# Study on EEMD-Based KICA and Its Application in Fault-Feature Extraction of Rotating Machinery
## Theory
### Kica  
The mathematical description of BSS is: \(\mathbf{X}=\mathbf{A}\mathbf{S}+\mathbf{N}\), where \(\mathbf{S}=[s_{1}(t),s_{2}(t),\cdots,s_{N}(t)]\) is a vector of N source signals, \(\mathbf{X}=[x_{1}(t),x_{2}(t),\cdots,x_{M}(t)]\) is a vector of M mixed signals, the mixing matrix \(\mathbf{A}\) is an M \(\times\) N dimensional matrix, and \(\mathbf{N}(t)\) is a noise vector. The meaning is that when the mixing matrix and source signals are unknown, we only determine the remove-mixing matrix \(\mathbf{W}\) based on the observed data, and the output \(\widehat{\mathbf{S}}=\mathbf{W}\mathbf{X}\) is the estimation of source signals. KICA is not a simple nucleation of independent component analysis (ICA), but a new ICA method. The idea of kernel technology is to use nonlinear mapping \(\phi:R^{m}\to R\), and map the nonlinear variable \(y^{i}\in R^{m}(i=1,2,\cdots,N)\) of original input space into a kernel feature space \(R\) to linearize it, and then analyse the mapped data in this feature space. Thus, the linear blind source separation in the space \(R\) is equivalent to nonlinear blind source separation in the original space. One of the important characteristics of this technology is that a kernel function can be used instead of an inner product between two vectors to realize nonlinear transformation without specific form being considered. The kernel function of the radial basis function (RBF) \(K(x_{i},y_{i})=\exp(-\parallel x_{i}-y_{i}\parallel/\delta^{2})\) is chosen here. The characteristic of KICA is to use a nonlinear function in reproducing kernel Hilbert space (RKHS) as a contrast function; the signal is mapped from low-dimensional space to the high-dimensional space, and the kernel method is used to search the minimum value of contrast function in this space. This function has a certain correlation with mutual information and has better mathematical properties. Moreover, this function space is suitable for various source signals. Therefore, compared with traditional ICA, KICA has better flexibility and robustness. The KICA's contrast function is constructed by measuring the correlation of a set of random variables directly. Let \(F\) be a real vector function space, for simplicity, \(s_{1}\) and \(s_{2}\) are two unary random variables of space \(F\). Define the correlation coefficient \(\rho_{F}\) of \(s_{1}\) and \(s_{2}\) as the maximum correlation coefficient between random variables \(f_{1}(s_{1})\) and \(f_{2}(s_{2})\),  
\[\rho_{F}=\max_{f_{1},f_{2}\in F}corr(f_{1}(s_{1}),f_{2}(s_{2}))=\max_{f_{1},f_ {2}\in F}\frac{\mathrm{cov}(f_{1}(s_{1}),f_{2}(s_{2}))}{\left(\mathrm{var}f_{ 1}(s_{1})\right)^{1/2}\left(\mathrm{var}f_{2}(s_{2})\right)^{1/2}}. \tag{10}\]  
\(\rho_{F}\) is also called a contrast function between random variables. Obviously, if \(s_{1}\) and \(s_{2}\) are independent, then \(\rho_{F}=0\). If space \(F\) is large enough and \(\rho_{F}=0\), then \(s_{1}\) and \(s_{2}\) are also independent of each other. The procedure of KICA is as follows:  
Input: data vector \(x_{1},x_{2},\cdots,x_{n}\) and kernel function \(k(x,s)\). Step 1: Whiten data vector \(x_{1},x_{2},\cdots,x_{n}\). Step 2: Use Cholesky decomposition to find the Gram matrix \((K_{1},K_{2},\cdots,K_{m})\) of original independent data, where \(s_{i}=\mathbf{W}x_{i}\) (\(\mathbf{W}\)is the remove-mixing matrix). Step 3: Define \(\lambda_{H}\) to be the maximum eigenvalue of Equation (11). \[\left( \begin{array}{ccccc}K_{1}K_{1}&K_{1}K_{2}&\cdots&K_{1}K_{m}\\ K_{2}K_{1}&K_{2}K_{2}&\cdots&K_{2}K_{m}\\ \vdots&\vdots&\ddots&\vdots\\ K_{m}K_{1}&K_{m}K_{2}&\cdots&K_{m}K_{m}\\ \end{array} \right)\left( \begin{array}{c}\alpha_{1}\\ \alpha_{2}\\ \vdots\\ \alpha_{m}\\ \end{array} \right)=\lambda\left( \begin{array}{ccccc}K_{1}K_{1}&0&\cdots& 0\\ 0&K_{2}K_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&K_{m}K_{m}\\ \end{array} \right)\left( \begin{array}{c}\alpha_{1}\\ \alpha_{2}\\ \vdots\\ \alpha_{m}\\ \end{array} \right) \tag{11}\]  
Step 4: Minimize \(M_{\lambda_{H}}=-0.5\log\lambda_{H}\) for \(\mathbf{W}\).