## 4 Strength Prediction Model
### Establishment of Strength Prediction Model  
Through the analysis of the above independent variables, it can be certain that the expression of the strength prediction model of sandstone-like materials is a binary function with the percentage of macropores and fractal dimension of macropores as independent variables, that is:  
\[\mathrm{M_{c}}=\mathrm{f(D_{mac},V_{mac})} \tag{7}\]  
In Formula (7), \(\mathrm{M_{c}}\) is the compressive strength of the material, \(\mathrm{D_{mac}}\) is the fractal dimension of macropore aperture, and the \(\mathrm{V_{mac}}\) is the percentage of macropore. To make the strength prediction model more reliable and operational, this study carried out the linear fitting for compressive strength, macroporous pore volume and fractal dimension of macroporous pore size respectively, to infer the concrete expression of strength model. From the Figures 5 and 6 of linear fitting results, the correlation coefficient among compressive strength, macropore volume ratio and pore size fractal dimension of fitting curve are 0.8947 and 0.8854. That is to say, the independent variables have a good correlation with compressive strength. However, the variation trend of different independent variables and strength is different. The compressive strength increases with the growth of pore size fractal dimension, but decreases with the growth of pore volume. Figure 5: Fitting relationship between fractal dimension of macropores and compressive strength. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fractal Dimension** & **Fractal Dimension of** & **Fractal Dimension of** & **Fractal Dimension of** \\
**Parameter** & **Micropore** & **Mesopore** & **Macropore** \\ \hline Fractal dimension of & & & \\ micropore & 1 & 0.8126 (**) & 0.7777 (**) \\ Fractal dimension of & & & \\ mesopore & 0.8126 (**) & 1 & 0.9555 (**) \\ Fractal dimension of & & & \\ macropore & 0.7777 (**) & 0.9555 (**) & 1 \\ \hline \hline \end{tabular} Note: (**) indicates that the correlation test is significant at a significance level of 0.01.  
\end{table} 
Table 9: Analysis of pore size fractal dimension. Based on the fitting relationship between the above independent variables and compressive strength, it is assumed that the expression of the predicted strength model is expressed as follows:  
\[\mathrm{M_{c}}=\beta_{0}+\beta_{1}\mathrm{D_{mac}}+\beta_{2}\mathrm{V_{mac}}+ \beta_{3}\mathrm{D_{mac}}\times\mathrm{V_{mac}} \tag{8}\]  
The multivariate regression analysis of the hypothetical expression is carried out, and the results are shown in Table 10.  
Table 10 shows that the correlation coefficient of the strength model regression equation reaches 0.943081, the \(p\)-value of the equation and the dependent variable are far less than the significant level 0.05, and the correlation coefficient of model expression is also greater than that of the single factor and the strength equation. The data show that regression results of the strength prediction model are significant, and the correlation coefficient of the model expression is better than that of the single factor expression. Therefore, Formula (8) is selected as the strength prediction model equation for sandstone-like materials. The complete expression is shown in Formula (9):  
\[\mathrm{M_{c}}=-36148.5+12082.66\mathrm{D_{mac}}+40771.45\mathrm{V_{mac}}-1362 1.8\mathrm{D_{mac}}\times\mathrm{V_{mac}} \tag{9}\]  
The mean absolute percentage (MAPE) is adopted to evaluate the degree of closeness of the model prediction results to the real data, it can reflect the true situation of the predicted value error of the model. If MAPE is smaller, it means that the model fitting effect is better and the model prediction accuracy is higher. [PERSON] [48; 49] combined MAPE and multiple regression method statistical methods to analyze the dependence of concrete tensile strength, flexural strength and compressive strength, as well as the durability study of concrete with limestone powder instead of fine aggregate under the condition of chemical erosion.