## 3 Test Result Analysis
### Vehicle Acceleration Analysis
#### 3.2.2 Acceleration Statistical Analysis  
In order to make a statistical analysis of the vehicle vibration acceleration of 5 sections, the probability density distribution histogram of the acceleration at the first test is drawn in Figure 8. Through the Kolmogorov-Smirnov test, the vehicle acceleration data are found to be normally distributed since the \(p\)-value is larger than 0.05. The acceleration probability density fitting graphs of the four tests are drawn in Figure 9.  
According to Figure 9, for the continuously reinforced composite pavement (Sections 1, 2 and 4), the probability density distribution is relatively concentrated, and the curve is steeper. The probability density distribution is more scattered, and the curve is gentler for the semi-rigid base asphalt pavement (Section 5). For the junction of the two pavements (Section 3), the probability density distribution map is located between the two. From the perspective of the longitudinal time series, compared with the previous test, the probability density distribution curve of the latter test has a lower height and a larger width.  
To quantify the above description, we calculated the width and height of the probability density distribution curve. The height is easy to understand, that is, the maximum probability density. Suppose the width is defined as the width of the abscissa axis, that is, the difference between the maximum acceleration and the minimum acceleration. In that case, it is easily affected by some abnormal values. Therefore, the width here is defined as the width of two inflection points on both sides of the probability density distribution curve. The point of inflection is the point where the second derivative of the function is zero.  
Acceleration datum \(x\) obeys normal distribution; \(\mu\) is the mean, and \(\sigma\) is the standard deviation. The probability density distribution function is as follows:  
\[f(x)=\frac{1}{\sqrt{2\pi\sigma}}\exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right) \tag{1}\]  
Find the second derivative of \(f(x)\) as follows:  
\[f^{\prime\prime}(x)=\frac{1}{\sqrt{2\pi\sigma}}\exp\left(-\frac{(x-\mu)^{2}}{ 2\sigma^{2}}\right)\left[\left(\frac{x-\mu}{\sigma^{2}}\right)^{2}-\frac{1}{ \sigma^{2}}\right] \tag{2}\]  
Let \(f^{\prime\prime}(x)=0\), and we can find that the function has two solutions:  
\[x_{1}=\mu+\sigma,\ x_{2}=\mu-\sigma \tag{3}\]  
In this study, the mean value of the vibration acceleration is calculated as 0; therefore, the two solutions of the function are as follows:  
\[x_{1}=\sigma,\ x_{2}=-\sigma \tag{4}\]  
Figure 9: The normal fitting diagram of the probability density of vehicle acceleration. (**a**) The first test. (**b**) The second test. (**c**) The third test. (**d**) The fourth test.  
Therefore, on both sides of the probability density distribution curve, there are two symmetrical inflection points, that is, the width of the curve is \(2\sigma\).  
By substituting \(x=\mu\) into Equation (1), the peak probability density (PPD) can be computed as follows:  
\[\text{PPD}=\frac{1}{\sqrt{2\pi\sigma}} \tag{5}\]  
It can be seen that the height and width of the probability density distribution curve are only related to \(\sigma\); hence we only need one parameter. The height is more evident in the figure than the width, and the final parameter selected is the peak height of the probability density distribution curve, that is, PPD.