## 2 Methods
### Speed Prediction  
Gaussian Process Regression (GPR) methods have shown great potential in dealing with the uncertain relationship between inputs and outputs compared to existing popular nonparametric forecasting methods such as neural networks [27; 28]. In [29], the results showed that traffic flow prediction using Gaussian models outperformed the commonly \begin{table}
\begin{tabular}{c c} \hline
**No.** & **Characteristic Parameters** \\ \hline
1 & Vehicle current speed (m/s) \\
2 & Relative change of horizontal coordinates (m) \\
3 & Relative change of vertical coordinates (m) \\ \hline \end{tabular}
\end{table} 
Table 1: Table of horizontal characteristic parameters. \begin{table}
\begin{tabular}{c c} \hline
**No.** & **Characteristic Parameters** \\ \hline
1 & Maximum Acceleration (m/s\({}^{2}\)) \\
2 & Minimum Acceleration (m/s\({}^{2}\)) \\
3 & Range of Acceleration (m/s\({}^{2}\)) \\ \hline \end{tabular}
\end{table} 
Table 2: Table of vertical characteristic parameters. used methods. In [30], a conditional linear Gaussian model is proposed for multistep prediction of leading velocity, and the model is trained to estimate the probability distribution of leading velocity using actual measurement data. Authors of [31] developed a multistep Gaussian process regression model to predict the driver's future power demand using vehicle-vehicle and vehicle-road information. Authors of [32] used a Gaussian process approach to estimate future vehicle motion from vehicle-to-vehicle and vehicle-to-infrastructure information. However, the above papers require the use of a priori roadwork information in vehicle speed prediction. Because the vehicle-to-vehicle information can be obtained without identifying the driving intention of the previous vehicle, the accurate prediction of the speed of the previous vehicle cannot be made when the data sources are limited, especially when the vehicle-to-road information and vehicle-to-vehicle information are not available, but there is room for further improvement. GPR considers \(f(x)\) to be related to some specific models compared to Gaussian processes, which can represent \(f(x)\) indirectly but rigorously by allowing the data to represent the input-output relationship of the object. Gaussian process regression is a form of supervised learning method with fewer parameters. GPs extend multivariate Gaussian distributions to an infinite number of dimensions. Formally Gaussian processes distribute the data in regions where any finite subset follows a multivariate Gaussian distribution. For all \(x=[x_{1},x_{2},\ldots,x_{n}]\), \(f(x)=[f(x_{1}),f(x_{2}),\ldots,f(x_{n})]\) which obey a multivariate Gaussian distribution, \(f\) is said to be a Gaussian process, denoted as:  
\[f(x)\,\sim N((\mu(x),c(x,x))) \tag{4}\]  
where \(\mu(x)\) is the mean value function and \(c(x,x)\) is the covariance matrix. The Gaussian noise model is used to represent the measured and noise values during data observation, which means:  
\[y=f(x)+N(0,\sigma_{n}^{2}) \tag{5}\]  
where \(x\) is the vector of independent variables, i.e., the eigenvector. \(f(x)\) assumes that there is a Gaussian process prior to this, which means:  
\[f(x)\,\sim GP(0,C) \tag{6}\]  
The effect of noise is also introduced in the covariance function, as in Equation (7):  
\[c(x-x^{\prime})=\sigma_{f}^{2}\exp[\frac{-\left(x-x^{\prime}\right)^{2}}{2l^{2 }}]+\sigma_{n}^{2}\delta(x-x^{\prime}) \tag{7}\]  
where \(\sigma_{f}^{2}\) is the maximum covariance, and \(l\) and \(\sigma_{f}\) are the hyperparameters determined during model training using the maximum likelihood estimation method. The covariance function can be understood as a measure of similarity between two \(x\). \(\delta(x-x^{\prime})\) is Kronecker delta function, whose independent variables are usually two integers, and its output value is 1 if they are equal, otherwise it is 0. With the covariance function, the covariance matrix is calculated for all input observations as in Equation (8):  
\[C=