## 1 Introduction
However, the selection of basisfunction is not uncontrolled because there are limited basis functions in the library. Furthermore, any inappropriate wavelet basis function employed in the special engineering application will directly lower the accuracy of the condition monitoring and fault diagnosis. Therefore, it is a vital step to select an appropriate wavelet basis function for vibration data processing. In fact, no fixed basis function that is related to the special vibration data can match a data feature entirely in all applications. Moreover, no scalar wavelet in the wavelet basis function library can possess orthogonality, symmetry, compact support and higher order of vanishing moments simultaneously. Unfortunately, these properties are significant for describing vibration data comprehensively and precisely. In addition, multiwavelet transform only pays attention to multi-resolution analysis in low frequency band, which may omit useful condition feature information. In order to reduce these limitations of the scalar wavelet transform, ensemble multiwavelet analysis method is developed. Multiwavelet transform, as the newer development of the traditional wavelet transform theory, was firstly developed by [PERSON] _et al._[14]. Multiwavelet not only possesses the ability of multi-resolution analysis but also simultaneously grasps such important properties as orthogonality, symmetry, compact support and higher order of vanishing moments that traditional scalar wavelet basis function does not [15]. Since possessing multiple wavelet basis functions, multiwavelet transform does well in identifying signal feature with multiple kinds of shapes for the fault detection. Moreover, ensemble multiwavelet is constructed to avoid a mistake in the selection of the basis functions in a special application. Lifting scheme [16] is a useful method for the wavelet basis function construction developed by Sweldens in recent years. Due to not relying on the Fourier transform and the whole construction step derived in the time domain, it provides much more freedom and flexibility for the construction of biorthogonal wavelet. The performance of any existing wavelet basis function from the basis function library can be enhanced by the lifting scheme according to the actual requirement in the special engineering application, which brings about the possibility of constructing a customized wavelet. [PERSON] _et al._[13] developed customized lifting multiwavelet based on the common lifting scheme for the condition identification of mechanical equipment. However, the mentioned method has the severe shortcoming of low adaptivity on the developed multiwavelet basis, which limits the ability to customize lifting multiwavelet on signal feature extraction. [PERSON] _et al._[17] and [PERSON] _et al._[18] developed customized lifting multiwavelet based on multiwavelet symmetric lifting scheme for the fault detection of mechanical equipment. Ensemble multiwavelet transform is performed to achieve the multi-resolution analysis in the entire signal frequency band, which is beneficial with describing the vibration data precisely and comprehensively. The remaining important problem to solve is how to reflect and evaluate the current state of mechanical equipment by a simple rule based on the vast customized ensemble multiwavelet transform coefficients. Information entropy was first proposed by [PERSON] in 1948 as a useful indicator to reflect and evaluate the uncertain degree of a system [19]. Based on the information entropy theory, the largest entropy value means the most uncertain probability distribution and the smallest entropy value equals the most certain probability distribution. From then on, more and more scholars have joined in the study of information entropy for special application. [PERSON] _et al._[20] proposed the contents of wavelet entropy and relative wavelet entropy to detect structural damage.