# A Dynamic Model Updating Method with Thermal Effects Based on Improved Support Vector Regression
## 5 The Numerical Examples  
### A Numerical Example of Two Parameters  
According to a nonlinear function named Branin rcos, the precision of the global approximate function constructed by SVR-Polynomial, SVR-Gauss and improved SVR based on the hybrid basis function (improved SVR) are compared. There are two variables in this function:  
\[f(x)=(x_{2}-\frac{5.1x_{1}^{2}}{4\pi^{2}}+\frac{5x_{1}}{\pi}-6)^{2}+10(1-\frac{ 1}{8\pi})\cos(x_{1})+10 \tag{32}\]  
where \(x_{1}\in[-5,10]\) and \(x_{2}\in[0,15]\).  
The full factorial DOE is used to solve the above small-scale problem, which defines the normalized parameters \(x_{1}\) and \(x_{2}\) as \(\overline{x}_{1}\) and \(\overline{x}_{2}\). The sample points and objective function values are shown in Table 1.  
The surrogate models using SVR based on different basis functions and the real function are shown in Figure 3, the number of sample points is 1600.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Number** & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \(\mathbf{f}\) & **Number** & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \(\mathbf{f}\) \\ \hline \(1\) & \(-1\) & \(-1\) & 308.1 & 14 & 0 & 0.5 & 60.57 \\ \(2\) & \(-1\) & \(-0.5\) & 80.12 & 15 & 0 & 1 & 22.17 \\ \(3\) & \(-1\) & 0 & 10.31 & 16 & 0.5 & \(-1\) & 47.98 \\ \(4\) & \(-1\) & 0.5 & 20.8 & 17 & 0.5 & \(-0.5\) & 22.38 \\ \(5\) & \(-1\) & 1 & 10.96 & 18 & 0.5 & 0 & 73.23 \\ \(6\) & \(-0.5\) & \(-1\) & 193.3 & 19 & 0.5 & 0.5 & 122.6 \\ \(7\) & \(-0.5\) & \(-0.5\) & 32.75 & 20 & 0.5 & 1 & 69.96 \\ \(8\) & \(-0.5\) & 0 & 3.156 & 21 & 1 & \(-1\) & 17.51 \\ \(9\) & \(-0.5\) & 0.5 & 26.62 & 22 & 1 & \(-0.5\) & 59.39 \\ \(10\) & \(-0.5\) & 1 & 2.501 & 23 & 1 & 0 & 150.5 \\ \(11\) & 0 & \(-1\) & 106.6 & 24 & 1 & 0.5 & 212.8 \\ \(12\) & 0 & \(-0.5\) & 13.51 & 25 & 1 & 1 & 145.9 \\ \(13\) & 0 & 0 & 24.13 & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: The full factorial experimental design based on 2 level.  
Figure 3: The contraction based on different basis function.  
From the figures above, we can find that SVR based on different basis functions can fit the real function according to the sample points. The difference is that the fitting results and accuracy of the constructor function based on SVR-Polynomial are the poorest and have severe overfitting phenomena. Even though the constructor function based on SVR-Gauss can describe the real function more accurately to some degree, from Figure 3c, we can find that there is obvious deviation between the constructor function and real function in some areas. However, the constructor function based on improved SVR can exactly describe the real function, so its accuracy is the highest. To illustrate it more directly, we apply the method of comprehensive factorial experimental design to create a series of test samples to test the accuracy of the constructor function.  
To further prove that the surrogate model based on improved SVR has the best fitting precision, a group of test sample points was constructed through the full factorial DOE method and the test sample points are listed in Table 2.  
According to Equations (27) and (28), \(R^{2}\) and RMSE can be obtained.  
Table 3 shows that the \(R^{2}\) value of the surrogate model based on improved SVR is larger than that based on SVR-Quad and SVR-Gauss, and the RMSE value of the surrogate model based on improved SVR is less than that based on SVR-Polynomial and SVR-Gauss. This means that among these three surrogate model construction methods, improved SVR obtains the best fitting precision and a more accurate description.