# Intelligent Prediction of Aeroengine Wear Based on the SVR Optimized by GMPSO
## 2 SVR Prediction Model  
SVR is derived from a classification application; the classification decision can be converted into a regression estimation depending on the \(\varepsilon\)-insensitive loss function. Assume that \(\mathbf{x}_{i}\in\mathbf{R}^{n}\) denotes the \(i\)th input vector, \(y_{i}\in\mathbf{R}\) denotes the observation value of \(\mathbf{x}_{i}\), and the \(\varepsilon\)-insensitive loss function can be defined as follows in Equation (1):  
\[L(y_{i},f(\mathbf{x}_{i}))=\left|y_{i}-f(\mathbf{x}_{i})\right|_{\varepsilon}=\max\{0, \left|y_{i}-f(\mathbf{x}_{i})\right|-\varepsilon\} \tag{1}\]  
where the insensitive loss coefficient \(\varepsilon\) is a positive number with minimized empirical risk, and \(f(\mathbf{x}_{i})\) is the predicted value of vector \(\mathbf{x}_{i}\). In the case of \(\left|\,y_{i}-f(\mathbf{x}_{i})\right|\leq\varepsilon\), the predicted values are free of loss. Furthermore, the regression optimization problem can be transformed as follows in Equation (2):  
\[\left\{ \begin{array}{l}\min\frac{1}{2}\left\|\mathbf{w}\right\|^{2}+C\frac{1}{I} \sum\limits_{i=l}^{l}\left|y_{i}-f(\mathbf{x}_{i})\right|_{\varepsilon}\\ \left\{\begin{array}{l}\left|y_{i}-f(\mathbf{x}_{i})\right|-\varepsilon,\;\left|y _{i}-f(\mathbf{x}_{i})\right|\geq\varepsilon\\ 0,\;\text{otherwise}\end{array}\right.\end{array} \right. \tag{2}\]  
where \(\mathbf{w}\) is the normal vector of the hyperplane; \(C\) is the penalty factor; and \(l\) is the number of input vectors. The insensitive loss coefficient \(\varepsilon\) can control the width of the insensitive region of the sample data influenced by the regression function and this can affect the number of support vectors. Furthermore, the penalty factor \(C\) reflects the penalty for these sample data exceeding the \(\varepsilon\) width, which can affect the complexity and stability of the SVR. A smaller \(C\) value can lead to a weaker penalty, which makes the training error larger. If \(C\) is too large, the learning accuracy can be improved, but the generalization ability of the SVR will become worse. For SVR prediction, the method used to construct the prediction function is important. In general, using the Lagrange function to construct and solve the convex quadratic programming problem, the prediction functions can be described in Equation (3):  
\[f(\mathbf{x})=\sum\limits_{i=1}^{n}\alpha_{i}\Phi(\mathbf{x})\Phi(\mathbf{x}_{i}^{*})+b \tag{3}\]  
where \(\Phi(\cdot)\) is a mapping transform from the original space to the high-dimensional feature space, so that the nonlinear regression in the original space can be transformed into the linear regression in high-dimensional space. \(\alpha_{i}\) is the weight connecting the feature space to the output space, \(b\in\mathbf{R}\), and \(b\) is a bias. \(\mathbf{x}\) is the unknown input vector, \(\mathbf{x}_{i}^{*}\) is the support vector determined through calculation, and \(n\) is the number of support vectors. Actually, the construction of \(\Phi(\cdot)\) is a complex theoretical problem. Moreover, the computation cost of the high-dimensional inner product is very large. In practice, the kernel function \(K(\mathbf{x}_{i}^{*})\), meeting the Mercer condition, can be used to combine the nonlinear mapping and inner-product of two vectors in feature space so that the nonlinear mappingis carried out implicitly. So \(K(\mathbf{x},\mathbf{x}_{i}^{*})=\Phi(\mathbf{x})\cdot\Phi(\mathbf{x}_{i}^{*})\), which is defined in the original space instead of the high-dimensional feature space. Furthermore, Equation (3) can be written as follows in Equation (4):  
\[f(\mathbf{x})=\sum_{i=1}^{n}\alpha_{i}K(\mathbf{x}\cdot\mathbf{x}_{i}^{*})+b \tag{4}\]  
The Gauss radial basis function can approach the train samples better when dealing with nonlinear problems [9], which is conducive to data prediction.