## 4 Preliminary Model of SVM  
The preliminary model of SVM is configured, as shown in Figure 4, where it consists of two phases, i.e., using a training algorithm (red background) and employing the model to classify new data (yellow background). This model is used to conduct a small scale study prior to a full-scale model. It is helpful to discover the knowledge of our preference in selecting the legitimate orbits and to observe how well the SVM might classify them.  
It all starts with a small portion of examples. We arbitrarily choose the dimensional parameters a and b to form a mesh of grids. Each of them possesses eight points uniformly distributed over [3; 5], i.e., [3; 3.2857, 3.5714, 3.8571, 4.1429, 4.4286, 4.7143, 5]. Then a mesh of grids formed by [a, b] consists of 64 points, each of which associates with an orbit as shown in Figure 5a where the rest of parameters are set to be c = 3, d = 3.1111, S\({}_{\text{x}}\) = 3, S\({}_{\text{y}}\) = 0.6667.  
At this preliminary stage, we could boldly categorize the orbits based upon our intuitiveness. Consequently, the orbits in Figure 5a labeled with 1, 2, 3, 9, 10, 11, 17, 18, 19, 25, 26, 27, 33, 34, 35, 41, 42, 43, 49, 53, 54, 55, 56, 60, 61, 62, 63, 64 are attributed to the legitimate class, while the others are defined as the illegitimate class. In Figure 5b, the blue and red dots stand for the legitimate and illegitimate classes, respectively.  
Obviously, this set of training data cannot be separated linearly. Hence, a Gaussian kernel is applied to achieve the nonlinear separation. The model of SVM is obtained by using fitcsvm, a command of Matlab, with KernelFunction, KernelScale, Sigma, and Standardize, setting to be rbf, auto, 0.8, and true, accordingly. The decision boundary results in two black curves, as shown in Figure 5b.  
To get a grasp of how the kernel trick works, we evaluate the Equation (18) with respect to \((a,b)\) and plot the family of its contours accordingly. In Figure 6, the surface that is generated by the score function contains all of the data where the points of the legitimate class are distributed over the uphill, while the points of the illegitimate class are distributed over the downhill. The decision boundary is exactly the contour at the level of zero, as shown in Figure 6. Hence, the nature of the kernel trick is to embed the input space into a higher-dimensional space, referred to as feature space or kernel space, in which the data can be linearly separable by a plane lying horizontally at level zero. The scatter diagram in Figure 5b is in fact the top view of the stereoscopic model, as shown in Figure 6.  
So far, the SVM has manifested its great strength in classification. Taking a close look of the orbits attributed to the legitimate class, it unveils that our preference in selecting the orbits possesses three characteristics, i.e., the obliqueness of the orbit being shallow, avoiding the orbit from being too slim or too thick, and no self-intersection.  
Figure 4: Preliminary Model of Support Vector Machine (SVM).  
Figure 5: **(a) Sixty four orbits as a set of training data (b) Scatter diagram in two-dimensions (2D) with decision boundary.**  
In this pilot study, the training data were prepared manually because they were inspected by our vision and judged by our intuitiveness. The judgement based on the qualitative descriptions is more or less involved with subjectivity rather than objectivity. It leads that the training data might be contaminated with some amounts of unconscious bias. How to well prepare the training data is always a vital step in the supervised machine learning techniques. Hence, the following two sections are devoted to turn those discriminant criteria from qualitative descriptions into quantitative definitions, like mathematical functions or rules, based on which the machine vision can take the place of human vision to label a large scale of orbits.