## 7 Advanced Model of SVM  
Obtaining a sufficient amount of training data that adequately reflects the characteristics of the field data is always the bottleneck of supervised learning. The breakthrough we made is to combine the image processing with the rule-based criteria, as shown in Figure 13, so that the machine vision can label a vast amount of training data as well as a full-scale of data. Figure 11: Some cases of non-self-intersection misjudged as self-intersection. Figure 12: Lemniscates recognized by (**a**) human vision and (**b**) machine vision. Figure 13: Advanced model of SVM. Note that Xp consists of properties \(\alpha\) and \(\beta\); Xd consists of six dimensional parameters, i.e., \(a,b,c,d\in[3,5]\), \(S_{x}\in[3,4]\), \(S_{y}\in[0,2]\); and, \(Y\) is the label set. Xd and Y carry the inputs and correct outputs of the training data for SVM2, so do Xp and Y for SVM1. If each dimensional parameter is uniformly sampled into \(n\) points, the number of training data associated with 6 parameters will be \(N=n^{6}\). Hence, we prepare four sets of training data with different numbers, i.e., 64(\(=2^{6}\)), 4096(\(=4^{6}\)), 46656(\(=6^{6}\)), 262144(\(=8^{6}\)), to train SVM1 and SVM2. The cross-validation is the method to evaluate the performance of the trained model. In Table 2, there is a list of losses that were obtained by 10-fold cross-validation. It shows that the SVM1 has a performance better than SVM2. As the number of training data is increased, the rate of loss is decreased. Since SVM1 is the model drawn from the training data [Xp, Y] and the input data Xp possess only two variables, its model can be visualized. However, it is not the case for SVM2 as its input data Xd possessing dimension higher than two. The scatter diagrams of SVM1 with four sets of training data are shown in Figures 14 and 15 where the blue and red dots stand for legitimate and illegitimate classes, respectively; the black cross-X stands for the lemniscates that were identified by machine vision. Note that the diagram with 64 data is insufficient to manifest its geometrical manifold, hence some contours are added for the sake of visualization. Recall the self-intersection addressed previously. Observing the Figures 14 and 15, most of lemniscates marked with X indeed aggregate at the region with a slenderness ratio below 0.25. Interestingly, the lemniscates whose slenderness ratio exceeds 0.25 will also be precluded from the legitimate class by the reason of their steepness. Evidently, both slenderness and obliqueness criteria are sufficient to categorize the data. As refining the sampling points on each dimensional parameter, either the number of training data or the computational time in property extraction prodigiously grows, as shown in Table 3. Comparatively, each dimension sampled into six points is more pragmatic than when it is sampled into eight points, because it can provide the SVM with sufficient amount of unbias training data at the cost of less computational time. In the framework of machine vision and SVM, as shown in Figure 13, two databases gathering all eligible data concerned with properties of orbits and dimensions of TJLs are established. With the aid of these two databases, a program can be developed to fulfill the inverse problem. It searches the orbits in compliance with the specification of \(\alpha\) and \(\beta\) from the database of properties and uses their correspondent identity numbers to list all candidates of TJLs from the database of dimensions. Undoubtedly, the database of properties has collected all the orbits with legitimate properties, i.e., \(\alpha\in\beta\), and \(\beta\). Figure 14: Scatter diagrams of SVM1 in cases of N = 64 and N = 4096. Figure 15: Scatter diagrams of SVM1 in cases of N = 46656 and N = 262144. \([-10^{\circ},+10^{\circ}]\) and \(\beta\in[0.25,0.45]\). However, what would these legitimate orbits look like? Let us randomly select the data from the database of dimensions and plot their orbits accordingly.