# Fault Diagnosis of Roller Bearings Based on a Wavelet Neural Network and Manifold Learning
## 3 Materials and Methods
### Wavelet Neural Network  
The wavelet neural network (WNN) is a forward network, which uses the wavelet basis function as the neuron activation function [22]. It adaptively adjusts the scaling and translation factors of the wavelet function, network connection weights, and the approximation function in batch mode. The fault diagnosis of the bearing based on the wavelet neural network is shown in Figure 3.  
where \(\left\{x_{1},\ x_{2},\ \dots,\ x_{N}\right\}\) is the bearing fault feature, which can obtain by the ILE method proposed. \(\left\{y_{1}y_{2}y_{3}y_{4}\right\}\) is the bearing state,where \(y_{1}\) is the normal state; \(y_{2}\) the ball failure; \(y_{3}\) the inner ring failure; and \(y_{4}\) is the outer ring failure four. \"1\" is in this state, while \"0\" is not in this state. \(\Psi\) is the wavelet base; its Fourier transform \(\hat{\Psi}\left(\omega\right)\) satisfies the admissible condition:  
\[C_{\Psi}\left(\omega\right)=\int_{-\infty}^{+\infty}\left|\omega\right|^{-1} \left|\hat{\Psi}\left(\omega\right)\right|^{2}d\omega<\infty \tag{7}\]  
By scale scaling and translating for wavelet base functions \(\Psi\left(x\right)\), the family of wavelet functions is obtained:  
\[\Psi_{a,b}\left(t\right)=\frac{1}{\sqrt{\left|a\right|}}\Psi\left(\frac{t-b}{ a}\right) \tag{8}\]  
Figure 3: Fault diagnosis model based on the wavelet neural network. Figure 2: Diagram of the steps in the method. where \(a,b\in R,\ a\
eq 0.\ a\) and \(b\) are scaled translation scale factors, respectively. The wavelet transform is:  
\[\left(W_{\Psi}f\right)\left(a,b\right)=\int_{R}f\left(t\right)\Psi_{a,b}^{*} \left(t\right)dt \tag{9}\]  
Discretizing \(a\) and \(b\), when \(a=a_{0}^{m},\ b=nb_{0}a_{0}^{m},\ n,m\in Z\), then:  
\[\Psi_{n,m}\left(t\right)=\left|a_{0}\right|^{-\frac{m}{2}}\Psi\left(a_{0}^{-m} t-nb_{0}\right),\ n,m\in Z \tag{10}\]  
Equation (10) is the discrete wavelet transform. The discrete wavelet transform is:  
\[\left(W_{\Psi}f\right)\left(a,b\right)=\left|a_{0}\right|^{-\frac{m}{2}}\int_ {-\infty}^{+\infty}f\left(t\right)\Psi^{*}\left(a_{0}^{-m}t-nb_{0}\right)dt \tag{11}\]  
Select the scaling parameters \(a>0,\ b>0\) to ensure that the wavelet system \(\left\{\Psi_{n,m}\left(t\right)\big{|}\left(n,m\right)\in Z^{2}\right\}\) satisfies the \(L^{2}\left(R\right)\) frame condition:  
\[A\|f\|^{2}\leq\sum_{\left(m,n\right)\in Z^{2}}|\left\langle f,\Psi_{n,m} \right\rangle|^{2}\leq B\|f\|^{2} \tag{12}\]  
In the formula, \(A,\ B\) are the frame boundaries. If \(\widetilde{\Psi}_{n,m},\ \Psi_{n,m}\left(t\right)\) are the dual function, then for any function \(f\left(t\right)\in L^{2}\left(R\right)\), \(f\left(t\right)\) can be expressed as a form of wavelet series:  
\[f\left(t\right)=\sum_{\left(m,n\right)\in Z^{2}}\langle f,\widetilde{\Psi}_{n,m}\rangle\Psi_{n,m}\left(t\right) \tag{13}\]  
The function can be expanded by a wavelet basis function and an approximate function:  
\[C_{\Psi}\left(\omega\right)=\int_{-\infty}^{+\infty}\left|\omega\right|^{-1} \left|\hat{\Psi}\left(\omega\right)\right|^{2}d\omega<\infty \tag{7}\]0  
The above equation can be realized by a neural network with a hidden layer, and by adjusting the weights \(a_{j}\) and \(b_{j}\) to an approximate function. In the formula, \(\Psi_{i}\left(t\right)\) is the wavelet basis function \(\left(i=1,\ 2,\ldots,n\right)\); \(y_{k}\left(k=1,\ 2,\ldots,n\right)\) is the output of the network; and \(w_{ij},\ \ w_{jk}\) is the connection weight between the input layer/the hidden layer and between the hidden layer/the output layer.