## 2 Related Concepts  
### SVM System and SVR  
SVM (Support Vector Machine) is a novel supervised learning algorithm that was first proposed by [PERSON] and [PERSON] [14] in 1995. It is based on the statistical theory concept of the VC (Vapnik Chervonenkis) dimension and can improve the generalization performance of learning machines by seeking structural risk minimization [15]. The VC-dimension of the set of indicator functions \(f(x,a),a\in\Lambda\), is the maximum number of a set of vectors \(x_{1},x_{2},x_{3},...,x_{h}\) that can be separated into two classes in all \(2^{\text{h}}\) possible ways using functions of the set. If for any \(n\) there exists a set of \(n\) vectors that can be shattered by the set of the set \(f(x,a),a\in\Lambda\), then the VC dimension is equal to infinity [16]. VC dimension measures the capacity of a hypothesis space. Capacity is a measure of complexity and measures the expressive power, richness or flexibility of a set of functions by assessing how wiggly its members can be. Notably, SVM can obtain good statistical laws under the premise of limited samples [17]. SVM is divided into support vector classification (SVC) and SVR, which are used to solve data classification and data regression problems, respectively. This paper uses an SVR model for vessel trajectory prediction. Given a set of sample data \(\{(x_{i},y_{i})|\,i=1,2,...,n\,\}\) for prediction, where \(x_{i}\in R^{n}\) are the input variables and \(y_{i}\in R\) are the output variables. The core of the SVR model is the kernel function. By introducing a kernel function, SVM subtly solves the inner product operation in high-dimensional feature space, which solves the problem of nonlinear classification and prediction. Commonly used kernel functions mainly include linear kernel functions, polynomial kernel functions, Radial Basis Function (RBF) and sigmoid kernel functions. Among them, RBF is the most widely used kernel function, which can realize nonlinear mapping [18; 19]. It has better performance for both large samples and small samples and has fewer parameters than polynomial kernel functions. In this study, RBF function is employed as the kernel function for the SVR model [20]:  
\[K(\mathbf{x},\mathbf{x}_{i})=\varphi(\mathbf{x})\cdot\varphi(\mathbf{x}_{i})=\exp\Bigl{(}-g\|\mathbf{ x}-\mathbf{x}_{i}\|^{2}\Bigr{)}, \tag{1}\]  
where \(K\) stands for kernel function; \(\mathbf{x}\) and \(\mathbf{x}_{i}\) are the vectors in initial low-dimensional feature space and \(g\) is the kernel function parameter; \(\varphi(\mathbf{x}_{i})\) is the nonlinear mapping function, which can map the input data into the high dimensional feature space; The operator \(\cdot\) is the inner symbol. Through RBF kernel function mapping, each sample point \((x_{i},y_{i})\) is fitted as closely as possible to the following SVR linear model:  
\[f(\mathbf{x})=\mathbf{\omega}^{T}\mathbf{x}+b, \tag{2}\]  
where \(\mathbf{\omega}\) is the weight vector, with \(\mathbf{\omega}\in R^{n}\); and \(b\) is the bias, with \(b\in R\). SVR takes an \(\varepsilon\)-insensitive function as a loss function. First, a constant \(\varepsilon\) is defined for a specific problem, where \(\varepsilon>0\). For a sample \((\mathbf{x}_{i},y_{i})\), if \(\bigl{|}y_{i}-\mathbf{\omega}^{T}\varphi(\mathbf{x}_{i})-b\bigr{|}\leq\varepsilon\), there is no loss; otherwise, the corresponding loss is \(\bigl{|}y_{i}-\mathbf{\omega}^{T}\varphi(\mathbf{x}_{i})-b\bigr{|}-\varepsilon\). That is, the \(\varepsilon\)-insensitive loss function can be written as:  
\[err(\mathbf{x}_{i},y_{i})=\left\{ \begin{array}{cc}0&\bigl{|}y_{i}-\mathbf{\omega}^{ T}\varphi(\mathbf{x}_{i})-b\bigr{|}\leq\varepsilon\\ \bigl{|}y_{i}-\mathbf{\omega}^{T}\varphi(\mathbf{x}_{i})-b\bigr{|}-\varepsilon& \bigl{|}y_{i}-\mathbf{\omega}^{T}\varphi(\mathbf{x}_{i})-b\bigr{|}>\varepsilon\\ \end{array} \right.. \tag{3}\]  
As shown in Figure 1, an interval band with \(f(\mathbf{x})\) as the center and a width of \(2\varepsilon\) is established.