# Nonlinear Blind Source Separation and Fault Feature Extraction Method for Mining Machine Diagnosis
## 2 The Proposed Method
### Wavelet-Nonlinear ICA Model  
The vibration signal of a mining machine collected by the accelerator is generally a nonlinear mixture of various vibration sources, such as machine base movement, gear meshing, shaft rotating and connector vibration sources. In addition, the fault source is submerged in strong background noise. Therefore, the fault feature extraction is significantly influenced by the noise and disturbance components, which may make incorrect fault identification. ICA is widely applied to information fusion of multi-channel sensor signals to extract the fault source [1]. However, the traditional ICA \begin{table}
\begin{tabular}{l l} \hline \hline
**Step:** & **Implementation** \\ \hline
1. & Training: Establish a fault detection model \\
1. & Measure multi-channel vibration signals of the mining machine \\
1. & Train a radial basis function neural network (RBFNN) classifier using the features \\
2. & Testing: Validate the trained model using test datasets \\
2. & Extract fault features using the wavelet-nonlinear ICA and WPD preprocessing \\
2. & Identify the fault patterns via the trained RBFNN \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detailed implementation of the proposed method. Figure 1: Process of fault diagnosis principle of mining machines. WPD, wavelet package decomposition; RBF, radial basis function; ICA, independent component analysis. algorithm is based on the linear mixing assumption, which assumes a linear mixing processing of different source signals [13]. Equation (1) shows the traditional ICA model:  
\[\textbf{X}=\textbf{AS}, \tag{1}\]  
where, \(\textbf{X}=[\textbf{x}_{1},\,\textbf{x}_{2},\,\ldots,\,\textbf{x}_{m}]^{\text{T}}\) denotes the measurements of \(m\) sensors, \(\textbf{S}=[\textbf{s}_{1},\,\textbf{s}_{2},\,\ldots,\,\textbf{s}_{n}]^{\text{T}}\) denotes \(n\) unknown sources, and **A** is the mixing matrix. As can be seen, the unknown source matrix **S** is a linear transformation of the measurement matrix **X** via **A**. ICA aims to find the inverse matrix of **A** to recover the sources by \(\textbf{S}=\textbf{W}\textbf{X}\), where **W** is the inverse of **A**. However, real vibration measurements do not satisfy the linear mixing assumption. In addition, Equation (1) does not consider the additive noise. The following nonlinear ICA model is developed to solve these issues:  
\[\left\{ \begin{array}{l}\textbf{X}=\textbf{AS}\\ \textbf{T}=f(\textbf{X})+v\\ \textbf{Z}=g(\textbf{T})\\ \textbf{Y}=\textbf{WZ}=\textbf{W}g(f(\textbf{X})+v)\approx\textbf{S}\end{array} \right., \tag{2}\]  
where, \(v\) is the additive noise, **T** is the observation matrix after the nonlinear mixing process, \(f(\cdot)\) is the nonlinear mixing function, **Z** is the observation matrix after nonlinear de-mixing process \(g(\cdot)\) is the nonlinear de-mixing function and **Y** is the separated source matrix. Unlike the linear ICA model in Equation (1), the nonlinear ICA model in Equation (2) aims to find \(f(\cdot)\),\(g(\cdot)\), **A** and **W**. Since \(f(\cdot)\) and **A** are unknown, one cannot directly find \(g(\cdot)\) and **W**. In order to obtain the approximation of \(f(\cdot)\) and **A** and their inverses, wavelet de-noising is used in this paper, and the maximum mutual information criterion based on the radial basis function neural network (RBFNN) is used to solve the de-mixing function and the separation matrix **W**. The separation process is shown in Figure 2.  
The estimation of de-mixing function and the separation matrix based on RBFNN is briefly introduced. RBFNN consists of an input layer, a hidden layer and an output layer. The essence is to convert the input variables from one space to another, thus achieving an infinite approximation of the objective function.