# Rotate Vector Reducer Fault Diagnosis Model Based on EEMD-MPA-KELM
## 3 Theoretical Basis  
### Kelm  
Extreme Learning Machine (ELM) is a neural network algorithm composed of single hidden layer. Because the hidden layer of the input weight kernel is randomly biased and fixed in the network, compared with the traditional neural network, the parameter setting is lower, the learning speed is faster, and the generalization ability is stronger.  
Suppose that the input of the model's training set is \(N\)-dimensional \(X=\{x_{1},x_{2},\ldots,x_{n}\}\), the ideal output is \(I\) dimensional \(T=\{t_{1},t_{2},\ldots,t_{i}\}\), when the number of hidden nodes \(L\) and the excitation function \(G(x)\) are determined, the calculation formula of the actual output value of ELM network is:  
\[y_{i}=\sum_{i=1}^{L}\beta_{li}G(W_{nl}\cdot x_{n}+b_{l})\quad n=1,2,\ldots,N \tag{9}\]  
In formula, \(W_{nl}\) is the input weight between node \(n\) of ELM input layer and node \(l\) of hidden layer; \(b_{l}\) is the bias of the \(l\) th hidden layer node; \(\beta_{li}\) is the output weight of hidden layer node \(l\) and output layer node \(i\). \(G(x)\) is the activation function.  
When the input sample is trained by ELM network model, the output data \(Y=\{y_{1},y_{2},\ldots,y_{k}\}\) can be infinitely approximated to an ideal output \(T=\{t_{1},t_{2},\ldots,t_{k}\}\), there is a definite combination of \(W\), \(b\) and \(\beta\), such that \(Y=T\), then the above formula can be converted into a matrix form:  
\[H\cdot\beta=T \tag{10}\]  
In formula, \(H\) is the output matrix of the hidden layer; \(\beta\) is the weight matrix of the output layer; \(T\) is the expected output matrix.  
Using the least square method to solve \(\beta\), we can obtain:  
\[\beta=H^{+}T=H^{T}\Big{(}HH^{T}\Big{)}^{-1}T \tag{11}\]  
In formula, \(\mathbf{H^{+}}\) is the generalized inverse matrix of matrix \(\mathbf{H}\).  
Due to the randomness of the setting of input layer weight and hidden layer bias in ELM, the state of the model is extremely unstable. Therefore, [PERSON] and others from Nanyang Technological University introduced kernel function idea from the Support Vector Machine (SVM) into ELM and proposed Kernel Extreme Learning Machine (KELM). The algorithm not only has the same excellent running speed as the ELM neural network, but also has more stable performance and generalization ability as SVM. Kernel function is used to map the input space to higher dimensional space to enhance the stability of the model. That is, kernel function matrix \(\mathbf{\Omega_{ELM}}\) is used to replace \(\mathbf{E}\) hidden layer random matrix \(\mathbf{HH^{T}}\). Based on Mercer condition definition, we can obtain:  
\[\begin{cases}\Omega_{ELM}=HH^{T}\\ \Omega_{ij}=h(x_{i})\cdot h\big{(}x_{j}\big{)}=K\big{(}x_{i},x_{j}\big{)}\end{cases} \tag{12}\]  
RBF kernel function is adopted in this paper:  
\[K\big{(}x_{i},x_{j}\big{)}=exp\Bigg{(}-\frac{\left\|x_{i},x_{j}\right\|^{2}}{ 2\sigma^{2}}\Bigg{)} \tag{13}\]  
In formula, \(\sigma\) is the width parameter of RBF kernel function.  
The regularization coefficient \(C\) and the identity matrix \(\mathbf{I}\) are introduced into the ELM neural network random matrix \(\mathbf{HH^{T}}\), and the evaluation formula of \(\beta^{\prime}\) is as follows:  
\[\beta^{{}^{\prime}}=H^{T}\bigg{(}\frac{I}{C}+HH^{T}\bigg{)}^{-1}T \tag{14}\]  
Based on the above formula, the output function of KELM neural network can be written:  
\[f(x)=\begin{bmatrix}K(x,x_{1})\\ \vdots\\ K(x,x_{N})\end{bmatrix}\bigg{(}\Omega_{ELM}+\frac{I}{C}\bigg{)}^{-1}T \tag{15}\]