# Vehicle Acceleration Prediction Based on Machine Learning Models and Driving Behavior Analysis
## 3 Personalized Driving Behavior Analysis Model
### Similarity Evaluation for the Driving Behavior  
Grouping drivers by similarity can facilitate the design of human-centric driver assistance systems. For example, drivers with similar driving behaviors can share the same driving behavior analysis model and driver assistance system [28]. However, human drivers are somewhat random and dynamic, and daily driving is highly nonlinear. Thus, the direct calculation of drivers' observed statistical indicators (e.g., the mean and standard deviation) does not evaluate behaviors well [29]. In this study, a KS test was used for driving behavior similarity evaluation.  
The Kolmogorov-Smirnov test (KS test) is a non-parametric test; it is based on the limiting distribution of the statistic \(\sqrt{n}D_{n}\) as \(K(x)\). For the hypothesis problem \(H_{0}\), The random variable \(X\) has a distribution function of the form \(F_{0}(x)\), and is tested for significance. The steps are as follows [30]:  
Step 1: The original hypothesis \(H_{0}:\ F_{n}(x)=F_{0}(x)\) and alternative hypothesis \(H_{1}:\ F_{n}(x)\
eq F_{0}(x)\) are proposed, where \(F_{n}(x)\) is the empirical distribution function and \(F_{0}(x)\) is a certain distribution function of a known type.  
Step 2: The sample size \(n\) and the significance level \(\alpha\) are given.  
Step 3: According to the limiting distribution of \(D_{n}\), there is \(P\big{\{}\sqrt{n}D_{n}>K_{\alpha}(n)\big{\}}\approx\alpha\), where \(D_{n}=sup|F_{n}(x)-F_{0}(x)|\), \(-\infty<x<+\infty\), and \(K_{\alpha}(n)\) is the upper quantile of the Kolmogorov distribution, which can be given by checking the Kolmogorov table.  
Step 4: The rejection region \(\big{\{}(x_{1},x_{2},\cdots,x_{n})|D_{n}\big{\}}K_{\alpha}(n)/\sqrt{n}\big{\}}\) of \(H_{0}\) is given, i.e., if \(D_{n}>K_{\alpha}(n)/\sqrt{n}\), \(H_{0}\) is rejected; otherwise, \(H_{0}\) is accepted.  
In the above steps, the calculation of \(D_{n}\) is cumbersome. Because the empirical distribution function \(F_{n}(x)\) is a step function, we have  
\[D_{n}=\max_{1<k<n}\big{\{}|F_{n}(x_{k})-F_{0}(x_{k})|,|F_{n}(x_{k+1})-F_{0}(x_ {k})|\big{\}} \tag{17}\]  
In this study, the KS test is conducted using the driving behavior semantics, and then the similarity analysis is represented in heat maps. In Figure 6, crimson means a large  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multirow{2}{*}{**Total**} & \multicolumn{8}{c}{**Durations of Driving Behavior Semantic (s)**} \\ \cline{3-10}  & & & \(<\)1.0 & [1.0, 5.0) & [5.0, 10.0) & [10.0, 15.0) & [15.0, 20.0) & [20.0, 25.0) & [25.0, 30.0) & \(\geq\)30.0 \\ \hline HMM & 12949 & 0.0203 & 0.3468 & 0.2549 & 0.1162 & 0.0641 & 0.0345 & 0.0299 & 0.1333 \\ MHMM & 7213 & 0.0110 & 0.1305 & 0.1896 & 0.1481 & 0.1050 & 0.0628 & 0.0473 & 0.3056 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Statistical results of the durations of the driving behavior semantic.  
difference in driving behavior and burgundy means a small difference in driving behavior. In addition to this, \(D_{n}=0\) between the same drivers is represented in dark purple.  
As shown in Figure 6, the KS test results for driver #11 were significantly different from other drivers such as drivers #5, #7, #8, #9, #15, and #16. In addition, driver #21 was similar to others such as the drivers #2, #3, #4, #6, #14, and #19. Last but not least, drivers #18, #19, #20, #23, #24, and #26-#30 were significantly similar to each other, as their color patches have similar colors and combinations.  
By similarity evaluation, this study divided 10 similar drivers into group 1, and the other 20 drivers were classified as group 2. Based on the similarity evaluation results, this study built a vehicle acceleration prediction model in order to verify the effectiveness of the MHMM in dividing the semantics of driving behavior.