## 4 Validation of Proposed Model  
To validate the proposed model, 12 groups of ultrasonic-vibration-assisted grinding experiments on BK7 optical glass samples were carried out. The experimental specifications are shown in Table 5. In order to comprehensively validate the prediction accuracy of the proposed model, the least square support vector machine (LS-SVM) and Gaussian process regression (GPR) method were both used to develop the prediction model for the \(MSSCD\). The prediction results are shown in Figure 8.  
To evaluate the accuracy of the model, it is necessary to calculate the error between the predicted and experimental results, which can be defined as follows:  
\[E_{c}\%=|(SSD^{*}-SSD)/SSD|\times 100 \tag{16}\]  
The average relative error (\(E_{Ac}\)) of prediction results can be expressed as follows:  
\[E_{Ac}\%=\frac{1}{n}\sum_{1}^{n}E_{c} \tag{17}\]  
From Figure 8, it can be seen that all three types of prediction results have a similar trend as the experimentally measured values. Among them, 75% of LS-SVM predicted results are smaller than measured results, and 67% of GPR-based prediction results are bigger than the experimental results. That is, the LS-SVM and GPR models cannot match the experimental training data well, which causes most of the predicted results to deviate to one side. It can be easily seen from Figure 8 that half of the WLS-SVM prediction results are bigger than experimental results, which means all the WLS-SVM predicted results are evenly distributed about the actual measurement data. Table 6 shows the specific relative errors calculated by Equations (16) and (17). It can be seen from Table 6 that the LS-SVM method without weighting factors has the biggest maximum relative error (21.72%, validation experiment 11), while that of the proposed model is the smallest (10.54%). This is likely because the essence of LS-SVM is to solve the convex quadratic programming problem and obtain its optimal global solution; the  
Figure 8: Comparison between WLS-SVM, LS-SVM, and GPR predictions and experimental results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Serial Number** & \(D_{g}\) **(mm)** & \(d_{a}\) **(\(\mu\)m)** & \(N\) **(r/min)** & \(V_{f}\) **(mm/min)** & \(a_{p}\) **(\(\mu\)m)** & **A **(\(\mu\)m)** \\ \hline
1 & 4 & 76 & 11,000 & 110 & 50 & 5 \\
2 & 4 & 64 & 13,000 & 110 & 70 & 6 \\
3 & 6 & 76 & 15,000 & 110 & 50 & 7 \\
4 & 4 & 76 & 11,000 & 100 & 60 & 7 \\
5 & 4 & 64 & 11,000 & 120 & 60 & 7 \\
6 & 4 & 91 & 8000 & 110 & 60 & 7 \\
7 & 2 & 91 & 11,000 & 160 & 60 & 7 \\
8 & 4 & 76 & 11,000 & 110 & 100 & 7 \\
9 & 6 & 46 & 11,000 & 110 & 120 & 7 \\
10 & 4 & 76 & 11,000 & 110 & 140 & 7 \\
11 & 8 & 46 & 1500 & 110 & 60 & 7 \\
12 & 4 & 64 & 2000 & 110 & 60 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Experimental grinding and ultrasonic vibration parameters. LS-SVM and GPR method both are well adapted to solve the problems of small sample data: non-linearity. However, when isolated nodes exist in training data which were caused by the differential influences of training input data on the output data, the prediction accuracy would decrease. Another possible reason is the spindle rotation speed in validation experiment 11 differed significantly from the other experiments, i.e., the diameter of grinding wheel and average diameter of abrasive grains in validation experiment 11 were both the largest of all. Since any combination of process parameters is treated as coordinates in a high dimensional space, the LS-SVM model could not accurately fit the three extreme values of coordinates. According to the influence strengths listed in Table 4, the influence strength of spindle rotation speed is the smallest; thus, the proposed improved WLS-SVM model could reduce the disturbance of this process parameter combination. Furthermore, the proposed model has the smallest average relative error (5.59%).