## 3 Ensemble Multiwavelet Analysis Method
### Ensemble Multiwavelet Analysis Method  
Customized construction of multiwavelet basis function can enhance the ability of multiwavelet on the vibration date feature extraction. However, multiwavelet transform only pays attention to multi-resolution analysis in low frequency band, which may omit the useful abnormal state feature information. The decomposition step of the multiwavelet transform and the corresponding frequency bands are shown in Figure 2. In order to overcome this disadvantage, ensemble multiwavelet transform is performed based on the customized multiwavelet filter-bank. Then, the multi-resolution analysis can be realized in both low frequency band and high frequency band. Thus, ensemble multiwavelet analysis method is a more precise and comprehensive vibration data processing tool than the multiwavelet transform [18,32,33,34]. Let \(l\) be the transform level and \(\left(x_{{}_{I,i}}\right)\) be the ensemble multiwavelet transform coefficients at the \(l\) decomposition in the \(i\) frequency band. \(f_{{}_{N}}\) means the upper frequency limit of the signal \(x(t)\). Then, the decomposition step of the ensemble multiwavelet transform and the corresponding frequency bands are shown in Figure 3.  
The ensemble multiwavelet transform also obeys the energy conservation principle due to its biorthogonal basis. After the \(l\) transform, the \(2^{l}\) frequency bands can be obtained and each decomposed frequency band owns the same bandwidth. Let \(x_{{}_{I,i}}\big{(}k\big{)}\) be the ensemble multiwavelet transform coefficients at the \(l\) decomposition in the \(i\) frequency band, then its energy \(E_{{}_{I,i}}\) and relative energy \(\tilde{E}_{{}_{I,i}}\) are, respectively, calculated as follows:  
\[E_{{}_{I,i}} = \frac{1}{n-1}\sum_{{}_{k=l}}^{n}\left(x_{{}_{I,i}}(k)\right)^{2} \text{, \ }i=1,2,\cdots,2^{l}\text{, \ }k=1,2,\cdots,n\text{, \ }n\in Z \tag{17}\] \[\tilde{E}_{{}_{I,i}} = E_{{}_{\cup}}\left(\sum_{i=1}^{x^{2}}E_{{}_{I,i}}\right)^{-1} \tag{18}\]  
Obviously, \(\sum_{i=1}^{x^{2}}E_{{}_{I,i}}=1\), the summation of total relative energy equals to 1. Then, the definition of normalized ensemble multiwavelet information entropy \(Ent\) can be described as follows:  
\[Ent=-\sum_{i=1}^{x^{2}}\tilde{E}_{{}_{I,i}}\log_{2^{l}}\tilde{E}_{{}_{I,i}} \tag{19}\]  
Figure 3: **(a)** The decomposition step of ensemble multiwavelet transform and **(b)** the corresponding frequency bands. Figure 2: **(a)** The decomposition step of multiwavelet transform and **(b)** the corresponding frequency bands. In Equation (19), the base of logarithm is \(\,2^{{}^{\prime}}\,.\) The value \(Ent\) belongs to [0, 1] in this definition. We can find that if every frequency band has the same relative energy (\(2^{{}^{\prime}}\,\))\({}^{-1}\) (like equal probability distribution), then \(Ent=1\); if only one of the \(2^{{}^{\prime}}\,\) frequency bands concentrates the whole energy of vibration signal, its relative energy would equal to 1 (like the most certain probability distribution), and then \(Ent=0\). Thus, we can regard \(Ent\) as normalized information entropy. In this paper, the normalized ensemble multiwavelet transform information entropy is developed and introduced for assembly looseness fault detection in aero-engine rotors. The procedure of the developed method, called normalized ensemble multiwavelet transform information entropy, can be concluded in the following flow chart, as displayed in Figure 4. Meanwhile, the process of the normalized ensemble multiwavelet transform information entropy for the assembly looseness fault detection of aero-engine rotors can be summarized as follows:  
1. Choose the original multiwavelet with the vanishing moment \(p\,.\) Then, with the given vanishing moment \(p\,^{{}^{\prime}}\,\), the translations of \(\Phi\) and \(\Psi\,\) are symmetrically selected;
2.