity index (TSI). Many similarity indexes are based on distance measures e.g. Manhattan distance, Euclidean distance, Kullback-Leibler divergence or correlation analysis e.g. Pearson correlation coefficient. Neither can measure the trend similarity of two sequences. In order to measure the trend of a time series, what comes to mind first is to analyze the derivatives. Certainly, we need to deburr the complexities curves. There we use the smoothing spline for fitting. Fig. 10 shows the SampEn deburred by smoothing spline with the smoothing parameter s equals to 1e-6, 1e-7 and 1e-8. As we can see, when $s=$1e-6, the fitted SampEn exists a little fluctuation. When $s=$1e-8, the smoothed SampEn doesn’t have a good fitting at the end of the data.

Fig. 9The seven complexities of Example 1

a) ShEn and PermEn

b) ApEn

c) SampEn

d) FuzzyEn

e) LZC

f)$C_{0}C$

To define the trend similarity of two same length time series, a direct way is to consider the derivatives of them. Fig. 11(a) shows three functions, where $y_1=x^2$, $y_2=x$, $y_3=\sqrt{x}$, we can consider that they have the same trend, for they have positive derivatives. Fig. 11b) shows two functions, where $y_4=x^2$, $y_5=x^3\text{+}1$. We can deem that where $x>0$, they have the same trend, where $x<0$, they have the different trend. By defining the ratio of the same trend length to the entire length, we can obtain the TSI, where the TSI of $y_4$ and $y_5$ is 50 %. In Section 3, we have concluded that SampEn present the best performance of the seven complexities in simulation signals, and then, we can have each TSI based on the SampEn deburred by smoothing spline which is shown in Table 3. From the results, we can clearly find that ApEn have extremely high similarity. ShEn and PermEn have relatively low similarity to the SampEn. Though, fitting parameters can affect the results of TSI, it is not an obstacle to estimate which complexity has higher similarity to SampEn.

Fig. 10The SampEn of Example 1 deburred by smoothing spline with the three smoothing parameters

a) The smoothing parameter equals to 1e-6

b) The smoothing parameter equals to 1e-7

c) The smoothing parameter equals to 1e-8

Fig. 11The example of trend similarity

a)$y_1$, $y_2$ and $y_3$

b)$y_4$ and $y_5$

Intuitively speaking, the directly way to reflect the degradation of impact is the amplitude of defect frequency (ADF). The peaks at the envelope spectrum of impact of the Example 1 are calculated and make up the ADF as shown in Fig. 12. As we can see, there is a little fluctuation before #2750, and then there is a quick increasing. Accurately, the ADF measures the energy of the impact. It is a portion of the whole energy of the vibration signals. The root mean square (RMS) is a commonly used feature to measure the holistic energy of the signals. The RMS of the Example 1 is shown in Fig. 13 with SampEn. The increasing of RMS indicates the deepening of the deterioration. We can see that there is an opposite trend between the RMS and SampEn. When there is a pitting occurs on the rubbing surface, it will make energy concentration and the signal more periodical. So, the complexities go down. However, at the end of failure, the RMS increases quickly, but there is no sudden decreasing of SampEn. That’s can be explained when the bearing is closing to failure, there are more pitting on the rubbing surface, each pitting can cause the periodical signal, but the combined signal is not as periodical as the one caused by a pitting. So, the RMS and SampEn measure different properties of the signal.

Fig. 12The Example 1’s ADF

Fig. 13The RMS and SampEn of the Example 1

4.2. Example 2 (outer race fault)

Another run-to-failure data is used to verify the performance of the seven complexities. The data comes from the Intelligent Maintenance System (IMS) center [42]. The test rig is mounted four bearings on a shaft as shown in Fig. 14. In the test, four double row bearings typed Rexnord ZA-2115 were installed on the shaft. An accelerometer was installed on the test rig to monitor the vibration signal of the bearings. The parameters of the test and bearings are shown in Table 4. The bearing 2-1 which is the first bearing of the Set No. 2 is used as the Example 2. The failure type of the Example 2 is outer race defect which is shown in Fig. 15. The Example 2 has 982 files.

The complexities of Example 2 are shown in Fig. 16. We can see there is a similar trend. We use the SampEn as an example. Before #520, it presents stationary process and the bearing is in normal condition. We can see the complexity decreases linearly from #520 to #700 and the bearing is in slight defect condition. From #700 to #850, there is a peak and the condition is severe. From # 900 to the failure, the SampEn presents an increasing and the bearing is run to failure. On the whole, the SampEn is decreasing which is a same conclusion of Example 1. We can compare the RMS and ADF of the Example 2 to explore the reason why the trend of complexities is present like that.

Fig. 14The test rig of Example 2

Fig. 15The failure type of Example 2

Fig. 16The seven complexities of Example 2

a) ShEn and PermEn

b) ApEn

c) SampEn

1

e) LZC

f)$C_{0}C$

Fig. 17 and Fig. 18 show the RMS and ADF of the Example 2. The trend of RMS and ADF is similar. Before #520, the RMS presents stationary and it increases linearly to #700. Between #700 and #850, the RMS experiences decreasing and then increasing. This is so-called “healing” phenomenon and have been stated detail in Ref. [7, 43, 44]. Form #900 to the end, the environment of the bearing is becoming violent. The RMS rises quickly to the failure.

Take SampEn as a baseline, we can have the TSI of each complexity too, as shown in Table 5. Before calculating TSI, the SampEn must be deburred. Fig. 19 shows the SampEn deburred by smoothing spline with the smoothing parameter s equals to 1e-3, 1e-4 and 1e-5. From the results, we can still find that ApEn has extremely high similarity to SampEn.

Fig. 17The Example 2’s RMS

Fig. 18The Example 2’s ADF

Fig. 19The SampEn of Example 2 deburred by smoothing spline with the three smoothing parameters

a) The smoothing parameter equals to 1e-3

b) The smoothing parameter equals to 1e-4

c) The smoothing parameter equals to 1e-5