## 4 Analysis of Damage Model of Asphalt Mixture Based on Grey Theory  
The existed methods for damage prediction model of asphalt mixture after the freeze-thaw cycles mainly included: BP neural network method, support vector machine method, probability statistics and grey theory. The gray theory GM (1, N) model was a widely used prediction model [51; 52; 53]. It required less useful information and had higher prediction accuracy. In this study, the interaction between various properties of asphalt mixture and freeze-thaw damage was considered. The equal-pitch gray prediction model GM (1, 3) was used to predict the freeze-thaw damage of asphalt mixture.  
The establishment of the GM (1, N) model is described in the literature [51], and the final results are shown in Equations (5) and (6):  
\[\hat{x}_{1}^{(1)}(k+1)=\Bigg{[}x_{1}^{(0)}(1)-\frac{1}{\hat{\alpha}}\sum_{i=2 }^{n}\hat{b}_{i-1}x_{1}^{(1)}(k-1)\Bigg{]}\mathrm{e}^{-\alpha k}+\frac{1}{ \hat{\alpha}}\sum_{i=2}^{n}\hat{b}_{i-1}x_{i}^{(1)}(k+1),k=1,2,\ldots,n, \tag{5}\]  
\[\hat{x}_{1}^{(0)}(k+1)=\hat{x}_{1}^{(1)}(k+1)-\hat{x}_{1}^{(1)}(k),k=1,2, \ldots,n. \tag{6}\]  
In this study, GM (1, 3) prediction model was established with matlab program according to the reference [51]. The freeze-thaw data were fitted respectively, the stability, splitting tensile strength damage and stiffness modulus damage of asphalt mixture after freeze-thaw damage were taken as \(X_{1}^{(0)}\) sequence, \(X_{2}^{(0)}\) sequence and \(X_{3}^{(0)}\) sequence. The asphalt mixture specimens were tested and the prediction model was established after 0, 5, 10, 15 freeze-thaw cycles. The prediction was conducted based on 20 times GM (1, 3) model. Because the grey theory model conducted iterative prediction based on the previous data, this study selected the 15 th measured data for derivation. The 20 th prediction results were compared with actual data and the prediction accuracy was analyzed. Prediction damage and error are shown in Table 7.  
Table 7 indicated that the relative errors between the calculated results and experimental results were less than 15%. Comparing the predicted results of GM (1, 3) model for each performance index, it can be concluded that the predicted results of splitting tensile strength were the best and its absolute values were all small. The prediction result of indirect tensile modulus was better, while the prediction result of stability had a certain error. However, all were less than 15%. Therefore, the established GM (1, 3) gray prediction model could better predict the stability, indirect tensile stiffness modulus and splitting tensile strength of asphalt mixture under freeze-thaw cycles.  
The prediction curves of the loss ratio caused by freeze-thaw damage of stability, splitting tensile strength and stiffness modulus of four asphalt mixtures with GM (1, 3) model are shown in Figure 13.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Performances**} & **AM** & **NBAM** \\ \hline \multirow{3}{*}{Stability} & Predictive value (kN) & 5.02 & 8.40 \\  & Measured value (kN) & 4.40 & 7.38 \\  & Absolute error (\%) & 12.4 & 12.1 \\ \hline \multirow{3}{*}{Indirect tensile stiffness modulus} & Predictive value (MPa) & 6023.7 & 8523.0 \\  & Measured value (MPa) & 6144.2 & 8426.9 \\  & Absolute error (\%) & 2.0 & 1.1 \\ \hline \multirow{3}{*}{Splitting tensile strength} & Predictive value (MPa) & 0.31 & 0.48 \\  & Measured value (MPa) & 0.31 & 0.46 \\ \cline{1-1}  & Absolute error (\%) & 0.5 & 4.0 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Twentieth cycle prediction data of asphalt mixture.  
Figure 13: Prediction curves of loss ratio caused by damage: (**a**) loss ratio caused by damage of stability prediction curve; (**b**) loss ratio caused by damage of splitting strength prediction curve; (**c**) loss ratio caused by damage of stiffness modulus prediction curve.