## 3 Model Validation
There is no obvious distribution relationship between the prediction error and the modulus. In Table 4, the fitting parameter R\({}^{2}\) is greater than \(0.95\), which represents the proportion of the sum of squares of total errors that can be explained by the sum of squares of regression in linear regression analysis, indicating a good degree of fit. In contrast, model (1) cannot fit a set of modulus coefficients for convergence under different moisture content, and the fitting coefficients of different water contents are different. Therefore, model (1) is not suitable for the prediction of moisture change conditions, which also confirms the importance of considering the matric suction contribution to the effective stress in the study of the resilient modulus of unsaturated soils. After the introduction of (10), the fitting coefficient R\({}^{2}\) is still less than \(0.5\), which indicates that the fitting effect of model (10) on the resilient modulus data set of modified soil is not good and cannot be used for the M\({}_{\text{R}}\) prediction of modified soil. It also precisely illustrates the importance of constructing a specific resilient modulus prediction model for modified soil. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Valid Point** & **k1** & **k2** & **k3** & **k4** & **R\({}^{2}\)** \\ \hline Model (7) & 34 & \(6.13\times 10^{-7}\) & 7.2 & \(-4.793\) & 0.951 & 0.971 \\ Khalili&r-\(\theta\)Model & 34 & \(3.20\times 10^{-1}\) & 0.16 & \(-2.435\) & - & 0.406 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistical parameter. Figure 3: Residual distribution diagram. (3) Error analysis under freeze-thaw cycles  
As \(\eta_{n}\) proposed in the above section, the test data after 1, 3, 5, and 7 times of freezing and thawing were used for fitting, as shown in Figure 4. The fitting parameters are shown in Table 5  
It can be seen in Figure 4 that \(\eta_{n}\) can well fit the decreasing law of the \(\mathrm{M_{R}}\) of oil shale- and fly ash-modified soil after freeze-thaw cycles. It was introduced into model (7) for fitting and prediction calculations, and the predicted values were compared with the experimental observations. In addition, the errors of the predicted values of modulus under 1, 3, 5, and 7 freeze-thaw cycles were counted and discussed in conjunction with statistical parameters and images. The distribution of prediction errors is shown in Figure 5, and the error analysis results are shown in Table 6.  
As shown in the figure, the predicted value of the model has good regression performance with the measured value. In the observation interval, when the modulus is greater than 100, the prediction error is relatively small, and the predicted value is mostly smaller than the actual value. The statistics in Table 6 show that the prediction results of the model for the \(\mathrm{M_{R}}\) of 1, 3, 5, and 7 freeze-thaw cycles are close. Their maximum residuals are all around 10, and the maximum error of the predicted values within the survey range is 12.21. The RMSE of the model is between 6.54 and 7.15, with a slight increase in the number of freeze-thaw cycles. Under four cycles, the \(\mathrm{R^{2}}\) distribution range is 0.95 to 0.96, indicating that the predicted value of the model fits well with the measured value and that the combination of (7) and (9) can reliably predict the dynamic elastic modulus after freeze-thaw cycles. Figure 4: Fitting curves of freeze–thaw reduction coefficients. (**a**) \(\sigma\)3 = 50 kPa; (**b**) \(\sigma\)3 = 40 kPa; (**c**) \(\sigma\)3 = 30 kPa.