# 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  
Through the above analysis and calculation, a multi-transform domain combined feature set can be obtained, which can reflect the grinding processing state from many aspects. However, using these features directly as feature vectors may lead to a large amount of data, which makes it difficult to establish a mapping relationship between feature vectors and the grinding surface roughness of bearing rings. At the same time, these features may have redundant information, which will not only increase the number of parameters in model training but also affect the accuracy and training efficiency of the model. In order to solve this issue, this research employs principal component analysis (PCA) to synthesize the information of different signal sources, and it integrates the multi-transform domain features of vibration signal and acoustic emission signal extraction to explore potential data structures. The core idea of principal component analysis is dimensionality reduction, which reduces the dimensionality of the original feature space by transforming the original feature into a few comprehensive indicators (i.e., principal components), effectively eliminating the correlation between features and improving the learning efficiency and accuracy of the model. Principal component analysis can find the main direction of change in the dataset and eliminate redundant information in the data by keeping the main components and discarding the minor components. The feature set after dimensionality reduction is more representative and can better explain changes in the original data. Dimensionality reduction through principal component analysis can reduce the dimensionality of the dataset, thereby simplifying the model, improving training speed and generalization ability while reducing the risk of overfitting. The specific calculation steps for data fusion in principal component analysis are as follows:  
Step 1: Perform dimensionless processing on the signal feature set. Different features may have different dimensions and scales; for example, one feature may have a value range of tens to hundreds, while another feature may have a value of between 0 and 1. If no dimensionless processing is carried out, these scale inconsistencies will lead to unreasonable weight allocation in PCA, which will affect the extraction results of the PCA. Common non-dimensional processing methods include min-max normalization, z-score normalization, and normalization regularization. Among them, the z-score standardization method aims to standardize the original dataset into a dataset with a mean of 0 and a variance of 1 that is close to the standard normal distribution, thus eliminating the dimensional influence between the data. This makes the importance of each feature more equal to the impact of the model, converting the values of different features into values with the same scale so they can be compared and analyzed on the same scale. In order to ensure that the data of each dimension can play the same role in the distance calculation and avoid the significant impact of data of different dimensions on distance calculation, this study adopts the z-score standardization method to carry out dimensionless processing on the combined feature set. The main formula for z-score standardization is as follows:  
\[z_{ij}=\frac{x_{ij}-\overline{x_{j}}}{s_{j}} \tag{22}\]  
where \(i=1\), \(2\), \(3\), \(\ldots\), \(n\); \(j=1\), \(2\), \(3\), \(\ldots\), \(p\); \(\overline{x_{j}}\) denotes the mean of feature \(j\), and \(S_{j}\) denotes the standard deviation of the feature number, where:  
\[\overline{x_{j}}=\frac{1}{n}\sum_{i=1}^{n}x_{ij} \tag{23}\]\[s_{j}=\sqrt{\frac{1}{n-1}\sum\limits_{i=1}^{n}\left(x_{ij}-\overline{x_{j}}\right)^ {2}} \tag{24}\]  
Step 2: Solve the principal component and its variance contribution rate and cumulative contribution rate. The raw data matrix can represent \(X_{n\times p}\), where \(n\) represents the number of samples contained and \(p\) represents the feature number.