## 3 Proposed Method
### IMFE Design  
By evaluating the fuzziness and uncertainty of vibration signals, fuzzy entropy effectively extracts fault-related features. Compared to traditional fuzzy entropy algorithms, the multiscale fuzzy entropy (MFE) algorithm extracts multi-level signal features by calculating entropy values across different time scales, allowing potential fault patterns to be identified [26; 27]. However, as the scale factor of entropy increases, the length of the coarse-grained time series decreases significantly. This sudden reduction in sequence length causes sharp fluctuations in entropy values, complicating the extraction of key features. To address this issue, a novel time segmentation method is proposed in this article to replace traditional segmentation approaches, aiming to address the problem of high volatility in entropy values at large scale factors. Figure 2 illustrates the improved coarse-graining approach, unlike traditional coarse-graining that produces only a single subsequence of length 7, which preserves more local information essential for feature  
Figure 1: EMD of the signal. construction. The vibration signal sequence is denoted as \(x(t)\), and sequences \(\mu_{1}(t)\), \(\mu_{2}(t)\), and \(\mu_{3}(t)\) are defined based on a sliding window approach. Specifically, \(\mu_{1}(t)\) is obtained by averaging the first three elements of \(x(t)\), then shifting the sliding window one step to the right and averaging the subsequent three elements. Similarly, \(\mu_{2}(t)\) is derived by averaging the first three elements, followed by a two-step rightward shift of the window and averaging of the next three elements. \(\mu_{3}(t)\) is calculated by averaging the first three elements, shifting the window three steps to the right, and averaging the subsequent three elements. Assuming \(x(t)=[x_{1},x_{2},x_{3},\ldots,x_{n}]\), the following results are obtained: \(\mu_{1}(t)=\left[\frac{x_{1}+x_{2}+x_{3}}{3}\frac{x_{2}+x_{3}+x_{4}}{3},\frac {x_{3}+x_{4}+x_{5}}{3}\ldots\right]\), \(\mu_{2}(t)=\left[\frac{x_{1}+x_{2}+x_{3}}{3},\frac{x_{3}+x_{4}+x_{5}}{3},\frac {x_{5}+x_{6}+x_{7}}{3},\ldots\right]\), and \(\mu_{2}(t)=\left[\frac{x_{1}+x_{2}+x_{3}}{3},\frac{x_{4}+x_{5}+x_{6}}{3},\frac {x_{7}+x_{8}}{3},\ldots\right]\). For instance, when the sequence length is 21 and the scale factor is 3, the proposed offset coarse-graining method generates three sub-sequences with lengths of 19, 10, and 7, respectively, thereby reducing sudden entropy fluctuations and computational bias. This approach, combined with fuzzy entropy, forms an IMFE algorithm, with the algorithmic design shown in Figure 3. The main steps are described as follows:  
_Step 1_: The signal samples are input from the equipment, and set the coarse-graining scale factor sequence is set to \(scales=(scale_{1},scale_{2},\ldots,scale_{N})\), where \(1<scale_{i}<N\). _Step 2_: The offset factor is set to \(k=1\). When the scale factor is \(scale_{1}\), it increments from 1 until it equals \(scale_{1}\). At each distinct \(k\), coarse-graining is performed to obtain a signal, the fuzzy entropy of which is calculated. This process is repeated for each subsequent scale factor. _Step 3_: The fuzzy entropy is calculated [28]. Given an \(N\)-dimensional signal (\([u(1),u(2),\ldots,\)\(u(N)]\)), the phase space dimension (\(m(m\leq N-2)\)) and similarity tolerance (\(r\)) are defined. Then, the phase space is reconstructed as  
\[X(i)=[u\big{(}i\big{)},u\big{(}i+1\big{)},\ldots,u\big{(}i+m-1\big{)}]-u_{0} \big{(}i\big{)},i=1,2,\ldots,N-m+1 \tag{9}\]  
where \(u_{0}(i)=\frac{1}{m}\sum_{j=0}^{m-1}u(i+j)\). The fuzzy membership function is calculated by  
\[A_{ij}^{m}=\exp[-\ln(2)\left(d_{ij}^{m}/r\right)^{2}] \tag{10}\]  
where \(j=1,2,\ldots,N-m+1\) and \(j\
eq i\). The maximum absolute distance between window vectors \(X(i)\) and \(X(j)\) is obtained as  
\[d_{ij}^{m}=d[X(i),X(j)]=\max_{p=1,2,\ldots,m}(|u(i+p-1)-u_{0}(i)|-|u(j+p-1)-u _{0}(j)|) \tag{11}\]  
Figure 2: An improved coarse-graining method for a scale factor of 3.