## 2 FE Guided-VMD
### Fluctuation Entropy  
Takens has proved that if \(m\geq 2D+1\) is satisfied, where \(D\) is the dimension of the manifold of the attractor, the phase space reconstruction can ensure that the internal geometric structure of the prime mover corresponding to the time sequence remains unchanged [27]. We know that the local attenuation fluctuation of impulse signals is different from that of harmonic signals or white noise. The greater the fault degree is, the stronger the fluctuation is. The number of data points deviating from the mean value in the phase space is the specific standard of attenuation fluctuation. In view of this, we construct FE to quantitatively evaluate the decomposed modes. The calculation process of FE is briefly introduced as follows.  
For any arbitrary time series \(\{x(n),n=1,2,\ldots N\}\), according to the phase space reconstruction theorem, \(m\)-dimensional embedding matrix setting \(J=N-(m-1)\varsigma\) with time delay \(\varsigma\) is defined as:  
\[X=\begin{bmatrix}x(1)&x(1+\varsigma)&\ldots&x(1+(m-1)\varsigma)\\ x(2)&x(2+\varsigma)&\ldots&x(2+(m-1)\varsigma)\\ x(j)&x(j+\varsigma)&\ldots&x(j+(m-1)\varsigma)\\ \vdots&\vdots&&\vdots\\ x(J)&x(J+\varsigma)&\ldots&x(J+(m-1)\varsigma)\end{bmatrix} \tag{8}\]  
Then, for each subsequence \(\{x(j)\quad x(j+\varsigma)\quad\ldots\quad x(j+(m-1)\varsigma)\}\) of the reconstruction matrix, the mean values \(meanx(j)\) are calculated corresponding to each row in the matrix \(X\).  
\[meanx(j)=mean\big{[}x(j)\quad x(j+\varsigma)\quad\ldots\quad x(j+(m-1) \varsigma)\big{]},j=1,2,\ldots J \tag{9}\]  
We perform the following binary operations on the elements of each row, and the binary digits are determined as:  
\[S_{J,h}=\begin{cases}1,x(j)>meanx(j)\\ 0,x(j)\leq meanx(j)\end{cases} \tag{10}\]  
Thus, the original phase space \(X\) is transformed into binary phase space \(\overline{X}\). After binarization, we sum elements of each row in \(\overline{X}\) and perform histogram processing to enhance the distribution by the following expression:  
\[P_{h}=hist(sum(S_{J,h})) \tag{11}\]  
Consequently, the FE of a time series can be defined according to [PERSON]'s entropy as follows:  
\[H_{P,h}=-\sum P_{h}\ln(P_{h}) \tag{12}\]  
Meanwhile, if \(x(j)<meanx(j)\), we naturally get the following results:  
\[S_{J,l}=\begin{cases}1,x(j)<meanx(j)\\ 0,x(j)\geq meanx(j)\end{cases} \tag{13}\]  
and  
\[H_{P,l}=-\sum P_{l}\ln(P_{l}) \tag{14}\]For the sake of accuracy, the average form is usually used as follows:  
\[H_{P}=(H_{P,h}+H_{P,l})/2 \tag{15}\]  
It can be seen from the above calculation process that \(m\) and \(\varsigma\) are the two parameters that need to be set for FE. According to the vibration character of rolling bearing with local fault, we suggest that the value of \(m\) should be within the range of 6-9 so as to acquire better robustness. Actually, FE not only considers the number of amplitude deviations from the mean value but also incorporates the data distribution characteristics, so it can effectively evaluate the impulse fluctuation and fault information. Compared with the permutation entropy (PE), PE considers only the order of the amplitude values and hence some information regarding the amplitudes may be discarded [28]. The case proof will be provided in Section 5.1.