# Planetary Gearbox Fault Diagnosis Based on ICEEMD-Time-Frequency Information Entropy and VPMCD
## 2 Feature Extraction Method Based on ICEEMD-Time-Frequency Information Entropy
### A Description of Improved Complementary Ensemble Empirical Mode Decomposition (ICEEMD)  
The vibration signal of planetary gearboxes, with multiple components and nonlinear, nonstationary, and strong time-varying characteristics, is complex and easily buried by noise. However, traditional fault diagnosis methods fail to effectively diagnose the faults of planetary gearboxes. It is essential to find out a suitable method to suppress interference components in order to obtain intrinsic features of the vibration signal. Comparing complementary ensemble empirical mode decomposition (CEEMD) with ensemble empirical mode decomposition (EEMD) and empirical mode decomposition (EMD) [18], CEEMD is an improved algorithm that reduces the mode mixing problem [19]. However, the reconstructed components of CEEMD contain noise components that cannot be eliminated. In order to eliminate the influence of noise on feature extraction, we used multi-scale permutation entropy with signal noise reduction for the first time. Multi-scale permutation entropy combines multi-scale entropy with permutation entropy to analyze sequence information more effectively [20; 21; 22], which can measure noise characteristics from multiple scales rather than a single scale. Using this method, the noise components can be accurately identified. In this paper, the improved complementary ensemble empirical mode decomposition algorithm (ICEEMD) was used to analyze the vibration signal of planetary gearboxes for the first time. The specific ICEEMD algorithm steps are shown in Figure 1.  
Figure 1: Algorithm flowchart of improved complementary ensemble empirical mode decomposition (ICEEMD). Step 1: Use CEEMD to decompose the original signal. 1. The original signal is \(S(t)\). Add white noise signals \(n_{i}(t)\) and \(-n_{i}(t)\) with the mean value of zero onto the original signal. Then, the signal after adding white noise is obtained: \[S_{i}^{+}(t)=S(t)+a_{i}n_{i}(t)\] (1) \[S_{i}^{-}(t)=S(t)-a_{i}n_{i}(t)\] (2) where \(a_{i}\) is the amplitude of the white noise, \(i=1,2,\ldots,Ne\), and \(Ne\) is the logarithm of the added white noise. 2. Signal \(S_{i}^{+}(t)\) and signal \(S_{i}^{-}(t)\) are decomposed by EMD to obtain \(IMF_{i1}^{+}\) and \(IMF_{i1}^{-}\). Then, the integrated average value of \(IMF_{i1}^{+}\) and \(IMF_{i1}^{-}\) are obtained by Equation (3). \[IMF_{1}=\frac{1}{2 Ne}\sum_{i=1}^{Ne}\left[IMF_{i1}^{+}+IMF_{i1}^{-}\right]\] (3)
3. Repeat the above steps until the termination condition of EMD is met. Step 2: Reduce noise by using the MPE algorithm. The basic idea of multi-scale permutation entropy is the coarse graining of the original signal [23]. The specific steps for calculating the multi-scale arrangement entropy of IMF are as follows:  
1. Coarse graining of IMF. \(IMF_{p}=\{x_{i}\,i=1,2,\ldots,N\}\),\(1\leq p\leq M\), \(p\) is the number of IMFs. Coarse graining sequence: \[y_{j}^{s}=\frac{1}{s}\sum_{i=(j-1)s+1}^{j\text{s}}x_{i},\ j=1,2,\ldots\frac{N }{s}\text{ and }j\text{ is an integer}.\] (4) where \(s\) is the scale factor. 2. Time reconstruction of coarse graining sequence \(y_{j}^{s}\). \[Y_{l}^{s}=\left\{y_{l}^{s},y_{l+\tau}^{s},\ldots,y_{l+(m-1)\tau}^{s}\right\}\] where \(m\) is embedding dimension, \(\tau\) is delay time, and \(l\) is a reconstructed component, where the expression for \(l\) is \(l=1,2,\ldots,N-(m-1)\tau\). Arrange the \(Y_{l}^{s}\) in ascending order and calculate the probability \(P_{r}\) of each sequence, where \(r=1,2,\ldots,R,R\leq m\)! Calculate the multi-scale permutation entropy of each coarse graining sequence. \[IMF_{p}(m)=-\sum_{r=1}^{R\sum_{r}^{r}}P_{r}ln\] (5)
3. Eliminate noise components. The multi-scale permutation entropy of each IMF component is calculated and compared with the set threshold value. Then, eliminate noise components larger than the threshold value. Step 3: Second CEEMD decomposition. Reconstruct signal. The reconstructed signal \(X(t)\) is decomposed by CEEMD to obtain new intrinsic modal components (IMFs). Refer to step 1 for detailed steps.