## 2 The Driving Decision-Making Process of Autonomous Vehicle
### Support Vector Machine Regression Model  
SVR model is a kind of machine learning method based on statistical learning theory, which can improve the generalization ability of learning machine by seeking the minimum structural risk [16; 20]. So, SVR model has been widely applied and developed in the fields of pattern recognition, regression analysis, and sequence prediction [18; 21].  
Let \(SV=\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{m},y_{m})\}\) be a set of \(m\) training samples, each of samples \(x_{m}\) is the input variable, which is obtained from traffic environment features. \(y_{m}\) is the output driving decision corresponding to \(x_{m}\). These training samples are fitted by \(f(x)=\omega^{T}x+b\), and all of the fitted results must be satisfied with error accuracy \(\varepsilon\), i.e.,:  
\[\left|\omega^{T}x_{i}+b-y_{i}\right|\leq\varepsilon,i=1,\;2,\;\ldots,\;m \tag{1}\]  
According to the minimization criteria of structural risk, \(f(x)\) should make \(\frac{1}{2}\left\|\omega\right\|^{2}\) minimum. When considering the exiting fitted errors, the relaxation factors are introduced as \(\xi_{j}\geq 0\), \(\xi_{j}^{*}\geq 0\). The best regression result can be derived from the minimum extreme value of the following function:  
\[\mathbf{\Phi}(\omega,\xi_{j},\xi_{j}^{*})=\frac{1}{2}\|\omega\|^{2}+c\sum_{i=1}^{m }\left(\xi_{j}+\xi_{j}^{*}\right) \tag{2}\]  
Figure 3: Diagrammatic sketch of vehicle states.  
Figure 2: The detailed data processing of data processing program.  
where \(C\) is the penalty factor value, \(C>0\).  
Then, adopt the dual principle, and set the Lagrange multiplier \(\alpha\), \(\alpha^{*}\) to establish the Lagrange equation. Through drafting the parameters \(\omega\), \(b\), \(\xi_{j}\), \(\xi_{j}^{*}\) and making the drafted formulas equal to 0, the regression coefficient \(\omega\) and constant term \(b\) can be obtained:  
\[\left\{\begin{array}{c}\omega=\sum\limits_{i=1}^{m}{(\alpha_{i}-\alpha_{i}^{* })x_{i}}\\ \\ b=\frac{1}{m}\left\{\begin{array}{c}\sum\limits_{0<\alpha<C}{0<\alpha<C} \left[y_{i}-\sum\limits_{x_{j}\in\mathcal{SV}}{(\alpha_{j}-\alpha_{j}^{*}) \cdot(x_{i}-x_{j})}-\varepsilon\right]}\\ 0<\alpha^{*}<C\end{array}\right.\end{array}\right\} \tag{3}\]  
After that, the results are substituted into the function \(f(x)\) to get the regression function:  
\[f(x)=\sum\limits_{i=1}^{m}{(\alpha_{i}-\alpha_{i}^{*})(x_{i}\cdot x)}+b \tag{4}\]  
Finally, the original samples are mapped into a high-dimensional feature space with a kernel function \(K(x_{i},x)\), and calculate the parameters with the same method, as above. The obtained non-linear regression function is:  
\[f(x)=\sum\limits_{i=1}^{m}{(\alpha_{i}-\alpha_{i}^{*})K(x,x_{i})}+b \tag{5}\]  
The common kernels are showed as following:  
Polynomial kernel function: \(K_{\text{Poly}}(x,x_{i})={(a(x\cdot x_{i})+b)}^{d}\)  
Radial basis function: \(K_{\text{Rbf}}=exp(-\frac{\|x-x_{i}\|^{2}}{\sigma^{2}})\)  
Sigmoid kernel function: \(K_{\text{Sig}}=tan(\tau(x\cdot x_{i})-\delta)\)  
Where the dot denotes the inner-product operation in Euclidean space, \(d\) is the degree of polynomial kernel, \(\sigma\) is the constant term determining the width of RBF kernel [18]. With different kernels, it can be structured by different regression surfaces, then different training results may be gotten on driving decision-making. So, it is important to select the proper kernel function and kernel parameters in the SVR model.