# Prediction Research on Irregularly Cavited Components Volume Based on Gray Correlation and PSO-SVM
## 4 Parameter Optimization of Volume-Prediction Model Based on SVM
Since the volume-prediction of irregularly cavitied components belongs to nonlinear prediction, to improve the model's veracity we selected the radial-basis function (RBF) as the kernel function of the SVM, and used the particle-swarm-optimization method to optimize the initial parameters of the support vector machine, as in Equation (16). \[K\big{(}x_{i},x_{j}\big{)}=exp\bigg{(}-\frac{x-x_{i}}{2\sigma^{2}}\bigg{)} \tag{16}\]  
The correct selection of the penalty parameter \(C\) and the kernel parameter \(\sigma\) of the nonlinear SVM is very important for the support-vector algorithm; this not only affects the complexity of the model algorithm, but also has a great impact on the generalization performance and robustness of the model prediction, which further influences the prediction precision of the SVM model. Whether the selection of model parameters is appropriate or not, this will have a greater impact on the volume-prediction effect of irregular-cavitied components. The traditional SVM adopts the grid-search cross-validation method to optimize the parameters, but when using this method, subjective factors have a great influence, the search and verification process takes a long time, and a considerable number of training samples are wasted for verification. In the case of limited samples, its defects are more obvious. However, particle-swarm-optimization (PSO) is an optimization algorithm, which has the simplicity characteristics, strong global-search ability, fast convergence-speed and high solution-accuracy. In addition, it is easy to implement because it does not have to adjust too many parameters [25, 26]. In this paper, particle-swarm-optimization was used to optimize the parameters of SVM. For the volume-prediction process of PSO-SVM based on the grey correlation, see Figure 2.  
As an effective algorithm in the field of optimization, particle-swarm-optimization can achieve good results in the optimization process of SVM parameters. Particle-swarm-optimization (PSO) is a random-search algorithm-based group collaboration which works by simulating the behavior of birds foraging, where each \"bird\" represents a particle, and the \"food\" that the flock is looking for is the optimal solution. In the volume-prediction of the irregularly cavitied components, the PSO algorithm was used to initialize the two particles of \"penalty-factor C and kernel-function parameter \(\sigma\)\"; the coordinate of the \(i\)-th particle was \(x_{i}^{i}=(x_{i1}^{i},x_{i2}^{i},\cdots,x_{i\text{ID}}^{i},x_{i\text{ID}}^{i} \in[L_{D},U_{D}]\), and the coordinates of the D target solutions were the target solutions before optimization. The running speed of the \(i\)-th particle was \(v_{i}^{i}=\big{(}v_{i1}^{i},v_{i2}^{i},\cdots,v_{i\text{ID}}^{i}\big{)},v_{i \text{ID}}^{i}\in[v_{\text{min},D},v_{\text{max},D}]\), the optimal position of the \(i\)-th particle was \(p_{i}^{i}=\big{(}p_{i1}^{i},p_{i2}^{i},\cdots,p_{i\text{ID}}^{i}\big{)}\), and the optimal position of the population was \(p_{g}^{i}=\big{(}p_{g1}^{i},p_{g2}^{i},\cdots,p_{g\text{GD}}^{i}\big{)}\). Therefore, the iterative result of particle-motion velocity and particle coordinates in the s-th dimension can be expressed as in Equations (17) and (18). \[v_{i\text{D},s}^{i}=\beta v_{i\text{D},s}^{i}+c_{1}r_{1}\left(p_{i\text{D},s}^ {i}-v_{i\text{D},s}^{i}\right)+c_{2}r_{2}\Big{(}p_{g\text{D},s}^{i}-v_{g\text {D},s}^{i}\Big{)} \tag{17}\]  
\[x_{i\text{D},s}^{i+1}=x_{i\text{D},s}^{i}+v_{i\text{D},s}^{i+1} \tag{18}\]where \(v\) is the particle velocity, \(\beta\) is the inertia weight, \(c_{1}\), \(c_{1}\) are the learning factors, and \(r_{1}\) and \(r_{2}\) are random numbers in \([0,1]\). \(\beta\) has the ability to balance local and global searches. The larger the value of \(\beta\), the stronger the global-convergence ability; otherwise, it has a stronger local-convergence ability. The learning factors \(c_{1}\) and \(c_{2}\) control the ability of the particle to find the individual  
Figure 2: Flowchart of GRA-PSO-SVM model.