## 3 Ensemble Multiwavelet Analysis Method
### Symmetric Lifting Scheme  
The vital step in the ensemble multiwavelet transform is the customized construction of the multiwavelet basis function. In this section, the customized multiwavelet basis function is generated on the basis of the symmetric lifting scheme. Symmetry could guarantee the filter has linear phase or generalized linear phase, which is beneficial for perfect reconstruction. However, the symmetry is not realized in the traditional multiwavelet lifting scheme. To realize a symmetric multiwavelet lifting scheme, the vital factor is the appropriate selection of translation quantity \(k\) of multiwavelet basis functions [18]. Taking \(\Psi_{{}_{1}}\), for example, and supposing functions \(\omega_{{}_{i}}\) is symmetric or anti-symmetric at points \(a_{{}_{\omega_{i}}}\), respectively. The selection of translation quantity \(k\) should meet the following Equation:  
\[a_{{}_{\Psi_{1}}}-(a_{{}_{\omega_{i}}}+k_{{}_{\omega_{i},1}})=(a_{{}_{\omega_{ i}}}+k_{{}_{\Psi_{i},2}})-a_{{}_{\Psi_{1}}} \tag{12}\]  
Note that \(i\) = 1, 2, \(j\) = 1, 2, \(\cdots\), \(k\) \(\in Z\). The symmetry of the original multiwavelet functions and multi-scaling functions can be described as:  
\[B_{{}_{\mathrm{a}}}=\pm 1 \tag{13}\]  
where \(+1\) notes symmetry and \(-1\) notes anti-symmetry. Taking the symmetry conditions into account, Equation (11) turns into the following Equation:  
\[ \begin{bmatrix}\int\omega_{{}_{1}}(x+k_{{}_{\alpha_{1}}})x^{{}^{\prime}}dx& \int\omega_{{}_{1}}(x+k_{{}_{\alpha_{2}}})x^{{}^{\prime}}dx&\cdots\\ \int\omega_{{}_{1}}(x+k_{{}_{\alpha_{1}}})x^{{}^{\prime}+1}dx&\int\omega_{{}_{ 1}}(x+k_{{}_{\alpha_{2}}})x^{{}^{\prime}+1}dx&\cdots\\ \vdots&\vdots&\vdots\\ \int\omega_{{}_{1}}(x+k_{{}_{\alpha_{1}}})x^{{}^{\prime}-1}dx&\int\omega_{{}_ {1}}(x+k_{{}_{\alpha_{2}}})x^{{}^{\prime}-1}dx&\cdots\\ \end{bmatrix}\begin{bmatrix}1&&0\\ B_{{}_{\mathrm{a}}}B_{{}_{\mathrm{a}}}&&\\ \ddots&\ddots\\ 0&&\end{bmatrix}\begin{bmatrix}c_{{}_{1}}\\ c_{{}_{2}}\\ c_{{}_{k}}\\ \end{bmatrix} = \begin{bmatrix}-\int\omega_{{}_{0}}(x)x^{{}^{\prime}}dx\\ -\int\omega_{{}_{0}}(x)x^{{}^{\prime}+1}dx\\ \vdots\\ -\int\omega_{{}_{0}}(x)x^{{}^{\prime}-1}dx\\ \end{bmatrix}  \tag{14}\]The solutions of Equation (14) are the coefficients for the lifting operation \(\Psi_{1}\), while the lifting operation on \(\Psi_{2}\) is similar to \(\Psi_{1}\). Next, substitute the related lifting coefficients into the \"lifting coefficients equation\". Then, corresponding lifting matrices \(T\) and \(S\) can be obtained through \(Z\) transform. The direct presentation of the multiwavelet lifting scheme is displayed as follows:  
\[G_{new}(z)=T(z^{2})(G(z)+S(z^{2})H(z)) \tag{15}\]  
A new multiwavelet basis function with the symmetry property is successfully constructed based on the symmetric multiwavelet lifting scheme with the help of \(T(z)\) and \(S(z)\). In order to construct ensemble multiwavelet basis function with specified properties, the appropriate free parameters and optimization objective are needed to optimize this process. Equation (14) can be briefly described as \(MC=N\), where \(C=\left[c_{1},c_{2}\cdots,c_{k}\right]^{T}\), and matrix \(M\) means the related coefficient matrix of Equation (14). When the set of Equation (14) is underdetermined, there are \(N=(p\,\text{'}-p)-Rank(M)\) free parameters, which mean that ensemble symmetric lifting scheme can be conducted by the optimization of the free parameters. As described in the Section 1, information entropy is a useful indicator to evaluate the uncertain degree of a system. Thus, choose the minimum entropy principle as an evaluation objective to guide the customized lifting multiwavelet basis function construction. The mathematical definition of the information entropy is displayed as follows [19]:  
\[E=-\sum_{i}^{n}p_{i}\ln p_{i} \tag{16}\]  
where \((p_{1},p_{2},\cdots,p_{n})\) reflect the probability density functions of the processed data amplitude.