# A Dynamic Model Updating Method with Thermal Effects Based on Improved Support Vector Regression
### The Theory of Improved Support Vector Regression  
The selection of the kernel function has a great influence on the generalization ability of SVR. SVR with the quad kernel and Gaussian kernel is suitable for constructing surrogate model problems with strong nonlinear characteristics. If a real function has a low degree of nonlinearity or shows a more significant linear characteristic, considering the influence of \(\varepsilon\), SVR with these two types of kernel functions will have obvious overfitting. For this problem, SVR with a polynomial kernel function can obtain a better fitting effect. Since the selection of different kernel functions is related to the nonlinear characteristics of the fitting function itself, it is difficult to fit a general function with only a single kernel function. Aiming at this problem, this paper mixes the Gaussian kernel function or quad kernel function with a conventional polynomial kernel function and uses this new hybrid kernel function to improve the traditional SVR model. The improved SVR model is shown below:  
\[f(x)=\sum\limits_{i=1}^{l}\big{(}\overline{a}_{i}^{*}-\overline{a}_{i}\big{)} k(\mathbf{x}_{i},\mathbf{x})+\sum\limits_{j=1}^{n}c_{j}p_{j}(\mathbf{x}) \tag{21}\]  
Since the number of parameters to be calculated is more than the number of equations, the above formula cannot be definitively solved. Therefore, the orthogonality condition is introduced:  
\[\left[ \begin{array}{cc}\mathbf{K}&\mathbf{P}\\ \mathbf{P}^{\mathbf{1}}&\mathbf{0}\end{array} \right]\left[ \begin{array}{c}\mathbf{ \alpha}\\ \mathbf{c}\end{array} \right]=\left[ \begin{array}{c}\mathbf{f}\\ \mathbf{0}\end{array} \right] \tag{22}\]  
where the kernel function matrix of the improved support vector regression machine is a nonpositive definite matrix. In the process of constructing the surrogate model of SVR, when quadratic programming is used to solve the optimal coefficients, the kernel function matrix must be a positive definite matrix. Therefore, it is necessary to construct a suitable response function to solve the problem of quadratic programming. The new response function is defined as:  
\[g(\mathbf{x})=(k(\mathbf{x})-\mathbf{f})^{T}(k(\mathbf{x})-\mathbf{f}) \tag{23}\]  
where \(k(\mathbf{x})\) represents the kernel function and \(\mathbf{f}\) represents the original response value. Therefore, the new secondary planning problem is described as:  
\[\min\frac{1}{2}\mathbf{x}^{T}\mathbf{k}^{T}\mathbf{k}\mathbf{x}+\mathbf{x}^{T }(\mathbf{k}^{T}\mathbf{f}) \tag{24}\]  
Choosing the sample space. A suitable DOE is the key to sampling and determines the sample quantity and spatial distribution for constructing surrogate models. The normal DOE methods include the full factorial design of the experiment, central composite design of the experiment, orthogonal experimental design, uniform design of the experiment, random bit design and Latin square method. In this paper, the full factorial design of the experiment and central composite design of the experimental methods are used for sampling. Building the appropriate set of sample points based on the inputs and outputs and using a suitable experimental design method. The accuracy of the constructed surrogate model needs to be verified and ANOVA is a common method of verification. In this manuscript, the coefficient of determination \(R^{2}\) and the RMSE are introduced to examine the model precision:  
\[R^{2}=1-SSE/SST \tag{25}\]  
\[RMSE=\frac{\sqrt{SSE}}{n\cdot\overline{f}} \tag{26}\]  
where \(n\) represents the number of nonconstant in the surrogate model. Generally, there is a negative correlation between \(R^{2}\) and RMSE; additionally, there is a positive correlation between \(R^{2}\) and the model adaptability.