# Surface Roughness Prediction of Bearing Ring Precision Grinding Based on Feature Extraction
## 3 Grinding Surface Roughness Feature Extraction and Fusion
### Multi-Transformation Domain Feature Fusion
The covariance matrix is calculated, and the covariance matrix is obtained. Through eigenvalue decomposition, the eigenvalues of the covariance matrix and corresponding eigenvectors are obtained, and the eigenvalues are sorted from small to large. By obtaining \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\ldots\), \(\lambda_{\text{P}}\) and the corresponding eigenvector \(T_{1}\), \(T_{2}\), \(T_{3}\), \(\ldots\), \(T_{\text{P}}\), the decomposed principal component \(Y_{i}\) can be expressed as follows:  
\[Y_{i}=X\cdot T_{i},1\leq i\leq p \tag{25}\]  
Then, the variance contribution rate and the cumulative contribution rate of the eigenvalues are calculated. The variance contribution rate of principal component analysis refers to the proportion of total variance explained by each principal component, which reflects the contribution degree of each principal component to the degree of data variation. The specific formula is as follows:  
\[\varphi_{k}=\frac{\lambda_{k}}{\sum\limits_{i=1}^{p}\lambda_{i}} \tag{26}\]  
where \(\lambda_{k}\) denotes the eigenvalue corresponding to the \(k\)th principal component. Under normal circumstances, the cumulative contribution rate of the first m principal components of the sample can be calculated using the following formula:  
\[\Psi_{m}=\frac{\sum\limits_{k=1}^{m}\lambda_{k}}{\sum\limits_{k=1}^{p}\lambda _{k}}=\sum\limits_{k=1}^{m}\varphi_{k} \tag{27}\]  
If the variance contribution rate of the principal components is larger, this means that more information of the original data will be saved. Therefore, this study takes the calculated variance contribution rate and cumulative contribution rate as the basis for selecting the number of retained principal components to achieve the purpose of dimensionality reduction. Step 3: Solve the score coefficient. The score coefficient of the principal component in a principal component analysis is not only used to find the principal component but also to deeply understand the meaning of each factor for problem analysis. It is used to calculate the coefficient of the projected value of the original data on the principal component and to quantify the projection of each sample in the direction of the principal component. The score coefficient reflects the importance and relative position of the samples in the principal component direction, which is helpful to understand the distribution and mutual relationship of the samples in the principal component space. By analyzing the score coefficients, it is possible to obtain a clearer picture of how each sample performs in the principal component direction, thus better interpreting the meaning of the principal component and the structure of the data. The original data matrix is expressed as \(X_{n\times p}\), where \(n\) represents the number of samples and \(p\) represents the feature number. When the first \(m\) principal components are selected for dimensionality reduction, the dimension of the projection matrix \(V\) is \(p\times k\)Each column of the projection matrix \(\mathbf{V}\) is an eigenvector of a principal component, and the eigenvector contains the weight of each original feature on the principal component. Then, the calculation formula of the score coefficient matrix \(\mathbf{T}\) is as follows:  
\[\mathbf{T}=\mathbf{X}\mathbf{V} \tag{28}\]  
where the dimension of \(\mathbf{T}\) is \(n\times k\), and each row corresponds to the projected value of a sample on the selected principal component. Grinding parameters are the key control factors in the grinding process, which directly affect the removal of materials, the distribution of grinding heat, and the size of grinding force. Incorporating these parameters into the prediction model can more accurately capture the dynamic changes in the grinding process, thus improving the prediction accuracy of the model.