# Surface Roughness Prediction of Bearing Ring Precision Grinding Based on Feature Extraction
## 3 Grinding Surface Roughness Feature Extraction and Fusion
### Surface Roughness Prediction Model of Bearing Cylindrical Precision Grinding  
Support vector regression (SVR) is a machine learning method for regression analysis derived from the theory of support vector machines (SVMs). Similar to the SVM classification model, SVR fits the data by finding the optimal hyperplane. This non-probabilistic algorithm uses a kernel function to map the data into a high-dimensional space and find the optimal hyperplane in that space to maximize the interval between the training data and the hyperplane, thereby building a regression model. SVR is based on the structural risk minimization principle in statistical learning theory and aims to obtain the optimal solution with limited sample data, taking into account the empirical risk and confidence range. Compared with traditional regression methods, SVR pays more attention to the effective use of data and performs well with small sample sizes. This method uses a kernel technique to map raw data to high-dimensional space, which is helpful to deal with nonlinear relations. By using support vectors to define decision boundaries, SVR usually selects only a small number of sample points as support vectors, thereby obtaining sparse solutions, reducing the complexity of the model and effectively avoiding overfitting problems. Therefore, SVR is used to make regression predictions for the grinding feature set of the bearing ring. Suppose the current training dataset is \(\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\}\), where \(x_{i}\) is the eigenvector of the \(i\)-th sample and \(y_{i}\) is the corresponding target value. SVR maps the eigenvector \(\{x_{1},\)\(x_{2},\)\(\ldots,x_{n}\}\) nonlinearly in a high-dimensional space, followed by a linear fit in a high-dimensional space. Its basic model can be expressed as follows:  
\[y=f(x)=\omega^{\rm T}\Phi(x)+b \tag{29}\]  
where \(\omega\) denotes the corresponding weight vector after the eigenvector is mapped to the higher-dimensional space, \(\Phi(x)\) denotes the feature mapping function, and \(b\) denotes a bias term. In the calculation of SVR, the following formula is usually used as the loss function:  
\[E_{\varepsilon}(f(x)-y)=\left\{ \begin{matrix}0,&|f(x)-y|<\varepsilon\\ |f(x)-y|-\varepsilon,&|f(x)-y|\geq\varepsilon\end{matrix} \right. \tag{30}\]  
where \(\varepsilon\) denotes a tolerance parameter. If the fitted mathematical model is regarded as a curve in a high-dimensional space, a pipeline that envelopes this curve and the sample points is formed through the tolerance parameter; that is, the difference between the predicted value \(f(x)\) and the true value \(y\) is considered to lose 0 within a certain range. Finding the optimal hyperplane is equivalent to finding the maximum interval so that all sample points are within a pipe formed by two boundary lines. At this time, two relaxation variables \(\xi_{i}\geq 0\) and \(\xi^{*}\geq 0\) are introduced, where the former corresponds to the data point of \(y_{i}\geq f(x_{i})+\xi\) and the latter corresponds to the data point of \(y_{i}\geq f(x_{i})-\xi\). The optimal regression function can be obtained by finding the minimum extreme value of Formula (31):  
\[\frac{1}{2}\parallel\omega\parallel^{2}+C(\sum\limits_{i=1}^{i=1}\xi_{i}^{*}) \tag{31}\]  
where \(C\) denotes the value of the penalty factor, representing the degree of punishment for exceeding the error sample. Therefore, the issue of finding a regression function at this time translates into the following quadratic programming issue:  
\[\underset{\omega,b,\xi_{i},b\xi_{i}^{*}}{min}C\sum\limits^{i=1}(\xi_{i}+\xi_ {i}^{*})+\frac{1}{2}\parallel\omega\parallel^{2} \tag{32}\]  
\[ \begin{array}{l}[(\omega\cdot x_{i})+b]-y_{i}\leq\varepsilon+\xi_{i}\\ y_{i}-[(\omega\cdot x_{i})+b]\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i}\geq 0,\ \xi_{i}^{*}\geq 0,\ i=1,2,...,n\end{array}  \tag{33}\]  
As long as the optimization solution meets the KKT condition, the Lagrange dual problem solution is usually adopted.