# Prediction Research on Irregularly Cavited Components Volume Based on Gray Correlation and PSO-SVM
## 4 Parameter Optimization of Volume-Prediction Model Based on SVM  
The support vector machine (SVM) is based on statistical learning theory and structural-risk minimization [22]. The SVM algorithm has strong learning-functions and characteristics, good adaptability to small samples and limited samples, and strong stability and generalization-ability [23]. It can better deal with the problem of nonlinear regression. We used gray-correlation analysis to develop the support-vector-machine predictive modeling for the main influencing factors of multiple variables related to the volume of irregular-cavity parts after processing in this paper. In the given sample set \(\left\{\left(x_{i},y_{j}\right),\left(i=1,2,3,\cdots,m\right)\right\}\), \(x_{i}\) was a attribute vector of two-dimensional space, with characteristic input-values such as atmospheric pressure, atmospheric humidity, temperature before micro-compression, stable differential-pressure before micro-compression, stable differential-pressure after micro-compression, gas-equilibrium time, and temperature after micro-compression, etc. ; \(y_{j}\) was the corresponding predicted-target value, and m was the number of samples. For the volume model of irregularly cavited components in the low-dimensional nonlinear space [24], the sample data set, \(x\), was mapped to the high-dimensional linear space through the nonlinear function, \(\phi(x)\), and the decision function was established in the high-dimensional linear space, as in Equation (11). \[\phi(x)=\omega\cdot(x)+b \tag{11}\]  
where \(\omega\) is the weight vector, and \(b\) is the bias parameter. The algorithm uses the minimization-optimization model to establish a decision function, as in Equations (12) and (13). \[\text{min}\frac{1}{2}\omega^{2}+C\sum_{i=1}^{m}(\xi_{i}+\xi_{i}^{*}) \tag{12}\]  
\[S.T. \begin{cases}y_{i}-\omega\cdot\varphi(x)-b\leq\varepsilon+\xi_{i}\\ \left[\omega\cdot\varphi(x)\right]+b-y_{i}\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i},\xi_{i}^{*}\geq 0(i=1,2,3,\cdots,m)\end{cases}  \tag{13}\]  
where \(C\) is the penalty factor; \(\xi_{i},\xi_{i}^{*}\) are the relaxation factors; and \(\varepsilon\) is the upper limit of the error. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Serial** & **Variable** & **Input-Feature Parameters** & **Correlation Coefficient** \\ \cline{3-4}
**Number** & **Name** & & **Coefficient** \\ \hline
1 & \(x_{1}\) & Atmospheric pressure before & 0.9896 \\  & & micro-compression & 0.5735 \\
2 & \(x_{2}\) & & micro-compression & 0.5948 \\
3 & \(x_{3}\) & Temperature before micro-compression & 0.5948 \\
4 & \(x_{4}\) & & Stable differential-pressure after & 0.9989 \\
5 & \(x_{5}\) & & micro-compression & 0.8689 \\
6 & \(x_{6}\) & Atmospheric humidity & 0.5285 \\
7 & \(x_{7}\) & Gas-equilibration time & 0.9993 \\
8 & \(x_{8}\) & Atmospheric pressure after & & 0.8678 \\  & & micro-compression & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Correlation coefficients of the variables. We introduced the Lagrangian multipliers \(a_{i}\) and \(\alpha_{i}^{*}\), and then transformed the minimization-optimization model into a dual optimization, in order to solve it, by finding the maximum value of \(Q(\alpha)\), as in Equations (14) and (15). \[\max:Q(\alpha)=-\frac{1}{2}\sum_{i,j=1}^{m}(\alpha_{i}-\alpha_{i}^{*})\left( \alpha_{j}-\alpha_{j}^{*}\right)\cdot K\big{(}x_{i},x_{j}\big{)}+\sum_{i=1}^{m }y_{i}(\alpha_{i}-\alpha_{i}^{*})-\varepsilon\sum_{i=1}^{m}(\alpha_{i}+\alpha_ {i}^{*}) \tag{14}\]  
\[S.T.\bigg{\{} \begin{matrix}\sum_{i=1}^{m}\alpha_{i}=\sum_{i=1}^{m}\alpha_{i}^{ *}\\ 0\leq\alpha_{i},\alpha_{i}^{*}\leq C(i=1,2,3,\cdots,m)\end{matrix}  \tag{15}\]  
where \(Q(\alpha)\) is the dual objective function of the Lagrangian function, \(K\big{(}x_{i},x_{j}\big{)}=\varphi(x_{i})\cdot\varphi\big{(}x_{j}\big{)}\) is the kernel function when \(\alpha_{i}\times\alpha_{i}^{*}=0\) and \(\alpha_{i},\alpha_{i}^{*}\) are not \(0\) at the same time, and the corresponding \(x_{i}\) is a support vector (SV).