## 4 Proposed Method
### Fault Diagnosis Method
#### 4.4.2 PCA-SOM Algorithm Flow  
To improve the operation rate, clustering accuracy, and data processing ability of the SOM neural network, a PCA-SOM neural network is proposed by combining a PCA neural network with a SOM neural network. At the same time, feature standardization is introduced into the SOM neural network to balance the influence of different feature scales, and then the SOM neural network is further optimized. The network structure is mainly composed of an input layer and output layer. The input layer accepts high-dimensional data and transforms them into two-dimensional data visual output through a competitive learning mechanism. For the operation parameters of the equipment to be evaluated, the SOM neural network can output a two-dimensional topology after the monitoring data are processed, assuming that the number of neurons in the output layer is L. The specific algorithm process is as follows:  
(1) Data feature standardization processing  
Conduct feature standardization processing on the input training data matrix \(\mathbf{D}_{0}\), and the processed data matrix is \(\mathbf{D}\):  
\[\mathbf{D}=\frac{\mathbf{D}_{0}-\text{mean}(\mathbf{D}_{0})}{\text{std}(\mathbf{D}_{0})} \tag{31}\]  
where: \(\mathbf{D}_{0}\) (Nxd) is the training data, where N is the number of training samples, d is the dimension of sample data, mean (\(\mathbf{D}_{0}\)) is the mean value of matrix data, and std (\(\mathbf{D}_{0}\)) is the standard deviation of matrix data. (2) Determine parameters  
Determine the number of nodes X and Y in the output layer:  
\[\text{X}=\text{Y}=\sqrt{5\sqrt{\text{N}}} \tag{32}\]  
(3) PCA dimensionality reduction  
(1) Calculate the covariance matrix  
According to the data matrix \(\mathbf{D}\) after feature standardization, the corresponding covariance matrix is calculated as \(\mathbf{S}\):  
\[\mathbf{D}= \begin{bmatrix}\mathbf{x}_{11}^{*}&\mathbf{x}_{12}^{*}&\ldots&\mathbf{x}_{1D}^{*} \\ \mathbf{x}_{21}^{*}&\mathbf{x}_{22}^{*}&\ldots&\mathbf{x}_{2D}^{*}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{x}_{N1}&\mathbf{x}_{N2}^{*}&\ldots&\mathbf{x}_{ND}\end{bmatrix}  \tag{33}\]  
\[\mathbf{S}=\frac{1}{\mathbf{M}-\mathbf{1}}\mathbf{X}^{\text{T}}\mathbf{X} \tag{34}\]  
(2) Calculate the eigenvalues of the covariance matrix and corresponding eigenvectors  
The eigenvalues of the covariance matrix \(\mathbf{S}\) are decomposed, and then the eigenvalues and corresponding eigenvectors are calculated. \[\mathbf{S}=\mathbf{P}\mathbf{A}\mathbf{P}^{\text{T}}=\text{Pdiag}(\lambda_{1},\lambda_{2}, \ldots,\lambda_{D})\mathbf{P}^{\text{T}} \tag{35}\]  
\[\text{s.t.}\ \lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{D}\geq 0\]  
where: \(\mathbf{\Lambda}\) represents the diagonal matrix; \(\mathbf{P}\) represents the eigenvector matrix composed of corresponding eigenvectors in descending order. The largest eigenvalue and corresponding eigenvector can represent the variance and direction of the first principal component, and the smallest eigenvalue and corresponding eigenvector can represent the variance and direction of the last principal component. (3) The original feature is projected onto the selected feature vector to obtain the new K-dimensional feature after dimensionality reduction, and \(\hat{x_{k}}\) is the real-time sample vector after dimensionality reduction. \[ \begin{bmatrix}\hat{x_{1}}\\ \hat{x_{2}}\\ \vdots\\ \hat{x_{k}}\end{bmatrix} = \begin{bmatrix}P_{1}^{\text{T}}(\hat{x_{11}}&\hat{x_{12 }}&\ldots&\hat{x_{1D}^{\text{T}}})^{\text{T}}\\ P_{2}^{\text{T}}(\hat{x_{21}}&\hat{x_{22}}&\cdots&\hat{x_{2D}^{\text{T}}})^{ \text{T}}\\ \vdots&\vdots&\ddots&\vdots\\ P_{k}^{\text{T}}(\hat{x_{N1}}&\hat{x_{N2}}&\cdots&\hat{x_{ND}^{\text{T}}})^{ \text{T}}\end{bmatrix}