## 3 Principle of the Prediction Model
### Prediction Based on Support Vector Machine (SVM)  
In the SLS process, a seven-dimensional vector is obtained after the feature extraction and dimension reduction of each temperature image. There is a correlation between the key features of the temperature field at a certain time, and the actual value of the process parameters at the current time. Therefore, the actual value of the process parameters can be estimated based on the obtained seven-dimensional vector, and the quality of the SLS parts can be predicted. The premise is to establish a correlation model between the seven-dimensional vector and the actual value of the process parameters based on a large quantity of training samples. Since the process parameters are continuous quantities, to simplify the problem, in this paper, the process parameters are discretized and set to two categories as \"out of range\" and \"appropriate\". \begin{table}
\begin{tabular}{c c} \hline
**Component Number** & **Importance** \\ \hline
1 & 58.3\% \\
2 & 73.2\% \\
3 & 80.1\% \\
4 & 84.5\% \\
5 & 86.8\% \\
6 & 89.1\% \\
7 & 90.1\% \\ \hline \end{tabular}
\end{table} 
Table 1: Importance of Each Component after Orthogonal Transformation. However, based on the SVM, a classifier is constructed, and many data samples are generated by the SLS simulation model to train the classifier. Finally, the actual value classification prediction model of the process parameters is constructed. Assume that there are linear separable samples \((x_{i},\ y_{i})\), \(i=1,2,3,\ldots,\) where x is n-dimensional data and y is assumed to be a binary classification whose value is plus/minus one. The optimal classification hyperplane means that the hyperplane can not only separate two kinds of samples correctly--that is, make the training error zero--it can also maximize the classification interval. In d-dimensional space, the linear discriminant function is g(X) = \(\omega+b\), and the hyperplane equation is \(\omega\)X + b = 0. In the formula, \(\omega\) is an n-dimensional vector representing the hyperplane's normal, and \(b\) is a constant representing a classification threshold. To properly classify all of the samples, the following formula should be satisfied:  
\[y_{i}[(\omega X_{i}+b)]\geq 1,i=1,2,3,\ldots \tag{8}\]  
If the sample is linearly inseparable, there will be some samples that will never meet the above conditions. To balance these classification errors, a relaxation variable is introduced into the constraints. The formula becomes:  
\[y_{i}[(\omega X_{i}+b)]\geq 1-\xi_{i},i=1,2,3,\ldots \tag{9}\]  
When \(0<\xi_{i}<1\), sample points are classified correctly. When \(\xi_{i}\geq 1\), the point \(X_{i}\) is incorrectly classified. To solve this problem, a penalty factor is added, and a penalty term C \(\sum_{i=1}^{l}\xi_{i}\) is added to the objective function. The optimization problem can be expressed as:  
\[ \begin{array}{l}\min\varnothing(\omega,\xi)=\frac{1}{2}|\omega|+\text{C} \sum\limits_{i=1}^{l}\xi_{i}\\ y_{i}[(\omega X_{i}+b)]\geq 1-\xi_{i},i=1,2,3,\ldots\end{array}  \tag{10}\]  
\(\sum_{i=1}^{l}\xi_{i}\) represents the upper section of the wrongly classified samples' quantity in the sample set that is used to measure the deviation of sample data from ideal partitioning conditions. C is a penalty factor to control the punishment of the wrongly classified data. When the penalty factor value is small, the classification interval is large, and good generalization performance can be obtained. The above formula can be solved by applying the Lagrangian function to its dual form. For a nonlinear problem, it can be transformed into the linear data of a high-dimensional space by mapping, and then, linearly separable problems can be solved in a transformed space. According to the correlative functional theories, if the function satisfies [PERSON]'s condition, it corresponds to the inner product of a transformed space, which is called a kernel function K = \(\big{(}X_{i},X_{j}\big{)}\).